Systems Science and Modeling for Ecological Economics

S y s t e m s Science a n d M o d e l i n g for Ecological E c o n o m i c s Systems Science and Modeling for Ecological Economics A l e x e y V o i n...

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Systems Science and Modeling for Ecological Economics

Systems Science and Modeling for Ecological Economics Alexey Voinov

AMSTERDAM • BOSTON • HflDFl 3cRG • I ONCON • NfW YORK • OXFORD PARIS • SAN DIEGO • SAN fRANC'SCO • SINGAPORC • S^'DME* • TOKYO ELSFVIFR

Academic P'css is an mipuni C- Elsavie'

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ELSEVIER

| www.bookaid.org

| www.sabre.org

Sabre Foundation

To those who led - my parents, and to those who follow

Zoe and

Arkady;

- my sons, Anton and

Ivan

Contents

Preface

lx

Acknowledgements 1

2

3

4

5

xv

Models and Systems

I

1.1

Model

l

1.2

System

1.3

Hierarchy

1.4

T h e modeling process

16

1.5

Model classifications

21

1.6

Systems thinking

25

6 12

T h e A r t of Modeling

29

2.1

C o n c e p t u a l model

30

2.2

Modeling software

49

2.3

Model formalization

70

Essential M a t h

79

3.1

81

Tune

3.2

Space

3.3

Structure

105

99

3.4

Building blocks

108

Model Analysis

111

4.1

Sensitivity analysis

112

4.2

Model calibration

115

4.3

Model testing

I 29

4.4

Conclusions

134

Simple Model, C o m p l e x Behavior

139

5.1

Classic predator-prey model

140

5.2

Modifications of the classic model

146

5.3

Trophic chains

150

5.4

Spatial model of a predator-prey system

162

5.5

Conclusions

194

viii

Contents 6

7

Water M o d e l i n g as a hydrology p r i m e r

195

6.2

U n i t model

223

6.3

Spatial mode!

229

6.4

Conclusions

244

Adding Socio-Economics

249

7.1

Demographics

250

7.2

D y n a m i c s o n t h e market:

264

7.3

C o r p o r a t e rule

272

7.4

Sustainahihtv

278

7.5

T h e end ol c h e a p oil

287

7 . 6 T h e World 8

Optimi:ation

296 307

Introduction

30/

8.2

Resource management

315

8.3

Fishpond

324

8 4

Landscape optimization

338

8.5

O p t i m a l i t y principles

347

8.1

9

197

6.1

T h e Practice o f Modeling

555

9.1

W h y models d o n ' t work

355

9.2

Participatory a n d adaptive m o d e l i n g

.362

9.3

O p e n - s o u r c e , w e b t e c h n o l o g i e s a n d d e c i s i o n support

9 4 Conclusions

382 395

To Conclude

401

Index

407

Preface Why? A s 1 am finishing this book, Science magazine is running a special issue about the sequencing of the macaque genome. It turns out that macaques share about 93 perc e n t of their genes with us, humans. Previously it has been already reported that chimpanzees share about 9 6 percent of their genes with us. Yes, the macaque is our c o m m o n ancestor, and it might be expected that, together with the chimps, we c o n tinued with our natural selection some 23 million years ago until, some 6 million years ago, we departed from the chimps to c o n t i n u e our further search for better adaptation. Actually it was not quite like this. Apparently it was the chimps that departed from LIS; now that we have the macaques as the starting point, we can see that the chimp's genome has way more mutations than ours. S o the chimps are further ahead than we are in their adaptation to the e n v i r o n m e n t . How did that happen, and how is it then that we, and not the chimps, have spread around all the Earth? Apparently at some point a mutation put us on a different track. T h i s was a mutation that served an entirely different purpose: instead of adapting to the e n v i r o n m e n t in the process of natural selection, we started adapting the e n v i r o n m e n t to us. Instead of acquiring new features that would make us better suited to the e n v i r o n m e n t , we found that we could start changing the e n v i r o n m e n t to better suit us - and that turned out to be even more efficient. A n d so it went on. It appears that not that many mutations were needed for us to start using our brainpower, skills and hands to build tools and to design microenvironments in support of the life in our fragile bodies - certainly not as many as the chimps had to develop on their road to survival. Building shelters, sewing clothing or using fire, we created small cocoons of environments around us that were suitable for life. Suddenly the rate o f change, the rate of adaptation, increased; there was no longer a need for millions of years of trial and error. W e could pass the information on to our children, and they would already know what to do. W e no longer needed the c h a n c e to govern the selection of the right mutations and the best adaptive traits, and we found a better way to register these traits using spoken and written language instead of the genome. T h e human species really took off. O u r locally created comfortable microenvironments started to grow. From small caves where dozens of people were packed in with no particular comfort, we have moved to single-family houses with hundreds o f square meters of space. O u r c o c o o n s have expanded. W e have learned to survive in all c l i m a t i c zones on this planet, and even beyond, in space. As long as we can bring our cocoons with us, the e n v i r o n m e n t is good enough for us to live. And so more IX

X

Preface and more humans have been born, wtrh more and more space occupied, and more and more resources used co create our microcosms. W h e n microcosms are joined togecher and expand, chey are no longer " m i c r o . " Earch is no longer a big planet with infinite resources, and us, the humans. Now it is the humans' planet, where we dominate and regulate. As Vernadskii predicted, we have become a geological force that shapes this planer. He wasn't even talking about climate change at that time

Now

we can do even that, and are doing so. Unfortunately, we do not seem to be prepared to understand that. Was there a glitch in that mutation, which gave us the mechanism and the power but forgot about the self-control! 1 A r e we driving a car that has the gas pedal, but n o brake? O r we just have not found it yeL? For all these years, human progress has been and still is equated to growrh and expansion

W e have been pressing the gas to the floor,

only accelerating. But any driver knows that at high speed it becomes harder to steer, especially when the road is unmarked and Lhe destination is unknown. A t higher speeds, the price of error becomes fatal. But let us take a look at the other end o f the spectrum. A colony o f yeast planted on a sugar subsLrate starts to grow. It expands exponentially, consuming sugar, and then it crashes, exhausting the feed and suffocating in its own products of metabolism. Keep in mind that there is a lot of similarity between our g e n o m e and t h a t of yeasL. T h e yeast keeps consuming and growing; it c a n n o t predict or understand the c o n s e q u e n c e s of its actions. Humans can, but can we act accordingly based o n our understanding' W h i c h part of our genome will take over? Is it the part t h a t we share w a l l Lhe yeasL and which can only push us forward into finding more resources, c o n suming them and multiplying? O r is it going lo be the acquired part that is responsible for our intellect and supposedly the capacity to understand the more distant consequences ot our desires and the actions of today? S o far there is not much evidence in favor of the latLer. W e know quiLe a few examples o f collapsed civilizations, but there aie not many good case studies o f sustainable and long-lasting human societies. T o know, to understand, we need to model. Models can be different. E c o n o m i c s is probably one of the most m a t h e m a tized branches o f science after physics. T h e r e are many models m economics, but those models may not be the best ones to take into a c c o u n t the other systems that are driving Lhe economy. T h e r e is the natural world, which provides resources and takes care o f waste and pollution. T h e r e is the social system, which describes human relationships, life quality and happiness. T h e s e do not easily fit into the linear programming and game Lheory that are most widely used in c o n v e n t i o n a l economics. W e need other models if we want to add " ecological " to " e c o n o m i c s . " S o far our major c o n c e r n was how to keep growing. Just like the yeast population. T h e A n c i e n t Greeks c a m e up with theories of o i k o n o m i k a - the skills of household management. T h i s is what later became e c o n o m i c s - the s c i e n c e ot production, consumption and distribution, all for the sake of growth. A n d that was perfectly fine, while we were indeed small and vulnerable, facing the huge hostile world out there. Ironically, ecology, oikology - the knowledge and understanding of the household - came m u c h later. For a long rime we managed our household without knowing it, without really understanding what we were doing. A n d that was also O K , as long as we were small and weak. After all, what kind o f damage could we do to the whole big powerful planet? However, at some point we looked around and realized that actually we were not that weak any more. W e could already wipe out entire species, c h a n g e landscapes and turn rivers. W e could even change the climate on the planet.

Preface

xiii

It looks as though we c a n no longer afford " e c o n o m i c s " - management wtrhout knowledge. W e really need to know, to understand, what we are doing. A n d that is what ecological e c o n o m i c s is all about. W e need to add knowledge about our household to our management o f it. Understanding bow c o m p l e x systems work is c r u c i a l W e are part o f a c o m p l e x system, the biosphere, and we further add complexity to it by adapting this biosphere to our needs and adding t h e human c o m p o n e n t with its own complexities and uncertainties. Modeling is a fascinating tool that can provide a method to explore complex sysrems, to e x p e r i m e n t with them without destroying them at the same time. T h e purpose o f this book is to introduce some o f the modeling approaches that can help us to understand how this world works. I am mostly focusing o n tools and methods, rather than case studies and applications. I am trying to show how models can be developed and used - how they can b e c o m e a c o m m u n i c a t i o n tool that can take us beyond our personal understanding to j o i n t community learning and decision-making. Actually, modeling is pretty mundane tor all o f us. W e model as we think, as we speak, as we read, as we c o m m u n i c a t e - and our thoughts are mental models of the reality. S o m e people can speak w e l l clearly explaining what they think. It is easy to c o m m u n i c a t e with them, and there is less c h a n c e for misunderstanding, In contrast, some people mumble i n c o h e r e n t sentences that it is difficult to make any sense of. T h e s e people c a n n o t build good models o f their thoughts - the thoughts might be great, but they still have a problem. S o m e models are good while others are n o t so good. T h e good models help us to understand. Especially when we deal with complex systems, it is crucial that we learn to look at processes in their interaction. T h e r e are all sorts o f links, c o n n e c t i o n s and feedbacks in the systems that surround us. If we want to understand how these systems work, we need to leam t o sort these c o n n e c t i o n s out, to find t h e most important ones and then study them in more detail. A s systems become more complex, these c o n n e c t i o n s become more distant and indirecr. W e find feedbacks that have a delayed response, which makes it only harder to figure o u t their role and guess their importance. Suppose you start spinning a big flywheel. It keeps rotating while you add more steam to make it spin faster. T h e r e is no indication o f danger - no cracks, no squeaks it keeps spinning smoothly. A n engineer might stop by, see what you're doing and get very worried. He will tell you that a flywheel c a n n o t keep accelerating, that sooner or later it will burst, the internal tension will he too high, the material will n o t hold " O h , it doesn't look that way," you respond, after taking another look at your device. T h e r e is no evidence o f any danger there. But the problem is that there is a delayed response and a threshold effect. Everything is hunky-dory o n e minute, and then " b o o m ! " - the flywheel bursts into pieces, metal is flying around and people are injured. How can that happen? How can we know that it will happen? O h , we know, but we don't want to know. Is something similar happening now, as pait o f the global climate c h a n g e story and its denial by many politicians and ordinary people? W e don't want to know the bad news; we hate changing our lifestyle. T h e yeast colony keeps growing till the very last few hours. Models c a n help. T h e y can provide understanding, visualization, and important c o m m u n i c a t i o n tools. T h e modeling process by itself is a great opportunity to bring together knowledge and data, and to present them in a c o h e r e n t , integrated way. S o modeling is really important, especially if we are dealing with c o m p l e x systems that span beyond t h e physical world and include humans, economies, and societies.

xri

Preface

What? T h i s book originated from an on-line course that I started some 10 years ago. T h e goal was to build a stand-alone Internet course that would provide both access to t h e knowledge base and interaction between the instructor and the students. T h e web would also allow several instructors at different locations to participate m a collaborative teaching process. T h r o u g h their j o i n t efforts the many teachers could e v o l v e and keep the course in the public domain, promoting truly equal opportunity in education anywhere in the world. By c o n s t a n c y keeping the course available for asynchronous teaching, we could have overlapping generations o f students involved at the same time, and expect the more advanced students to help the beginners. T h e expectation was that, in a way that mimics how the open source paradigm works for software development, we would start an open education effort. Clearly, the ultimate test of this idea is whether it c a t c h e s on in the virtual domain. S o far it is still a work in progress, and there are some clear harbingers that it may grow LO be a success. W h i l e there are always several students from different countries around the world (including the U S A , C h i n a , Ireland, S o u t h Africa, Russia, e t c . ) taking the course independently, 1 also use the web resource in several courses I teach in class. In these cases I noticed that students usually started with printing out the pages from the web. T h i s made me think that maybe after all a book would be a good idea. T h e book has gone beyond the scope of the web course, with some entirely new chapters added and the remaining ones revised. Still, I consider the book to be a companion to the web course, which I intend to keep working and updated. O n e m a j o r advantage of web tutorials ts that new facts and findings can be incorporated almost as soon as they are a n n o u n c e d or published. It takes years to publish or update a book, but only minutes to insert a new finding or a U R L into an existing web structure. By the time a reader examines the course things will be different from what I originally wrote, because there are always new ideas and results tn implement and present. T h e virtual class discussions provide additional material for the course. All this can easily b e c o m e part of the course modules. T h e book allows you to work offline when you don't have your computer at hand. T h e o n - l i n e part offers interaction with the instructor, and downloads of the working models. A n o t h e r opportunity opened by web-based education can be described as distributed open-source teaching, which mimics the open-source c o n c e p t that stems from the hacker culture. A crucial aspect o f open-source licenses is that they allow modifications and derived works, but they must also be distributed under the same terms as the license of the original software. T h e r e f o r e , unlike simply free code that could be borrowed and then used in copyrighted c o m m e r c i a l distributions, the opensource definition and licensing effectively ensures that the derivatives stay in the open-source domain, extending and e n h a n c i n g it. Largely because of this feature, the open-source community has grown very quickly. T h e open-source paradigm may also be used to advance education. Web-based courses could serve as a core for j o i n t efforts of many researchers, programmers, educators and students. Researchers could describe the findings that are appropriate for the course theme. Educators could organize the modules in subsets and sequences that would best match the requirements o f particular programs and curricula, and develop ways to use the tools more effectively. Programmers could contribute software tools for visualization, interpretation and c o m m u n i c a t i o n . Students would test the materials and contribute their feedback and questions, which is essential for improvements o f both c o n t e n t and form.

Preface

xiii

S o m e o f ihts is st-ll in the future. Perhaps if you decide to read the hook and take t h e course o n - h n e , you could b e c o m e part of rhis open-source, open-education effort.

How? I believe that modeling c a n n o t be really taught, oniy learned, and that it is a skill and requires a lot of practice - just as when babies learn to speak they need to practice saying words, making mistakes, and gradually learning to say them the nghi way. Similarly, with formal modeling, without going through the pitfalls and surprises of modeling, it is not possible to understand the process properly. Learning the skiil must he a hands-on experience o f all the major stages of modeling, from data acquisition and building conceptual models to formalizing and itcrativdy improving simulation models. T h a t is why 1 strong', y recommend that you look on the web, get yourself a trial or demo version o f some of the modeling software that we are working wirh in rhis book, then download the models that we arc discussing. You c a n then not }ust read the book, but also follow the story widi t h e modek LV> t h e tests, change t h e parameters, explore or. your own, ask questions and try to find answers. It will be way more fun that way, and it will be much more useful. Best of all think of a topic that is of interest to you and start working on your individual project. Figure out what exactly you wish to find out. see what data are available, and then go through the modeling steps that we will be discussing in the book. T h e web course is a) hitp://www.[ikhei.com/AV7Simmod.himl, and will remain open to all. You may wish to register and take it. You will find where it overlaps with the book, you will be able to sent I your questions, get answers and interact with other students. A t the end of each chapter, you wil; find a bibliography. T n e s e books and articles may not necessarily be about models in a c o n v e n t i o n a l sense, but they show how c o m p l e x sysiems should he analysed -and how emergent properties appear from this' analysis. C h e c k out some of those references for more in-depth real-life examples o f different kind o f models, systems, challenges: and solutions. Best of all, learn to apply your systems analysis and modeling skills ;n your everyday life when you need to make small and big decisions, when you make your n e x t purchase or go to vote. Learn to look at the system as a whole, to identify the elements and the links, the feedbacks, controls and forcings, and to reahje how things are interconnected and how important it is to step back and see the big picture, t h e possible delayed effects and the cruical states.

Acknowledgements Many people have contributed to my understanding of modeling and to this effort. Professor Yuri Svirezhev, who passed away in 2 0 0 7 , was my teacher, and he certainly played a great role in shaping my vision of modeling - of what it should be, and what it can and what it can't do. My colleagues on many modeling projects in various parts o f the world helped me to learn many important modeling skills. I am grateful to my students, especially chose who took the on-line modeling course and contributed by asking questions, participating in o n - l i n e discussions, and letting me know what kind of improvements were needed. T h e G u n d Institute for Ecological E c o n o m i c s and its director, R o b e r t Costanza, provided a stimulating and helpful e n v i r o n m e n t for developing various ideas and applications. I very much appreciate that. For almost a decade I have been teaching a modeling course as part of the M S c Program

in Environmental

and Natural Resource

Economics at

Chulalongkorn

University in Bangkok. I am grateful to Jiragorn and N a n t a n a G a j a s e n i for inviting me and helping with the course. My thanks are due to the T h a i students who took the course and helped me improve it in many respects. Several people have reviewed various chapters o f the book and provided very useful c o m m e n t s . My thanks are due to Helena Vladich, Carl Fitz, Urmila Diwekar, Evan Forward and N a t h a n Hagens. I appreciate the suggestions I received from Andrey Ganopolski, Dmitrii O . Logofet, and Jasper Taylor. I am grateful to Erica Gaddis, who helped with several chapters and co-authored C h a p t e r 9. Joshua Farley encouraged me to write the hook, and has been the resource on all my questions on ecological economics. Finally, my thanks go to Tatiana Filatova, who diligently read the whole manuscript and provided valuable c o m m e n t s on many occasions.

XV

1 . M o d e l s and S y s t e m s 1.1

Model

1.2

System

1.3

Hierarchy

1.4

The m o d e l i n g process

1.5

M o d e l classifications

1.6

Systems thinking

SUMMARY W h a t ' s a model? W h y d o we model? How d o we model? T h e s e questions are addressed in this chapter. It is a very basic introduction to the trade. We shall agree on definitions - what is a system, what are parameters, forcing functions, and boundaries? W e will also consider some other basic questions - how do we build a c o n c e p tual m o d e l ' How are elements c o n n e c t e d ? W n a t are the flows of material, and where is it actually i n f o r m a t i o n ' How do interactions create a positive feedback that allows the system to run our o f c o n t r o l or, conversely, how do negative feedbacks manage to keep a system in s h a p e ' W h e r e do we get our parameters from? W e shall then briefly explore how models are built, and try to c o m e with some d i c h o t o m i e s and classes for different models.

Keywords C o m p l e x i t y , resolution, spatial, temporal and structural scales, physical

models,

m a t h e m a t i c a l models, N e p t u n e , emergent properties, elements, holism, reductionism, T h a l i d o m i d e , flows, stocks, interactions, links, feedbacks, global warming, structure, lunction, hierarchy, sustainability, boundaries, variables, conceptual

model,

modeling process.

1 . 1

Model We

A model is a simplification

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model

though

all rhe tune,

we don't

think

even aboat

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Models and Systems

3

N o t e t h a t t h e models we build a r e defined by t h e purposes t h a t they serve

It.

for e x a m p l e , you only wane to s h o w a friend h o w to get to your house, you will draw a very simple diagram, avoiding description of various places o f interest o n t h e way. H o w e v e i , if you want your friend to take n o t i c e o f a particular l o c a t i o n , you might also show her a p h o t o g r a p h , w h i c h is also

model. Its purpose is very different, and

so are t h e i m p l e m e n t a t i o n , t h e scale a n d t h e details. T h e best model, indeed, should strike a balance between

The best explanation is as simple as possible, but no simpler. Albert Einstein

realism

a n d simplicity. T h e

h u m a n senses s e e m to be e x t r e m e l y well tuned to t h e levels o f c o m p l e x i t y a n d resolution t h a t are required t o g i v e us a m o d e l o f t h e world t h a t is adequate to our needs. H u m a n s c a n rarely distinguish o b j e c t s t h a t a r e less t h a n 1 m m in si:e, but t h e n they hardly need to in

i h e i r everyday life. Probably for t h e s a m e reason, m o t e distant oh-octs are modeled with less detail t h a n are t h e c l o s e ones, if we could see all t h e details across, say. a 5 ' k m d i s t a n c e , t h e brain would be o v e r w h e l m e d by t h e a m o u n t o f i n l o r m a t i o n u would need to process. T h e ability o f t h e eye to focus o n individual o b j e c t s , while the surrounding picture b e c o m e s s o m e w h a t blurred a n d loses detail, probably serves t h e s a m e purpose o f simplifying t h e image t h e brain is currently studying. T h e model is m a d e simple, but n o simpler t h a n we need, li our vision is less t h a n 20/20, we suddenly realise t h a t there are c e r t a i n i m p o r t a n t features t h a t we c a n n o longer m o d e l . W e rush to t h e o p t i c i a n for advice o n how to bring our m o d e l i n g capabilities back to c e r t a i n standards. A s ill space, in t i m e we also register e v e n t s o n l y o f appropriate duration. S l o w m o t i o n cscapes o u r resolution c a p a c i t y W e c a n n o t see how a tree grows, and w e cannot register t h e m o v e m e n t o f t h e sun a n d t h e m o o n ; we have t o go b a c k to t h e s a m e o b s e r v a t i o n point to see t h e c h a n g e . O n t h e o t h e r h a n d , we do n o t o p e r a t e t o o well at very high process lates. W e do not see h o w the fly m o v e s its wings

E v e n driv-

ing causes problems, a n d quite o f t e n t h e h u m a n b r a m c a n n o t c o p e with t h e flow o f i n f o r m a t i o n when driving t o o fast. W h e n e v e r we are interested in more detail regarding time or s p a c e , we n e e d to e x t e n d t h e m o d e l i n g c a p a b i l i t i e s o f our senses and brain with some additional d e v i c e s microscopes,

telescopes, h i g h - s p e e d c a m e r a s ,

long-term

monitoring

devices, e t c .

T h e s e are required for specific m o d e l i n g goals, specific t e m p o t a l a n d spatial scales. T h e image c r e a t e d by Our senses is static; it is a s n a p s h o t of reality. It is o n l y c h a n g e d when t h e reality itself c h a n g e s , and as we c o n t i n u e observing we get a series o f snapshots t h a t gives us t h e idea o f t h e c h a n g e . W e c a n n o t modify this model to m a k e it c h a n g e in time, unless we use our i m a g i n a t i o n to play " w h a t if?" games. T h e s e are t h e m e n t a l e x p e r i m e n t s t h a t we c a n m a k e . T h e models we create outside our brain, physical models, allow us to study c e r t a i n features of t h e real-life systems e v e n without modifying t h e i r prototypes - for e x a m p l e , a model o f a n airplane is placed in a wind t u n n e l t o e v a l u a t e t h e a e r o d y n a m i c propeities o f t h e real a n p l a n e . W e c a n study t h e b e h a v i o r o f t h e airplane a n d its p a n s in e x t r e m e c o n d i t i o n s , we c a n m a k e them actually break w i t h o u t risking t h e plane itself - w h i c h is, o f course, m a n y times more e x p e n s i v e t h a n its m o d e l . (For e x a m p l e s o f wind t u n n e l s a n d h o w they are used, see http://wte.larc.nasa.gov/.) Physical models are very useful in t h e "what if?" analysis. T h e y have been widely used in engineering, hydrology, architecture, e t c In FigureL.t we see a physical model developed to study stream flow. Ir. mimics a real c h a n n e l , and has sand and gravel to

20 Systems Scienrf? and Modeling fo' Ecologicst Economics

A physical model to study stream flow in the Main Channel Facility at the St Anthony Falls Laboratory (SAFL) in Minnesota. The model is over 80m long, has an intake from the Mississippi River with a water discharge capacity of 8 5m 3 per second, and is configured with a sediment (both gravel and sandl recirculation system and a highly accurate weigh-pan system for measuring bedload transport rates (hnp://www need umri edu/streamlab06_seri_xpnrt)

represent the bedforms and allow us ro analyze how c h a n g e s in t h e b o t t o m profiles c a n affect t h e flow o f water in rhe stream. Physical models a r e quire e x p e n s i v e ro c r e a t e a n d m a i n t a i n . They arc also very hard ro modify, so c a c h n e w device ( e v e n if it is fairly similar to t h e o n e already studied) may require rhe building of an entirely new physical model. M a t h e m a t i c s offers a n o t h e r tool for m o d e l i n g . O n c e we h a v e derived a n a d e q u a t e m a t h e m a t i c a l r e l a t i o n s h i p for a c e r t a i n process, we c a n start analyzing it in

Models and Systems

5

many different ways, predicting t h e behavior o f the real-life o b j e c t tinder varying conditions. Suppose we have derived a model o f a body moving in space described by the equation S = V -T where S is the distance covered, V is the velocity and T is time. T h i s model is obviously a simplification o f real movement, which may occur with varying speed, he reciprocal, e t c . However, this simplification works well for studying the basic principles o f motion and may also result in additional findings, such as the relationship T = — V

A n important feature o f mathematical models is that some o f the previously derived mathematical properties can be applied to a model in order to create newmodels, at no additional cost. In some cases, by studying the mathematical model we can derive properties o f the real-life system which were not previously known. It was by purely mathematical analysis o f a model of planetary motion that Adams and Le Verner first predicted the position o f Neptune in 1845. Neptune was later observed by G a l l e and d'Arrest, on 2 3 September 1 8 4 6 , very near to the location independently predicted by Adams and Le Verner. T h e story was similar with Pluto, the last and the smallest planet in t h e Solar System (although, as o f 2 0 0 6 , Pluto is no longer considered to be a planet; it has been decided that Pluto does not comply with the definition o f a planet, and thus it has been reclassified as a "small planet' 1 ). Actually, the model that predicted its existence turned out to have errors, yet it made Clyde Tombaugh persist in his search for the planet. W e can see that analysis of abstract models can result in quite concrete findings about the real modeled world. All models are wrong because they are always simpler than the reality, and thus some features of real-life systems get misrepresented

AH models are wrong ... Some models are useful.

or ignored in the model. W h a t is the use o f modeling, then? W h e n dealing with

William Deming

some-

thing complex, we tend t o study it step by step, looking at parts of the whole and ignoring some details to get the bigger picture.

T h a t is exactly what we do when building a model. Therefore, models are essential to under stand the world around us. If we understand how something works, it becomes easier t o pred/ct its behavior under changing conditions. If we have built a good model that takes into a c c o u n t the essential features o f the real-life o b j e c t , its behavior under stress will likely be similar to the behavior o f t h e prototype that we were modeling. W e should always use caution when extrapolating t h e model behavior to the performance o f the prototype because of the numerous scaling issues that need be considered. Smaller, simpler models do not necessarily behave m a similar way to the real-life o b j e c t s . However, by applying appropriate scaling factors and choosing t h e right materials and media, some very useful results may be obtained. W h e n the o b j e c t performance is understood and its behavior predicted, we get additional information to control

the o b j e c t . Models can be used to find the most sen-

sitive c o m p o n e n t s o f the real-life system, and by modifying these c o m p o n e n t s we can efficiently tune the system into the desired state or set it on t h e required trajectory.

yjiiftms Soeoco a n d Modeling fof Ecoiog-ca' Econo^vcs

6

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26 Systems Scienrf? and Modeling fo' Ecologicst Economics

energy (tight, heat, electricity, etc.), money, e t c . It is something that c a n he measured and tracked. Also, if an e l e m e n t is a donor ot this substance t h e amount o f substance in this element will decrease as a result of the exchange, while at the same time the amount o f this substance will increase in the receptor element

T h e r e is

always a mass or energy conservation law in place. N o t h i n g appears from nothing, and nothing can disappear to nowhere. T h e second type nf e x c h a n g e is an informal i
Models and Systems 27

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'negative feedback' T h t f i w h e n . after I * u s h t h e toilet, t h e

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r>aw« a p r o t t e m - - j n l e w it's out money in the bank Sometr.ing gets

out 01 con«o< T»iete • w> i M O w i i m »

»ioo n » a t w - a w r situation It s c o n e * . it $ on

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t's ' t h e m o t e - m e m o r e ' thai « kind* scary. These processes iusi don't k r o w

h o w t o stop They . r - W i n g maman!-.i»« until they tun out o» sleom

or mmply blow up

e,ery!hing. N o v tr-;5 is vihat m a k e s t h e oattema of globol d i m a t o c N w g * look pteRy dim. The onty k i r d o* proce'.*^s m « : w e find m the climate iystiim a t e "the more - tho mora" ones Here's an e « a m c i e W h e n it's h o t I prefer l o d t f i s in white

I do>-1 « e w e a r n g K u c t hecause i1 |U51

'cete much h o r a ' i n bdack W h i l e -.earns to influct more sunshine K»d I don't q « i 9 0 hot This is called atoedo. W h i t e ties high albedo: block has tow elbeco So the Arctic has lost almost 3 0 c * c e * t o» its ice b e c a u s e r o w it's warmer. But this m e a n s that there is leas w h i t e over there and m o r e dark - but t h e darker it gets the hottin
Well you can see, "the more

th# m o r a " Not good, a n d not going

10 stop until
t h e r e are all

those huge Cogs, vvtveh used to be I r o w n Nov. they are melting and releasing huge amounts ot s o m e gases n t o the atmosphero and appaiently these o * » * s hoppan t o b e just the ones ttvit are causing th» teinpeiature 10 r i « i T h e t i e i i i e c a i i e o g t i * f>ou»o g»«?» The ti^nperature 3 t a r t e o to rise becousc w e h i w e put loo m u d i ot tf'oso

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Models and Systems

Figure 1.3

13

Hierarchies in systems Systems may be presented as interacting subsystems Systems themselves interact as parts of supra-systems. There are various hierarchical levels that can be identified to improve the descriptions of systems in models Elements in the same hierarchical level are usually presented in the same level of detail in the space-time-structure dimensions.

same level

H o w e v e r , lower levels of t h o s e similar systems are hardly i m p o r t a n t lor

rhis system. T h e y e n t e r t h e higher levels in terms o f t h e i r f u n c t i o n ; rhe individual e l e m e n t s may he negligible but their e m e r g e n t properties are what m a t t e r . F i e b l e m a n describes this m Ins theory o f integrative levels as follows: " F o r a n organism at any given level, its m e c h a n i s m lies at t h e level below and its purpose ar r h e level a b o v e " (Fiebleman, 1954: 6 1 ) . For e x a m p l e , c o n s i d e r a student
It does n o t m a t t e r what individual students

had tor breakfast, or w h e t h e r they are tall o r shorr. O n t h e o t h e r h a n d , t h e i r individual ability t o learn is affected by t h e n individual properties. If a student h a s a h e a d a c h e after t h e party o n t h e night before, h e or she probably will n o t be able to study as well as a n e i g h b o r w h o went t o t h e gym instead. T h e class as a w h o l e may be characterized by a c e r t a i n degree of a c a d e m i c a c h i e v e m e n t t h a t will be different from the t a l e n t s a n d skills o f individual students, yet that will be t h e b e n c h m a r k that t h e t e a c h e r will c o n s i d e r w h e n working with t h e class. E a c h s t u d e n t affects this emerg e n t property to a c e r t a i n e x t e n t , but n o t entirely. O n t h e contrary, t h e class average affects e a c h individual s t u d e n t , setting t h e level o f i n s t r u c t i o n chat is t o be offered by t h e t e a c h e r . Different classes are assembled i n t o a s c h o o l , w h i c h is t h e next level in t h e hierarchy. S c h o o l s may be e l e m e n t s in a R e g i o n a l D i v i s i o n , a n d s o o n . A t t h e o t h e r end o f this hierarchy, we c a n start by " d e c o m p o s i n g " e a c h individual student, looking a t his o r h e r body organs a n d c o n s i d e r i n g t h e i r f u n c t i o n s - and so o n , until we get to m o l e c u l e s a n d a t o m s . T h e r e a r e m a n y ways we c a n carry out t h e d e c o m p o s i t i o n . Instead of c o n s i d e r i n g a student as a n e l e m e n t of a class, we may look at t h a t student as a n e l e m e n t ol a family and build t h e hierarchy m a different way. A s with m o d e l i n g in general, t h e type o f h i e r a r c h y t h a t we c r e a t e is very m u c h driven by t h e goals o f our study T h e h i e r a r c h i c a l a p p r o a c h is essential in order to p l a c e t h e study o b j e c t w i t h i n t h e c o n t e x t of t h e macro- a n d micro-worlds - t h a t is, rhe super- and subsystems - relative t o it

30 Systems Scienrf? and Modeling fo' Ecologicst Economics A c c o r d i n g t:> T. S a a t y ( 1 9 8 2 ) , " h i e r a r c h i e s a r c a f u n d a m e n t a l tool o f t h e h u m a n mind. T h e y i n v o l v e identifying t h e e l e m e n t s of a p r o b l e m , grouping t h e e l e m e n t s i n t o h o m o g e n e o u s sets, and arranging those sets in different l e v e l s . " T h e r e may be a variety o f h i e r a r c h i e s , t h e simplest of w h i c h are linear - such as universe —> galaxy constellation

— solar system

—> planet

—...—»

molecule

— atom

— nucleus

—

— proton.

T h e more c o m p l e x o n e s a r e networks of i n t e r a c t i n g e l e m e n t s , with multiple levels affecting e a c h o f t h e e l e m e n t s . It

is i m p o r t a n t

to remember

t h a t there are n o real h i e r a r c h i e s in

Hierarchies

dc wrf ex.Ul. Wt make

t h e world we study. H i e r a r c h i e s a t e

to under U and a. iyUem

always c r e a t i o n s of our brain and

ow muierUtvuiituj

and to

tfvem up c&*u*uuuca£e

to txtktn.

are driven by our study. T h e y are just a useful way t o look at t h e system, t o understand it, t o put it in t h e c o n t e x t o f scale, of o t h e r c o m p o n e n t s t h a t affect t h e system

T h e t e is n o t h i n g o b j e c t i v e about

t h e h i e r a r c h i e s that we d e v e l o p . For e x a m p l e , c o n s i d e r t h e hierarchy t h a t c a n he assumed w h e n looking at t h e Earth system. Clearly, t h e r e a r e e c o l o g i c a l , e c o n o m i c a n d social subsystems. N e o classical e c o n o m i s t s may forget about t h e ecological subsystem a n d put t o g e t h e r their t h e o r i e s with only t h e e c o n o m i c a n d social subsystems in mine!. T h a t is h o w you would e n d up with t h e C o b b - D o u g l a s p r o d u c t i o n f u n c t i o n that c a l c u l a t e s output as a f u n c t i o n of labor (social s y s t e m ) and capital ( e c o n o m i c s y s t e m ) . E n v i r o n m e n t a l e c o n o m i s t s would certainly recognize t h e i m p o r t a n c e o f t h e e c o logical system. T h e y would want to take i n t o a c c o u n t all three subsystems, b u t would think a b o u t rhem a? if they were acting side-by-side, as equal c o m p o n e n t s o f t h e whole (Figure M A ) . For t h e m , t h e production function is a product of population (labor), resources ( l a n d ) and capital. A l l three are equally important, representing t h e social, natural ( e c o l o g i c a l ) and e c o n o m i c subsystems, respectively. T h e y are also substitutable: you c a n c i t h e r work more o r invest more to get t h e s a m e result You c a n also c o m e up

Ecological

Social

Social

Economic

Ecological Economic

Ecological Social

Figure 1.4

Different ways to present the ecological-economic-social hierarchy in the Earth system. Hierarchies are subjective and serve particular purposes of the analysis.

mv Models and

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• 15

Systems

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I R N ' O

W e vrtB bo considenng sustainability and sustainable development in more detail in Cnapter 7. Here, let us use this notion to demonstrate how systems ond hierarchies may be a useful tool for some far-reaching conclusions The Wtorjd Commission on Environment and Development (WCED. 1987) introduced the idea of sustainability several decades ago. but there is still no single ag
Most would agree that it implies that a system is to be main-

tained at 0 certain level, held within certain limits SustJunot-iity denies run-away growth, but also precludes any substantial set backs ot cuts While most - probably all - natural systems go through a renewal cycie. where growth is f d o w e d by decline and eventual disintegration, sustmneoility in h way has the goal of preventing the system from declining end collapsing. Onginalfy the Bruncand Commissio" came up with the concept of susta-^bJUy at the global level, as a way to protect our biosphere from becoming uninhabitable by humans, and human lives becoming full of suffering and turmoil because of the lac* of natural resources and assimilative capacity of the planet However, somehow in the environmental movement the goal of sustainability was translated into thB regional and local levels. Indeed, tne famous Schumacher iosa of "Think globally - act locally" appDrertiy means that the obvous path to global sustanab lity is through m a b n g sure that ouf local systems are sustainable. Is that really the case? Lei us apply seme of the ideas about hiera-chies and systems Keep in m*nd that renewal etows 'or readjustment and adaptation

However, it is the

next hierarchical level that benefits from this adaptation Renewal in components helps a syst e m to persist; therefore *or a hierarchical system to extend
and

wt

MJ\I:I

space f
za'.lon end adaptation. Costanza and Patten <1995 196). looking at sustainability m terms ot component longevity cr existence time, recognaed that "evolution cannot occur unless there is I mitec longevitvof -he component parts so that new alteratives can be selected' Sustainability of a system borrows from sustainabiioy of a supra-svstern and rests on lack of sustasnabtftY in subsystems. This mxjht be 'isrd to perceive, because at first glance it seems that a s>/stem made of sustainable, lasting components should be sustainable as well. However, in systems theory It has been song recognized that "the whole is more than the sum of parts" I von Bertalanffy. 'i968: 55}. r a t a system function is not provided only t v the functions of its components, and therefore, in fact, system sustainability is not a product of sustainable parts and vice versa. This is especially true fo' living dynamic-aity evolv.ng

16

S v J t o m n S n o n r f t ar.c* M o d e l i n g frvr E c o l o g i c a l E r o ^ o r r r c n

systems '"fou cannot i u m up thu behavior o l me w M M I ' o m t h e tvxateo p * t »

j >o.

havo to tako into Account the* rulatlons between the v u o u i subold*"-alod systems a r c the systemi which are Buaer-CHdinatea to !h»-n n ofdet to understand m e c m b t . c of p * t s

(woo

flettalaoHv I 9 6 0 1481 One way to rosolve this contradiction between s u s w n a M i t y of a socioeconomic eco*ogh cai s y s l e m and its components la 10 ea<»e thai V «ne a 0n*v o n e s y s J - • fo» wt :h -. .-.miiv ariiity will be sought. ond that l» the top level *vat e m - w h i d i m this case is t r e bosohere as a whole The global scnle in ih.s context seems to b e the rnanmai that humans can influence .it t h e prosent lev.4 of their dave'opme^t

It ir, a l i o t h * sca:« that n 4 e c t s t - « h u m a n l y

whole, the system ihat 4 shared by all people and shoUd t h e r e t o e be o» n v t V ccnee?to ail *obabfy m e famous Schgmochor slogan CThm*. globally - act tocety* I s-oM atso mc j d e ' W h e n acting locaSy - keep thinking globaay" W e do not want locally sustarfwW* s y o e m s leu •r 1 countiei. regons. fotms. Industries! W e want to let them tenew. so that at n e gccai can i i v « material for adaptation and evolution. wtifdi is essentu for Jui'.n nab-'ty

Exercise 1.4 1. Consider a tree in a foest and describe tne o l e v a r : r iei«irv. > V d : mOuO t*"above" anil " b e l o w ' ' H<>.v do vOu i l w d r what h> mw.rfr 2

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1.4

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M o d e l s and Systems

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18

Systems Scienrf? and Modeling fo' Ecologicst Economics alt o f rhem, or some o f rhem are entirely unknown? W h i c h are the limiting ones, where are the gaps in our knowledge? W h a t are che interactions between t h e elements? W e might already need 10 go back and forth from the goals t o the data sets. If our knowledge is insufficient for che goal in mind, we need eicher to update the data sets to better comply with the goals, or t o redefine t h e goals to make them more feasible at che exiscing level of knowledge. By answering che basic questions about space, time a n d structure, we describe the conceptual

model

of the system. A conceptual model may be a mental model, a

skecch or a flow diagram. Building che righc concepcual model leads us halfway co success. In t h e conceptual model, che following componencs o f che system should be clearly identified. 1. Boundaries.

T h e s e distinguish che system from the oucsLde world m both cime and

space. T h e y are important in deciding whac material and information flows into and out o f the system, which processes are internal (endogenous) and which are external (exogenous). T h e outside world is something that we assume is known and do noc cry co explore in out model. T h e outside world matcers for the model only in terms o f its effects upon che syscem that we are studying. 2.

Variables.

T h e s e characterize the elements in our system. T h e y are the quantities

that change in che system thnc we analyze and report as a result o f the modeling exercise. A m o n g variables, rhe following should be distinguished: • S t a t e variables, or output variables. T h e s e are che outputs from t h e model. T h e y are determined by inpucs that go into the model, and by the model's internal organization or wiring. • Intermediate or auxiliary vaiiables. T h e s e are any quantities defined and c o m puted in the model. T h e y usually serve only for intermediate calculations; however, in some cases looking at them c a n help us to understand what happens "under the h o o d " in the model. 3 . Parameters.

T h e s e are generally all quantities that are used to describe and rim a

model. T h e y do n o t need to be c o n s t a n t , bur all their values need to be decided before the model runs. T h e s e quantities may be further classified into the following categories: • Boundary conditions. T h e s e describe the values along the spatial and temporal boundaries o f a syscem. For a spatially homogeneous system we have onlyinitial conditions, which describe the state o f the variables at time t = 0 when we start the model, and the length o f the model run. For spatially distributed systems, in addition we may need to define che conditions along the boundary, as well as the geometry o f che boundary itself. • C o n s t a n t s or parameters in a narrow sense. T h e s e are the various coefficients and constants measured, guessed ot found. W e may want to distinguish between real constants, such as gravity, g, and, say, the half-saturation coefficient, K, in the M i c h a e h s - M e n t e n function that we will consider in the next chapter. W h i l e both o f them take on c o n s t a n t values in a particular model run, g will be always the same from o n e run co another, but K may change quite substantially as we improve the model. Even if fC cornes from observations, it will normally be measured with certain error, so the e x a c t value will n o t be really known. • Forcing functions. T h e s e are parameters that describe the effect o f the outside world upon the system. T h e y may c h a n g e in time or space, hut they d o

Models and Systems

19

not respond to changes within t h e system T h e y .ire external to it. driven by processes in tile lusher hierarchical levels. C l i m a t i c c o n d i t i o n s (rainfall, leirtperature, e t c ) certainly affect the growth o f tomatoes in my garden, but the tomatoes hardly affect the temperature or the rainfall pattern;. It we build a model of tornjtet. bvery nine the knob is dialed to a certain position, but we know that it may vary and will result m a different performance by the system. N o t e that 111 some te xis parameters wt]! he assumed only in the narrow sense of constants that may sometimes change, like the growth rate or halt-saturation coefficients. However, this may be somewhat confusing, since forcing functions are also Such parameters if rhey are fixed. Suppose we want to run a model with the temperature held c o n s t a n t and equal to t h e mean over a certain period of time - say, t h e 6 months of the growth season for a crop T h e n suppose Liter on we want to feed into the model the actual data that we have measured tor temperature; Temperature is now no longer a c o n s t a n t , but changes every day according to the recorded time series. Does this mean that temperature will n o longer be a parameter' For any given moment u will still be a constant. It will only c h a n g e from time to time according to t h e data available

Probably, ic won Id make sense sti I to treat it as a parameter,

e x c e p t now it will K n o longer constant hut will change accordingly. Suppose now that we approximate the course of temperatures by a function with some c o n s t a n t that control the form ot this function. Suppose we use rhe sine function and have parameters for the amplitude and the period. Now temperature will n o longer be a parameter. Note [hat we n o longer need to define all the values lor temperature before we hit the " R u n " button. Instead, temperature will b e c o m e an intermediate variable, while we will have two new parameters in the sine function that now specifies temperature - o n e parameter (D = 4 ) will make the period equal to 6 months, the oilier parameter ( A ) will define the amplii ude and make the temperature c h a n g e from a minimal value ( 0 ! co the maximal value ( 4 0 , if A = 2 0 } and back over this period ol time, as in che function: Temperature = A * SIN ;

5 365

2

+ A

where I is time, r. is a c o n s t a n t 7t = > 14, and A and B are parameters. It B = 2, then the period will change from 6 to 12 months. Both A a n d B are set before we start running the model. I here may be a number of ways t o determine model parameters, including the following 1. Measurements in situ T h i s is probably the best m e t h o d , since t h e measurements dehne the value ol exactly what is assumed in the modei. However, such measuiements are the most labor- a n d cost-intensive, and they also c o m e with large margins o f error. Besides, in many eases such measurements may n o t be possible at all, ii a parameter tepiesents some aggregated value or an extreme condition that may not occur in reality { t o r example, the maximal temperature for a population to tolerate - this mav differ from o n e organism to another, and such conditions may be hard t o find in reality).

20

Systems Scienrf? and Modeling fo' Ecologicst Economics 2 . Experiments in the lab (in vitro). T h e s e are usually performed when in siru experiments are impossible. S a y we take an organism and expose it to high temperatures to find out the limits o f its tolerance. W e c a n create such conditions artificially in a lab, but we c a n n o t change the temperature for the whole ecosystem. 3 . Values from previous studies found from literature, web searches or personal communications. If data are available for similar systems, it certainly makes sense to use them. However, always keep in mind that there are no two identical ecosystems, so it is likely that there will be some error in the parameters borrowed from a n o t h e r case study. 4- Calibration (see C h a p t e r 4 ) . W h e n we know what the model output should look like, we c a n always tweak some o f the parameters to make the model perform at its best. 5 . Basic laws, such as conservation principles and rherefore mass and energy balances. 6 . A l l o m e t r i c principles, stoichiometry, and other c h e m i c a l , physical, e t c . , properties. Basic and derived laws may help to establish relationships between parameters, and therefore identify at least some o f them based o n the o t h e r ones already measured or estimated. 7 . C o m m o n sense. T h i s always helps. For example, we know that population numbers c a n n o t be negative. S e t t i n g this kind o f boundary o n certain parameters may help with the model. N o t e that in all cases there is a considerable level of uncertainty present in the values assigned t o various model parameters. Further testing and tedious analysis o f the model is the only way t o decrease the error margin and deal with this uncertainty. C r e a t i n g a c o n c e p t u a l model is very much an artistic process, because there c a n hardly be any e x a c t guidelines for that. T h i s process very much resembles that o f perception, winch is individual to every person, T h e r e may be some recommendations and suggestions, but eventually everybody will be doing it in his or her own personal way. T h e same applies to the rest o f the modeling process. W h e n a conceptual model is created, it may be useful to analyze it with some tools borrowed from mathematics. In order to do this we need to formalize

the model - that

is, hnd adequate mathematical terms t o describe our concepts. Instead o f concepts, words and images, we need to come up with equations and formulas. T h i s is not always possible, and o n c e again there is no o n e - t o - o n e correspondence between a conceptual model and its mathematical formalization. O n e formalism can turn out to be better for a particular system or goal than another. T h e r e are certain rules and recommendations, but no ultimate procedure is known. O n c e rhe model is formalized, its further analysis becomes pretty much t e c h n i cal. W e c a n first compare the behavior o f our mathematical o b j e c t with the behavior o f the real system. W e starr solving the equations and generate trajectories for the variables. T h e s e are to be compared with the data available. T h e r e are always some parameters that we do not know exactly and that can be changed a little to achieve a better fit o f the model dynamics to the o n e observed. T h i s is t h e so-called

calibra-

tion process. Usually it makes sense to first identify those parameters that have the largest effect o n system dynamics. T h i s is done by performing semiwit^ analysis o f the model. By incrementing all the parameters and c h e c k i n g out t h e model inpur, we c a n identify to which ones the model is most sensitive. W e should then focus our attention on these parameters when calibrating the model. Besides, if the model has already been tested and found to be adequate, then model sensitivity may be translated into system

Models and Systems

21

sensitivity: we may conclude rhar rhe system is most sensitive ro certain parameters and therefore the processes that these parameters describe. If the calibration does not look good enough, we need to go back to some o f rhe previous steps of our modeling process (reiterate). W e may have got the wrong conceptual model, or we did nor formalize it properly, or there is something wrong in the data, or the goals do not match the resources. Unfortunately, once again we are plunged into the imprecise "artistic" domain o f model reevaluation and reformulation. If the ht looks good enough, we might Once you

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behaves as well on a part of the data that was

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want to do another test and check if the model not used in the calibration process. W e want

uvodei

to make sure that the model indeed represents

tfrtuqht

the system and not the particular case that was described by the data used to tweak the param-

circlet.

eters in our formalization. T h i s is called the validation

process. O n c e again, if the fit does not match our expectations we need to

go back to the conceptualization phase. However, if we are happy with the model performance we can actually start using it. Already, while building the model, we have increased our knowledge about the system and our understanding o f how the system operates. T h a t is probably the major value o f the whole modeling process. In addition ro that we c a n start exploring some of the conditions rhat have not yet occurred in the real system, and make estimates of its behavior in these conditions. This is the "what if?" kind of analysis, or rhe scenario analysis T h e s e results may become important for making t h e right decisions.

1.5

Model classifications T h e r e may be several criteria used to classify models. W e will consider examples of many o f the models below in much more detail in the following chapters. Here we give a brief overview of the kinds of models that are out there, and try to figure ways to put some order in their descriptions. Among many ways of classifying the models we may consider the following: 1. Form: in which form is the model presented? • Conceptual (verbal, descriptive) - only verbal descriptions are made. Examples include the following. - A description of directions to my home: Take. Road 5 for 5 miles East, then take a left to Main Street and follow house is 33.3.3 on the left

through ru-'o lights. Take a right to Cedar

Lane. My

T h i s is a spatial model of my house location relative

to a certain starting point. I describe the mental model o f the route t o my house in verbal terms. -

A verbal portrait o f a person: He is tail with red hair and green eyes, his cheeks are pale and his nose is pimpled.. His left ear is larger than the right one and one of his front teeth is missing. T h i s is a static verbal model of a person's face.

-

Verbal description of somebody's behavior: When she wakes ing, she is slow and sleepy until she hc.s her first cup of coffee. to move somewhat

up in the morn-

After thai she starts

faster and has her bowl of cereal with the second cup of coffee.

Only that brings her back tional verbal model.

to her normal

pace of life. This is a dynamic condi-

22

Systems Scienrf? and Modeling fo' Ecologicst Economics -

A verbal description o f ' a rainfall e v e n t : Ramfall

every ngw and then. //

occurs

is M i n e Q ° f C j (-32 F) the rain is called snow and it. is accumulated

feni/ierdfuve

snow or ice en tfie terrain. Otherwise into the subsurface stays on [lie surface

as

it comes m liquid form and part of it in fib rates

layer and adds to the unsaturated storage underground. T V ' r « r as surface

teaier.

C o n c e p t u a l ( d i a g r a m m a t i c ) - in some cases a g o o d dtawing may be worth a thousand words. E x a m p l e s include t h e following. -

A diagram

thai

may explain

your

model e v e n better t h a n words. -

A drawing or an image is also a model.

In some cases it c a n offer

much

mole

information

we are here

than t h e

verbal d e s c r i p t i o n , and may b e also

5 rniies

easier t o understand a n d c o m m u n i c a t e a m o n g people. A l s o n o t e t h a t in some cases a diagram c a n e x c l u d e some of t h e uncert a i n t i e s t h a t may c o m e From t h e verbal description For e x a m p l e , the verbal model cited a b o v e m e n t i o n e d the left ear, hut did n o t specify w h e t h e r it is t h e person's left car ot the person's left ear as s e e n by t h e o b s e r v e r T h i s ambiguity disappears when t h e image is offered, -

Dyriairuc features c a n be included in a n a n i m a t i o n or a cartoon.

-

A c o n c e p t u a l model ot t h e bvdrologic cycle.

Transpiration

& Rrecipitalion \ \ \ \\ \\

Evaporation

\ \ \\

\\\w

Snow Over lane

How Surtace water a4

• Physical -

A Serf ace • saturated exchanges Saturated wate-

Infiltration i ' Unsaturated water a Percolation

& upflow Grounciwater flow

a r e c o n s t r u c t i o n of t h e real o b j e c t at a smaller scale.

Examples

include t h e following. -

M a t c h b o x toy cars. R e m e m b e r those m a n n e q u i n s t h e y put in cars to crash t h e m against a brick wall and see what

happens to the passengers' W e l l , those arc m o d e l s o f

Models and Systems

23

humans. T h e y are no good for studying 1Q, but they reproduce certain features of a human body that are important to design car safety devices. - A n airplane model m a wind tunnel. -

A fairly large (about 5 0 - m long) model was created in the 1970s to analyze currents in Lake Balaton (Hungary). Large fans blew air over t h e model and currents were measured and documented.

-

A physical model to study stream flow (see Figure 1 . 1 ) .

• Formal ( m a t h e m a t i c a l ) - that is when equations and formulas reproduce the behavior of physical objects. Examples include the following. - Q = m C ( t | — t?) — a model o f heat emitted by a body o f mass m, when cooling from temperature t| to temperature ii. C is the heat capacity parametei. -

V = YQ*

- a model o f an exponentially growing population, where YQ is

the initial population and d is doubling time. 2.

Time: how

is time treated in the model?

• D y n a m i c vs static. A static model gives a snapshot of the reality. In dynamic models, time changes and so do the variables in the model. Examples include the following. -

A map is a static model; so is a photo.

-

A cartoon is a dynamic model.

-

Differential or difference equations are dynamic models.

• C o n t i n u o u s vs discrete. Is time incremented step-wise in a dynamic model, or is it assumed to change constantly, in infinitesimally small i n c r e m e n t s ' Examples include t h e following: -

Y o n m a y w a t c h a t o y c a r roll d o w n a w e d g e . Ir will h e a p h y s i c a l m o d e l w i t h continuous time.

-

Generally speaking, systems of differential equations represent continuous

-

A difference equation is a discrete model. T i m e c a n change, but it is incre-

-

A movie is a discrete model. M o t i o n is achieved by viewing separate images,

time models. mented in steps (1 minute, 1 day, I year, e t c . ) taken at certain intervals. • S t o c h a s t i c vs deterministic. In a deterministic model, the state o f the system at t h e n e x t time step is entirely defined by the state o f the system at the current time step and the transfer functions used. In a stochastic model, there may be several future states corresponding to the same current state. E a c h o f these future states may o c c u r with a certain probability. 3 . Space:

how is space treated in the model ?

• Spatial vs local (box-models). A point model assumes that everything is homogeneous m space. Either it looks at a specific locality or it considers averages over a certain area. A spatial model looks at spatial variability and considers spatially heterogeneous processes and variables. Examples include the following. -

A demographic model o f population growth in a city. A l l the population may be considered as a point variable, the spatial distribution is not o f interest, and only the total population over the area o f the city is modeled.

- A b o x model o f a small lake. T h e lake is considered to be a well-mixed c o n tainer, where spatial gradients are ignored and only the average c o n c e n t r a tions o f nutrients and biota are considered. -

A spatial hydrologic model. T h e watershed is presented as an array o f cells with water moving from o n e cell to another downhill, along the elevation gradient.

40 Systems Scienrf? and Modeling fo' Ecologicst Economics • C o n t i n u o u s vs discrete. Like time, space may be represented either as c o n t i n u ous or as a mosaic o f uniform objects. Examples include the following. - A painting vs a mosaic. Both represent a spatial picture and both look quite similar from a distance. However, at close observation it is clear that smooth lines and color changes in a painting are substituted by discrete uniform elements in the mosaic., which change their color and shape in a stepwise manner, -

Differential equations or equations in partial derivatives are used for c o n tinuous formalizations.

-

Finite elements or difference schemes are used to formalize discrete models.

4 . Structure: how is the model structure defined? • Empirical (black-box) vs process-based (simulation) models. In empirical models, the output is linked to the input by some sort of a mathematical formula or physical device. T h e structure o f the model is not important as long as the input signals are translated into the output ones properly - that is, as they are observed. T h e s e models are also called black-box models, because they operate as some closed devices o n the way o f the information flows. In process-based models, individual processes are analyzed and reproduced in the model. In any case, it is n o t possible to go into all the details or to describe all the processes in all their complexity (it would not be a model then). Therefore, a process-based model may be considered as being built from numerous black boxes. T h e individual processes are still presented as closed devices or empirical formulas; however, their interplay and feedbacks between them are taken into account and analyzed. • Simple vs complex. T h o u g h qualitatively clear, this distinction might turn out to be somewhat hard to quantify. It is usually defined by the goals o f the model. Simple models are built to understand the system in general over long time intervals and large areas. C o m p l e x models are created for detailed studies of particular system functions. T h e increased structural complexity usually has to be compensated by coarser temporal and spatial resolutions. 5 . Method:

how is the model formulated and studied?

• A n a l y t i c vs computer models. A n a l y t i c a l models are solved by finding an a n a lytical m a t h e m a t i c a l solution to the equations. M a t h e m a t i c a l models easily become too complex to be studied analytically. Instead, numerical methods are derived that allow solving equations o n a computer. • Modeling paradigm. -

Stock-and-flows or systems dynamics models assume that the system c a n be represented

as a collection

of reservoirs

(that

accumulate

biomass,

energy, material, e t c . ) c o n n e c t e d by pipes ( t h a t move the material between reservoirs). -

Individual- (or a g e n t - ) based models. T h e s e describe individual organisms as separate entities that operate in time a n d space. T h e r e are rules that define the behavioi o f these agents, theii growth, m o v e m e n t , e t c .

- Network-based models. -

Input/output models.

-

Artificial neural networks.

6 . Field-related

classification:

what field is the model in (e.g. ecology)?

• Population models. T h e s e are built to study the dynamics and structure of populations. A population is easily characterized by its size, w h i c h may be why population ecology is piobably the most formalized branch o f ecology.

Models and Systems

25

• C o m m u n i t y models take several populations and explore what happens when they interact. T h e classic predator-prey or host-parasite systems and models o f trophic interactions are t h e most prominent examples. • Ecosystem

models attempt to represent the whole ecosystem, not just some

components o f it. For example, a model has been developed for the wetland ecosystem in the Florida Everglades (http://my.shvmd.gov/pls/portal/url/page/PG_ SFWMD_HESM/PG_SFWMD_HESM_ELMMavpage=elm).

It includes t h e

dynamics o f water, nutrients, plants, phvtoplankton, zooplankton and fish. T h e goal is to understand how changes in the hydro-period affect the biota in that area, and how the biota (plants) affects hydrology. 7. Purpose:

what is t h e model built for?

• Models for understanding would normally be simple and qualitative, focusing o n particular parts or processes o f a system - for example, the predator-prey model that we consider in C h a p t e r 5. • Models for education or demonstration. T h e s e are built to demonstrate particular features o f a system, to educate students or stakeholders. For example, the well-known Daisy World model is used to demonstrate how the planet c a n selfregulate its temperature, using black and white daisies ( S e e http://www.mformatics.sussex.ac.uk/research/projects/daisyworld/daisyworld.html for more about the model or http://library.thinkquest.org/C003763/flash/gaial.htm

for a nice

Flash a n i m a t i o n ) . • Predictive models are detailed and scrupulously tested simulations that are designed t o make real decisions. A perfect example is a weather model that would be used for weather forecasts. • Knowledge bases. Models c a n serve as universal repositories o f information and knowledge. In this case, the model structure puts various data in a c o n t e x t providing conceptual links between different qualitative and quantitative bits of information. For example, the Multi-scale Integrated Models o f Ecosystem Services (MIM.ES - http://www.uvm.edu/giee/mimcs/) organizes a n extensive body o f information relevant to ecosystem services valuation in five spheres: anthroposphere, atmosphere, biosphere, hydrosphere, and lithosphere.

Systems thinking In more recent years, people have really started to appreciate the importance o f the systems approach and systems analysis. W e are now talking about a whole new mindset and worldview based o n this understanding o f systems and the m t e r c o n n e c t e d ness between c o m p o n e n t s and processes. W i t h systems we can look at c o n n e c t i o n s between elements, at new properties that emerge from these c o n n e c t i o n s and feedbacks, and at the relationships between the whole and the part. T h i s worldview us referred to as "systems thinking." T h e roots o f systems thinking go back to studies on systems dynamics at M I T led by Jay Forrester, who was also the inventor o f magnetic-core memory, which evolved into the random access memory used in all computers today. Even though back in 1 9 5 6 he never mentioned systems thinking as a c o n c e p t , the models he was building clearly chiseled out the niche that would he then filled by this type o f holistic, integrative, cross-disciplinary analysis. W i t h his background in electrical and computer engineering, Forrester has successfully applied some of the same engineering principles

42 Systems Scienrf? and Modeling fo' Ecologicst Economics co social, e c o n o m i c and environmental problems. You can find a certain resemblance between electric circuits and systems diagrams that Forrester has introduced. T h e titles of his most famous books, Industrial World Dynamics

Dynamics

( 1 9 6 2 ) , Urban

Dynamics

( 1 9 6 9 ) and

( 1 9 7 3 ) , clearly show the types of applications that have been studied

using this approach T h e main idea is to focus on the system as a whole. Instead o f traditional analytical methods, when in order to study we disintegrate, dig inside and study how parts work, now the focus is on stud=ying how the whole works, how the parts work together, what the functions are, and what the drivers and feedbacks are. Forrester's works led to even more sophisticated world models by Donella and Dennis Meadows. T h e i r book, Limits to Growth

( 1 9 7 2 ) , was published in paperback

and became a national bestseller. Systems dynamics got a major boost when Barry R i c h m o n d at High Performance Systems introduced Stella, the first user-friendly icon-based modeling software. Despite all t h e power and success o f t h e systems dynamics approach, it still has its limits. As we will see later on, Stella should n o t be considered to be the ultimate modeling tool, and there are other modeling systems and modeling paradigms that are equally important and useful. It would b e wrong to think that systems approach and the ideas o f systems thinking are usurped by t h e systems dynamics methods. Systems can be described m a variety o f different ways, not necessarily using the stock-and-flow formalism o f Stella and the like. Systems thinking is more than just systems dynamics. For example, the so-called Life Cycle Assessment ( L C A ) is clearly a spin-off of systems thinking. T h e idea o f L C A is that any e c o n o m i c production draws all sorts o f resources from a wide variety of areas. If we want to assess the true cost o f a certain product, we need to take into a c c o u n t all the various stages o f its production, and estimate the costs and processes that are associated with the different o t h e r products that went into t h e production of this o n e . T h e resulting diagrams become very complex, and there are elaborate databases and e c o n o m e t r i c models now available to make these calculations. For example, to resolve the ongoing debate about the efficiency o f corn-based ethanol as a substitute for oil, we need to consider a web o f interactions (Figure 1 . 6 ) that determine t h e so-called Energy Return o n Energy Invested ( E R O E I ) . T h e idea is that you always need to invest energy to derive new energy. If you need to invest more than you get, it becomes meaningless to run the operations. T h a t is exactly why we are not going to run out o f oil, W h a t will happen is it will become more expensive in terms o f energy to extract it than we c a n gain from the product. T h a t is when we will stop pumping oil to burn it for energy, but perhaps will still extract it lor o t h e r purposes, such as the c h e m i c a l industry or material production. S o if e o l l t is t h e amount o f energy produced and e i n is the amount o f energy used in production, then E R O E I , e = e oul /e m . In some cases the net E R O E I index is used, which is the amount of energy we need to produce to deliver a unit o f net energy to the user: e ' = e o m /(e ( H i r - e i n ) . O r e ' = e / ( e - 1). As we unwind t h e various chains o f products and processes that go into the production o f energy from corn, the E R O E I dramatically falls. T h e current estimate stands at about 1.3, and there are still some processes that have not been included in this estimate. A true systems thinking approach would require that we go beyond t h e processes in Figure 1.6 and also look at social impacts as well as further ecological impacts, such as the e m i n e n t deforestation that is required for expanded corn production, and the loss o f wildlife that will follow. Taking all that into account, the question arises: with an E R O E I o f 1,3 or less, is it worth it? T o compare, the E R O E I for crude oil used to be about 1 0 0 ; nowadays it is falling to about 10.

Models and Systems

Energy to build and maintain machinery

A " "

Energy to mitigale land erosion and soil depletion

/

Trucks and tractors (horses 9 )

27

Energy io produce and deliver v & P

tf

Fertilizers ..

Pesticides (F S P)

( L a n d cultivation)

Energy lor machines

\

"-• - — —

Energy for s u d a c e and groundwater treatment

X

Energy 10 apply F & P

Crop growth

)

Energy (or crop yield a n y transportation x

Water run-off

Net e n e r g y —

Refinery

Water supply Energy for pumps

"t

——-...

Water for refinery j

Canals and pipelines

Energy lor refinery

Figure 1 . 6

•

;

j

Energy to build and maintain infraslruclure

output ) -

Energy lor delivery, storage arid distribution of tueis

Main flows of energy =£>

Flows of waler

-£>

Flows ol auxiliary energy

The iifecycle ol energy fmrn biomass production.

Further reading v o n Rerialnnffy, L. ( 1 9 6 8 } . Geriei;if Svilfrr? Theory. systems theory, AND one of the EIIMICJ of this

George 13iazilkr - This is an introduction to

approach.

H a l l , C . and Day, J ( 1 9 7 7 ) . Ecosystem Mixieling in Tfn;CTT tinci Piitciict- A n Imvyduaiov: with C i u e Htsrorfes. W i l e y . • A n evcellent caivcacn

of papers mi iht' iheoiy

of mofietirc); illujcrored fiy

a variety r>f rt:$$e!;. ft urn. wrry Ji/jfeient n'ers of life. Furd, A . ( 1 9 9 9 ) . MrjciL'lmg the Enwroni)u')ir. Island P r e " . - A n e n t i r d y Sue Hit based Gilo's u. nciii mfrotiuetion to modelirlg. ifie way it car. be perfimntd gOWK on fjeiii'iUh rfit' .Wl/ii imerftinf.

u'll/tuut stilly

The book is per feci for a mtirhematic/iily

B c r h n s k i , D. ( 1 9 7 8 ) . O a J ^ j t e i r j i A n a b i ; s .

textbook

kji:niin>; what is tRackier

A n Ess;:} CtnirennfYg i f t f Limiwtiuns of Sumt

M a r t i n i wrier;! Methods in ihe Social, Political, and Biological Sciences. M I T Press. - A carious collection ofentujucs of some very famous

models. M t i ) be ' t ' c o i n m e n ^ i l tu belter

mode.is are not the tif;imare snftitton, thai thivr^ are oliwrjy ifiese fiTftSjfc'iis i f W U tie extite::-, There is some c o n n w e r s } akoui

made l>art of the

model.

inho acrwi./i> said, " A l l mocieis an wrong

/iiJ ,: . Acciii ding io some texts it .iytts

nntttritarui tfict

i i m f t i n n a j m their us? and thut

Souk1 iitufels me u j t -

Dealing; ;» feast tfiui i? icftof M e C o j claims m his collec-

tion: of i^tiores M c C o y , R. ( 1 9 9 4 ) . Tfte Best of Deramg.

Statistical Prucev. C o n r m l Prc«. H o w e v e r ,

others a n i i b u t e i t t o G e o i ^ e E.P. Box i n "Robustness i n the stiateyv o f scientific model building", page 202 ut flotusmess i n Statistics ( 1 9 7 9 ) , Launer, R.L. a n d W i l k i n s o n C M . , KdnotSA c a d e m i c Press. A t t t a all it doesn't really matter wlio said i i hist; ii is c e r t a i n l y very true.

28

Systems Scienrf? and Modeling fo' Ecologicst Economics S a a t y . T . L . ( 1 9 8 2 ) . Decision is not exactly decision

related

making.

Saaty

hierarchies

systems

structural

properties

m building

between

are decomposed models.

color,

Saaty

In

in descending order

pans

functional

hierarchies

that are created

accord-

S o m e philosophical interpretations o f hierarchies c a n be found in H a u g h t , J F. ( 1 9 8 4 ) .

The

Cosmic

Adventure•

and help in conflict

process

Science,

Religion

are essential

to useful the

making

His hierarchies

most to analyze

decision

the elements.

in

structural

according

age, etc This is the type of hierarchies

is analyzing

between

This

o( hierarchies and their applications

and functional hierarchies.

structural

into their constituent

such as size, shape, relationships

Lifetime Learning Publications, p . 2 8 -

j or Leaders.

but gives a good analysis

distinguishes

process-based

ing to the essenaal

Making

to modeling

resolution

and the Quest for Purpose,

Paulisr Press (also available o n l i n e

ac hccp\//www.religion-onlme,org/sho\vchapcer.asp.'tide = 1 9 4 8 & . C — 1 8 1 4 ) A n interestinganatyiis of hierarchical

levels and then interaction

Levels. The Bnash Journal There

on sustainabihty

is more

tion to the concept: Common sented for

by Fieblernan, J. ( 1 9 5 4 ) . T h e o r y o f Integrative

is performed

for the Philosophy in Chapter

oj Science,

7 . The Bnmtland Commission report gives a good

introduc-

W C E D ( W o r l d C o m m i s s i o n o n E n v i r o n m e n t and D e v e l o p m e n t , 1 9 8 7 ) , O u r O x f o r d University Press. Some

Future.

5 (17): 59-66.

to hierarchy theory

of the issues related

are pre-

by von Berralanfy, L. ( 1 9 5 0 ) . A n O u t l i n e o f G e n e r a l S y s t e m T h e o r y . The British

the Philosophy

and eventually

and how different

15, 1 9 3 - 1 9 6 . For an overview

Economics:

it can be in different

sustainability:

on how sustainabilicy

relates

to

Journal longevity

see: C o s t a m a , R , a n d Patten, B. ( 1 9 9 5 ) . Defining and predicting

- to hierarchies

sustainability, Ecological communicating

1 ( 2 ) - 1 3 4 - 1 6 5 . F o r more

of Science,

hierarchical

global versus regional

of sustainability,

us

definitions,

levels see V o m o v , A . ( 2 0 0 7 ) . Understanding and perspectives.

E n v i r o n . Dev. and S u s t a i n . (Keep://

www.springerlink,com/content/e77377661 p 8 j 2 7 8 6 / ) . If you want

to see how this can be related to

discounang, see V o m o v , A . a n d Farley, J . ( 2 0 0 7 ) . R e c o n c i l i n g Sustainability, S y s t e m s T h e o r y and D i s c o u n t i n g . Ecological Economics, 6 3 ; 1 0 4 - 1 1 3 . The modeling jyrocess (2006).

Environmental There

is very

T e n iterative Modelling

sreps

well described

and Software-

a good

and evaluation

R A, and Norton J . P

of e n v u o n m e n t a l

models.

21 ( 5 ) : 6 0 2 - 6 1 4 .

are quite a few web sites on Systems

s t h i n k i n g l e a r n . h t m l gives

by J a k e m a n , A ).. L e t c h e i

in d e v e l o p m e n t

overview

Thinking, h ttp://ww\v.t hesys terns thinker, com/systemof the field. Another

good

introduction

is available

at

htcp://www. t h i n k 1 n g . n e t / S y s t e m 5 _ T h i n k i n g / I n t r o _ c o _ S T / r n t r o _ t o _ s t . h t m l The

several

classic

books

that led to many

concepts

of systems

thinking

are by I.Forrester: ( 1 9 6 9 ) .

U r b a n Dynamics. Pegasus C o m m u n i c a t i o n s , I n c . ; ( 1 9 6 2 ) Industrial Dynamics, T h e M . I . T , Press ) o h n W i l e y &. S o n s . ; ( 1 9 7 3 ) W o r l d D y n a m i c s , W r i g h t - A l l e n Press; and a general his 1968 book.

Principles

Another

classic

is the book

Growth.

S i g n e t . Jti paperback

ited in the 2004

of Systems,

overview

in

Pegasus C o m m u n i c a t i o n s .

by M e a d o w s D . H , , R a n d e r s J . , a n d M e a d o w s D . L, ( 1 9 7 2 ) , Limits edition

was a bestseller

at that lime.

edition. Limits to Growth; The 3 0 - Y e a r Update,

More recently

the topic was

C h e l s e a G r e e n , 3 6 8 p.

to

revis-

2. T h e Art of M o d e l i n g 2.1

Conceptual model

2.2

Modeling software

2.3

Model formalization

"How to avoid false

proof?

1. allow no hasty and predetermined judgment; 2. decompose each difficult problem into simple ones that you can resolve; 3. always start with simple and clear, and gradually move on to more complex; 4. make complete surveys of all done before and make sure that nothing is left aside." Descartes

SUMMARY T h e r e is really a lol of nrt in building .1 good model. T h e r e are n o clear rules, only guidelines for good practice. T h e s e are constantly modified when required by the goals of modeling, the data available, and the particular strengths and weaknesses of the research team. In many cases it is possible to achieve the same level of success c o m i n g from very different directions, choosing different solutions. However, there are certain steps or stages that are c o m m o n to most models. It is important to understand these and learn to apply them. A n y system can be described in the spatial, temporal and structural c o n t e x t . It is important to be clear about these three dimensions in any model, to avoid inconsistencies or even errors. W e start with a conceptual model describing the system in general terms, qualitatively. If needed, we will then find the right quantitative formalizations for the processes involved. W e may apply theoretical knowledge or rely on data from another similar system to do this, or we can base our search o n data that are available only for the particular system we are studying and try to reproduce these in our equations A s a result, we will get either process-based or empirical " b l a c k - b o x " models. T h e y both have their strengths and weaknesses. A brief introduction to Stella, a systems dynamics modeling package, is presented,

with step-by-step instructions for model

building using this

formalism.

29

46 Systems Scienrf? and Modeling fo' Ecologicst Economics It is important to have a version of this or other similar software ( M a d o n n a , Simile, Vcnsim, or the like) and scart practicing, sincc modeling is like playing a piano - it is hard to learn to do n only by reading books and listening to lectures. You have to get your hands dirty and do it yourself.

Keywords Time, space, structure, conceptual models, resolution, Superfiind, biological

titne,

grids, black box models., empirical models, process-based models, Boftnim's paradox, systems dynamics, software, Stella, stocks and flows, exponential growth, limiting factors, M i c h a e l i s - M c n t e n functions *

A

T h e r e is n o predefined prescription for how to build a good model

h is the model-

buildniK piocess itself that is most valuable for a better understanding of a system, for exploring the interactions between system components, and for identifying the possible effects of vari.01.1s forcing functions upon the system. O n c e the model has been built n is a useful tool to explain the system properties, and in some cases may lead to new luuimgs about the system, but u is clearly the process of modeling thai adds most to our knowledge about and understanding of the system. Even (hough we do not know the ultimate model-making algorithm, we are aware of some key rules that are always useful to keep in mind when creating a model, by adhering to them, a great deal of frustration and various crises can be avoided T h e list of such rules c a n be quite long, and varies slightly for every modeler and every modeled system. Therefore, as in arc in modeling - experience is probably the most valuable asset, a n d there is n o way t o avoid all eirors, W e c a n only try to decrease their number

1

Conceptual model In most cases, the modeling process starts with a conceptual model. A conceptual model is a qualuai ive description of the system, and a good conceptual model is hall the modeling effort. To create a c o n c e p t u a l model, we need t o study the system and collect as much information as possible both about the system irseli and about similar systems studied elsewhere. W h e n creating a conceptual model, we start with the goal of the study and then try to explain the system that we have in terms that would march the goal. In designing the conceptual model, we decide what temporal, spatial and structural resolutions and ranges are needed for our study ro reach the goal. Reciprocally, the conceptual model eventually becomes important t o refine the goal o f in. del development. In many cases the goal of t h e study is quite vague, and it is only after the conceptual model has been created and the available data sets evaluated t h a i the goals of modeling can b e c o m e clear. Modeling is an essentially

iterative

pioe-ess. W e c a n n o t prescribe a sequence of steps that take us t o the goal; it is an adaptive process where rhe target 1$ repeatedly adjusted and moved as we go along, depending both o n our modeling progress and o n the external conditions that may be changing the scope of rhe study. It is like shooting at a moving target - we c a n n o t make the target stop to take a good aim and then start the process; we need to learn to readjust, to refine our model as we go. Building a good c o n c e p t u a l model is an important slep o n this path.

The Art of Modeling

31

Temporal domain In the temporal domain, we first figure out the specific rates (resolution) of the main processes that we are to model and decide for how long (range) we want to observe the; system. If we are looking at bacterial processes with microorganisms developing and changing the population size within hours, it is unlikely that we would want t o track such a system foi over a hundred years. O n the o t h e r hand, if we are modeling a forest we can probably ignore the processes that are occurring within an hour, but we would want to watch i his system for decades or even centuries. If there ;s little change registered over the study period, the model may not need to be dynamic. Ir may be static and focus on other aspects of the system. For example, a photo can be a snapshot that captures the state of the system at a particular moment, or a series of snapshots can be averaged over a certain time interval. In a way every photo

is like

since

it

is

that, never

really instantaneous. S o m e time needs t o pass

between t h e

moment

the shut-

ter opens and t h e moment it closes. A picture o n a photo can be just a little blurred,

repre-

senting the

bit

change

in the system while the

shutter

was

open, and we may not even notice it if the exposure was short

enough.

In

some photos where the

exposure was

not

set right

comes

out

A photo as a snapshot

may

nol be an instantaneous

model. Il may actually r e p i e s e n t averages over time for certain s y s t e m c o m p o n e n t s

this quite

clearly, and in most cases these photos end up in the trash bin, except when we actually wanted to see the trajectory o f the o b j e c t while it was moving and intentionally kept the shutter open for a while. T h a t would be a static representation of a dynamic system, in a way showing its average stare. If temporal c h a n g e is important, we need to identify how this change occurs. In reality, time is continuous. However, in s o m e cases it may be useful to think of time as being discrete and to describe the system using event-based formalism. O r we can think o f change in time as a sequence o f snapshots. A seiies of snapshots creates a better representation of dynamics. T h a t is how a movie is made, when by alternating many static images we e.ieate the feeling of moving objects. If a dynamic approach is selected and warranted to answer the questions posed in the study, we should start thinking about the appropriate resolution of our temporal model. T o have the movie smoothly rolling, we need to display at least 16 snapshots per second. In this model that we watch on t h e screen, t h e temporal resolution is then o n e - s i x t e e n t h of a second. T h i s resolution is dictated by the goal, which is

48

Sys turns Sciairo and d Modeling M o d e l " i g fcr f o * Ec'ological in A •

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m a n e / much more m a n w h a t happens ever other p e r « x » at » m e -

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s so huge mat

w e still cannot associate m u c * with n a n a think about
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car® less about systems thai evolve In considerably di"»-en» t r r e s c e t o s Largs m a m m a l s l*vs for 10 and more years, and w * care a o t mora about m e m . n a r

i * . about r s e c t s . w h o Ave

for months g oeat to us. nor tt Stapp-'^fl o n an ant While s o m e religions, such as Buddhism J H Quite concerned aeout Me tr general and consider .1 a sin to ki'i even an insect. they d o not r d cannot cere about m e myriads o* bacteria mat live abound us. the >»e cycles of w r . c n w e p-jrposefuty ot inedbertandy interfere w u h Just imagine h o w many bacteria you H i w f e n you lake a n u M t c s ' Smttarfy. < * * concerns soem to fade w f i e n w e start looting at »y»ge- v n e s e a t e s M o s t people certainty cars k r mail children, perhaps to a large extent Because the d»*fren's Wesoan ovc^aps considerably with our o w n Tne m o i e distant m e generation, m e less concerned w e s e e m k> b e about t h e m . H o w else w o u l d w e expram w

o d s e s s o n w f l h t h e rtea of economic growth which is almost

8Nv&y$ associated Wim further resource c o n j j m p a o r at one end and po»ufton at tne other? There was certainly a penod in K j m » r h ^ o r y w h e n • o v o g r o w t n c e a t e d mora poaaib*ties for future generations it led m e growing p c o j t a t c n o* p e c C e . t »>eiped to f>ght disease, and t has s i g m f c a n t y d e c i e a s e d m o r t a l * / There u s e d to s e d e a r correlation b e t w e e n the s j e of the economy, measured by say. the Grass Nauyvar Product iGHP) and t i e expectancy or quality of life m general But 4 thai K * t h e cave gr mm w e n o w e » c e e « n g m e carrying capacity and m o s r y borrowing from the future, f r o m o w chikfran. grandchildren and future generations' Clearly, w e aio taking m o r a resources then w e are r e t u r r w g t a c k to the pooL The global footprint g a e s way D e v w d m e s e e Of ifvs p u n e t . out fcnee w e are not used to thinking in m u l t d e twme«eies. w e d o not s e e m t o

about mat w e are »>eady reapino

m e consequences a t w r i t e r thoughtlessness by o r a ^ o u a generations. and investing millions and billions of dollars to d e a n u p m e m e s s that was left for us by our predecessors O n e Such e * a m p e is the Super^und - a n environmental program a s r i f c l a r e d to id«n care of the abandoned n a w d o u s w a s t e « e s

H d i»so the n a m e of the fund established by the

Comprehensive E m w o n m e n t a i Response. C o m p e n s a t e " a * d l a b i l i t y Act Of 1990 ( C E R C I & I This tow w a s enacted in t n e w a * e of r « crsccwery otf towc w a s t e dumps such as Love Canal and T i m e s Beach in m e 1 9 7 0 s ^he EPA uses r o Act and tf-e money t o clean up such situs, and to compel i « P 0 n s « 4 e p e r t * * to perform c t e w u p s or reimburse the government for E f i M e d d e a r v u o s At tfvs t i m e there are

S O T *

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and the numbe» is

steadrfy increasing. W t d c m e amount of money mat goes into m e fund varies from year to year im

m e program rece-ved S 1 4 3 fe
1? yedrs later •! rece«voo

5 1 2 5 b> oft) w e are S t * 'afcmg B o u i b o o n s and & u « n s of rsollars That is the legacy ol p r e vious g e r e r a t k y - e W e are a - e a d y a g e ^ - e ' i t c n m a t suffers from the unwise environmental deosicms ve" man W e s t e r n ovnlizationa. there used t o b e m o t e irterest m long-term effects The Great L aw of the Iroquois Confederacy

S y s t e m s S c i e n c e a n d M o d e l i n g f o r E c o i o g ' c s ' tco , -»om«cs

34

states "In o v ovary doSbciston w e must cons«cJci l t v impact of oui o e c s M X s

ne*t

seveo gerefaaons." Are there any modern societies w h e r e decisions * r c m a d * b w r J on m o d e f t witr- j « c h a temporal extent? A c t ^ l l y . in some systems time evolves In quite a diffetent way tt-a- at? 4x9 u->ed ••: Fey c e r t i i processes in p^nis. for example, "he number ot olapsod minutes a n d days not meOer Plants do not d o something oimoV boCUJMJ It t» 8 o'clock in ' ' • « iTVfmog wti.it ••iggers their processes is the total number of fair days w h e n temperature is - M-' th»~. a c e n * - r v e s n o l d - say. 5"C In such systems it mokes m o r e J W W to m«nk in time.

m 0* v . />

ateo ca'ied biological time, to figure out w h a n the plants siart to grow, w h e n s H d i

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T -

m o * (0, f

6)

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m x h o n ^ e - f & r * n e c e s s a r y active t e m p e i a t u i o to a c c u m u l a t e T e m c e i a t u i e in Ota %yit*m b e c o - e s t m e T i m e i* m e a s u r e d a s t e m p e r a t u r e Cleoriy it is a totally difltxwn t i m » K « l a in 0 - y no< m«i t . m « o n a c c e l e r a t e or s l o w d o w n deppixfcnfl on climatic c o n d i t i o n s

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35

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kely get a sharper image

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in t r » case this a stipulated by chemt-

v x h .»- computer models, you will find t i n v u r

a , , «'vpomertr. b u t then .: w « b e m e c c n x m t * < t p e e d . for example, that will likely bmit your ability to h a v e h a h temcorai and spatial resolution at The s a m e r . m e

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36

Systems Scienrf? and Modeling fo' Ecologicst Economics

Uniform grid of equal square cells A watershed is represented by different larduse types. The grid simply mimics the raster information that comes from a landuse map. The cell is given the attribute of the landuse that covers the largest proportion of the area of a cell.

impossible t o work with. A g a i n , d e p e n d i n g upon t h e system specifics and t h e goal o f o u r model, we would want to use different spatial resolutions. It is n o t just t h e sixe o f t h e grain t h a t is i m p o r t a n t ; t h e form o f t h e grain also m a t t e r s S h o u l d we use a grid oi u n i f o r m , e q u a l - s u e d square c e l l s , as we would d o o n a rasterued m a p (Figure 2 . 3 ) . ' O r perhaps t h e cells should n o t be uniform, representing t h e a c t u a l configuration of t h e ecosystem.' A n d w h e r e d o we draw t h e b o u n d a r y in this case - especially tl t h e r e is an e x c h a n g e ot material across t h e boundary, as is t h e c a s e at t h e o u t l e t of a bay ' A n d h o w small should r h e cells b e (Figure 2 . 4 ) ? Perhaps a triangulated grid would be b e t t e r (Figure 2 . 5 ) ; T h i s certainly works b e t ter if we h a v e n o n uniform spatial c o m p l e x i t y a n d need t o describe c e r t a i n spatial e n t i t i e s in more detail t h a n o t h e r s . S u p p o s e we model a watershed. It m a k e s sense to h a v e finer resolution along rhe river t o c a p t u r e s o m e o f t h e effects o f t h e riparian zone. O n t h e o t h e r hand, vast s t r e t c h e s o f forest o r agricultural land may b e presented as spatially h o m o g e n e o u s e n t i t i e s - there is n o n e e d t o subdivide t h e m i n t o smaller areas. T h e boundaries may also n e e d a h i g h e r resolution. A triangular grid serves t h e s e purposes really well. BLI t h e n o t h e r c o n s i d e r a t i o n s also c o m e i n t o play: W h a t data d o we h a v e ? blow m u c h c o m p l e x i t y c a n we afford with t h e type of c o m puter t h a t we h a v e at oui disposal.' W h a t a r e t h e visualization tools rhar we h a v e t o m a k e t h e most of rhe model results? M a y b e instead o f triangles w c prefer t o use h e x a g o n s (Figure 2 . 6 ) . T h e s e c a n d o a b e t t e r iob describing dispersion, s i n c e they measure a b o u t t h e same d i s t a n c e from

Figure 2.4

A non-uniform grit) of quadrangles used to model the Chesapeake Bay. There is also the third, depth dimension (as in Figure 2 2), w h i c h is not

visible In this picture. A. The original gr d of 4,000 cells. B. A r advanced grid of 10,000 cells, w h i c h also expanded into the ocean to give a better description of the boundary conditions. C. The latest 14,000-cell grid, w i t h finer resolution in tributaries. Ironically, c h a r g i n g Irom one grid to another did not make the model any more accurate, however n did help incorporate more processes.

38

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Polygons as spatial compartments.

The area is described by much smaller number of entities. Flows between compartments need special attention. In most cases they are connected with some other processes, like river flow, for example t h e c e n t e r to t h e boundary a n d t h e y are s y m m e t r i c a l in terms of d i a g o n a l flows. N o marrer in w h i c h of t h e six d i r e c t i o n s we go, t h e links with n e i g h b o r i n g cells will be t h e same. T h i s is not so in t h e c a s e of square cells, if we want t h e m to c o m m u n i c a t e w i t h eight surrounding cells, assuming diagonal llows. Perhaps polygons could be used, as in t h e case of vector-based

Geographic

I n f o r m a t i o n S y s t e m s ( C I S ) such as A r c l N F O (Figure 2 7 ) H e r e t h e space is described by polygons, w h i c h are presented in terms of vectors o f c o o r d i n a t e s for all vertices o f l h e polygons. C o n v e r t i n g regulai c o n t i n u o u s geographic maps i n t o vector-based digital sets is usually performed in a tedious process of "digitizing," using special equipm e n t that registers t h e c o o r d i n a t e s of various points c h o s e n along t h e boundary. T h e more points you c h o o s e to describe t h e polygon that will a p p r o x i m a t e che area digitized, the higher t h e precision of t h e digital image anil the closer it is to t h e original. Polygons are good for m a p and image processing, s i n c e they c r e a t e a digital image t h a t is m o r e a c c u r a t e with far less i n f o r m a t i o n ro store. T o a c h i e v e t h e s a m e a c c u r a c y with raster maps, we would need m a n y more cells and t h e r e f o r e m u c h larger data sets. However, for purposes of m o d e l i n g , polygons are quite hard to h a n d l e if you want to s t r e a m l i n e processing

E a c h polygon is unique, and needs to be spe-

cially defined. If s o m e t h i n g c h a n g e s - say, land is c o n v e r t e d from o n e landuse type to a n o t h e r - t h e n t h e m o d e l may n e e d to be reinitialized. Triangular grids seem to present a good c o m p r o m i s e , offering greater flexibility: t h e size o f e a c h triangle might vary, yet it is still a triangle, with three b o u n d a r i e s and t h r e e vertices, a n d e a c h c a n be handled in a similar way in t h e m o d e l . C h o o s i n g t h e light spatial representation and designing a good spatial grid is a craft in its own right

As you h a v e seen, there are uniform and non-unifotm gtids,

40

Systems Scienrf? and Modeling fo' Ecologicst Economics triangular. hexagtirwl, square, e t c . p k l s . For a complex. model of a large spatial o b j e c t , say the C h e s a p e a k e Bay or art o c e a n , the design o f a grid c a n take many m o n t h s if not years. T h e r e are also software tonls that help to design the right grid. M o d e l performance and even results c a n c h a n g e substantially when switching from o n e grid t o a n o t h e r , so t h e importance ol this step in t h e model-building process should n o t be underestimated

Structural domain Finally, we decide h o w to represent t h e structure o f t h e system. O n e important dist i n c t i o n is b e t w e e n e m p i r i c a l .ind process-based m o d e l s (Figure l . b ) . A n e m p i r i c a l model may he c o n s i d e r e d as a " b l a c k b o x , " w h i c h takes c e r t a i n inputs a n d produces outputs in response to t h e inputs. Perhaps because we do not know, or d o n o t care, o r c a n n o t afford greater c o m p u t i n g resources or laxer d e a d l i n e s , we m a k e a deliberate d e c i s i o n n o t t o c o n s i d e r what happens a n d h o w inside t h e black hnx t h a t presents t h e system. T h e internal structure in this case is n o t analysed, a n d our o n l y goal is to find t h e appropriate f u n c t i o n to translate inputs intn outputs. T h i s is usually d o n e by statistical m e t h o d s . W e h a v e i n f o r m a t i o n about t h e data sets t h a t describe the inputs, and we h a v e t h e data regarding output values W e t h e n rry t o represent t h e n u m e r i c a l values o f outputs as m a t h e m a t i c a l f u n c t i o n s o f inputs. Below, we will c o n sider a n e x a m p l e of how this c a n he d o n e . In the case o f a process-based or m e c h a n i s t i c model, we a t t e m p t t o look inside t h e black b o x a n d try to identify s o m e of tin- processes t h a t iitcui in the system, a n a lyze t h e m a n d represent t h e m in 3 series o f e q u a t i o n s . Process-based models employ t h e additional i n f o r m a t i o n S h o u t t h e system that we may h a v e from previous studies of analogous systems, or a b o u t t h e individual processes t h a t we a r e looking a t . T h e y may use c e r t a i n t h e o r e t i c a l k n o w l e d g e c o m i n g from a vjiTiety o f disciplines. I n this sense, a process-based m o d e l m a y b e e v e n more useful than the i n f o r m a t i o n avail a b l e about t h e system studied. It should b e n o t e d , t h o u g h , t h a t all process-based models are still e m p i r i c a l , in a sense. W e c a n n e v e r describe all t h e details of all t h e processes in a system. It is

U Figure E f f l f X2a. l8U

A A black-box model, where the output is calcinated as a function of the inauts:

b, = /.{a,,a z ,a 3 ,a 4 ), without looking at what is liappenirg inside the system. B. A white-box model, where the structure of the system is analyzed and represented in the mocel.

The Art of Modeling

41

jusc that we go inro furcher d e p t h in t h e system, providing more detail about the processes in it. Yet we still end up with c e r t a i n black boxes, w h i c h we d o n o t wish to or c a n n o t consider in any more detail. If t h a t were n o t the case, we would hardly be a c c o m p l i s h i n g the m a j o r goal o f any m o d e l i n g effort, w h i c h is a simplification o f t h e system d e s c r i p t i o n . W e would be e n d i n g up with m o d e l s as c o m p l e x as t h e original systems, and t h e r e f o r e delivering little value for purposes o f synthesis. If we c h o o s e to build a process-based model, we may start describing the structure by using a diagram, r e p r e s e n t i n g the m a j o r c o m p o n e n t s o f the system: variables, forcing f u n c t i o n s and c o n t r o l f u n c t i o n s . W h e n deciding on the model structure, it is important to m a t c h t h e structural c o m p l e x i t y with the goals o f the study, the available data, and the appropriate temporal and spatial resolution. For e x a m p l e , if we are m o d e l i n g fish populations (Figure 2 . 9 ) , w h i c h grow over several years, there is little use in considering the dynamics o f bacterial processes, w h i c h have a specific rate of hours. In this case we may consider the bacterial population to be in equilibrium, quickly adapting to any c h a n g e s occurring in the system in "fish t i m e " , which is weeks or m o n t h s . W e may still want to consider the bacterial biomass for mass balance purposes, but in this case it makes perfect sense to aggregate it with the detrital biomass. H o w e v e r , c e r t a i n fast processes may h a v e a d e t r i m e n t a l effect upon t h e system. For e x a m p l e , it is well k n o w n that fish kills may o c c u r during n i g h t - t i m e and in t h e early m o r n i n g hours, w h e n there is still n o p h o t o s y n t h e s i s , but only respiration from algae in the system. A s a result, the o x y g e n c o n t e n t may fall below c e r t a i n threshold levels. T h e o x y g e n c o n c e n t r a t i o n s in this case vary from hour to hour, whereas fish

This model is not very detailed. It represents the chosen state variables and some of the forcing functions (fertilizers, feed). It is not clear w h a t t h e other forcing functions involved are, such as environmental, climatic conditions. There is also no indication of the spatial and temporal scale. Apparently this information is contained in the narrative about the model that usually goes together with the diagram.

42

Systems Scienrf? and Modeling fo' Ecologicst Economics

j

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r d Sediments

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Conceptual mociel of a lake ecosystem structure.

The model structure is different for the different spatial segments used in the model.

biomass c h a n g e s m u c h m o r e slowly. W e might want t o c o n s i d e r o x y g e n as part o f t h e system, t o m a k e sure that we d o not miss such critical regimes. In t h e lake e c o s y s t e m m o d e l s h o w n in Figure 2 . 1 0 , in addition t o t r o p h i c relations c e r t a i n spatial properties are present. T h e diagram shows h o w t h e model structure is presented in t h e t h r e e vertical s e g m e n t s that d e s c i i b e t h e pelagic part o f t h e lake. In t h e upper part t h r e e p h y t o p l a n k t o n groups ( A l , A 2 , A 3 ) are present; they are food for z o o p l a n k t o n ( Z ) a n d fish ( R ) . Various forms o f n u t r i e n t s ( o r g a n i c and i n o r g a n i c n i t r o g e n ( N O W , N I W ) a n d phosphorus ( P 0 W , P I W ) are supplied by d e c o m p o s i t i o n of detritus ( D ) . In t h e b o t t o m segments, t h e r e are n o b i o t a , o n l y n u t r i e n t s ( P I S , P O S , N O S , N I S ) a n d detritus. C o n c e p t u a l models m a y present m o r e t h a n flows o f material. Figure 2 . 1 ! shows a diagram used in a simple m o d e l developed t o analyze s u s t a i n a b l e d e v e l o p m e n t tn a s o c i o - e c o n o m i c a n d e c o l o g i c a l system. T h e model will be considered in m o r e d e t a i l in C h a p t e r 7. H e r e , n o t e t h a t , in a d d i t i o n t o t h e variables, t h e diagram also c o n t a i n s i n f o r m a t i o n a b o u t t h e processes a n d t h e i r causes. It describes b o t h t h e flows of m a t e rial a n d i n f o r m a t i o n in t h e system. W h e n m a k i n g all t h e s e d e c i s i o n s a b o u t t h e m o d e l structure, its spatial a n d t e m poral resolution, we should always k e e p in mind t h a t t h e goal o f any m o d e l i n g e x e r cise is to simplify t h e system, t o s e e k t h e most i m p o r t a n t drivers a n d processes. If t h e m o d e l b e c o m e s t o o c o m p l e x to grasp a n d to study, its utility drops. T h e r e is little a d v a n t a g e in substituting o n e c o m p l e x system t h a t we d o n o t understand w i t h a n o t h e r c o m p l e x system t h a t we also d o n o t understand. Even if t h e model is simpler than t h e original system, it is quite useless if it is still t o o c o m p l e x t o shed n e w light o n t h e system a n d t o add to t h e u n d e r s t a n d i n g o f it. Even if you c a n perform experim e n t s o n this m o d e l t h a t you might n o t be a b l e t o d o in t h e real world, is there m u c h value in t h a t if you c a n n o t e x p l a i n your results, figure o u t t h e causes, and h a v e any trust in what you a r e producing?

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Bonnini's paradox I t h o u g h t thnt thlo won en original way <0 phrase thi« the danger a n d low utility ol t u b ntiiuting ono c o m p l e x nyatem w t h another. But then I l e e m o d i h * r s o m e t h i n g sirr.far h a s boon alroadv described by [>utton and S t a r b u t * In 1971 a s the ' B o r r r r t p a r a d o x " A moot* 15 (n/iVf in (Vd6( ro acftieve uxwrarantfnp of an o t » a r > « d causer process. a n d t f w

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Tne rate and trie level are two main icons that can be used to put together more complex diagrams such as the one for the insect population (from; http://www.ento.vt.edij/~sh3rov/PopEcol/lec1/struct.htmri.

models for such systems as cities, industries and e v e n the whole world. S i m i l a r formalism was later used in several modeling software packages. O d u m c r e a t e d a n o t h e r set of symbols t o m o d e l systems based o n t h e energy flows t h r o u g h t h e m . H e c a l l e d t h e m energy diagrams, and used six m a i n i c o n s (see Figure 2.1 3). All systems are described in terms of energy, assuming t h a t for all variables a n d processes we c a n c a l c u l a t e t h e " e m b o d i e d " energy. In this case, e n e r g y works as a general c u r r e n c y t o measure all processes and " t h i n g s . " In many software packages (like s o m e of those considered in the next

para-

g r a p h ) , c o n c e p t u a l diagrams are used to input the m o d e l . For e x a m p l e , o n e o f t h e reasons t h a t systems d y n a m i c s software such as S t e l l a b e c a m e so popular in m o d e l i n g is that they are also handy tools to put t o g e t h e r c o n c e p t u a l diagrams, and, moreover, these diagrams are t h e n a u t o m a t i c a l l y c o n v e r t e d into n u m e r i c c o m p u t e r models. Figure 2 . 1 4 presents a sample c o n c e p t u a l m o d e l for a river system put t o g e t h e r o n t h e S t e l l a interface. Ir describes the river n e t w o r k as a c o m b i n a t i o n of Sub watersheds, river reaches and reservoirs. T h e S t e l l a interface c a n be used as a drawing board to put t o g e t h e r various c o n c e p t u a l diagrams and discuss t h e m w i t h o t h e r people in a process k n o w n as participatory m o d e l i n g . In this case, t h e m a j o r value o f t h e interface is that it is possible to easily add or delete variables and processes and i m m e d i a t e l y

The Art of M o d e l i n g

= "

45

D

Figure 2.13

Odum's formalism for energy-

based conceptual diagrams. A. Source of energy, B. Sink floss of energy from system), C. Storage tank, D. Production unit (takes in energy and information to create other quality of energy), E. Consumption unit, F Energy mixer or work gate.

see t h e impact o n model p e r f o r m a n c e . T h e m o d e l itself b e c o m e s a tool for deliberation a n d c o n s e n s u s building. Very similar diagrams c a n be put t o g e t h e r using o t h e r systems d y n a m i c s soft ware, s u c h as M a d o n n a , V e n s i m , Powersim or S i m i l e . In t h e s e software packages, " s t o c k - a n d - t l o w " formalism is used to d e s c r i b e t h e system. T h e diagrams are also k n o w n as flow diagrams, because t h e y represent h o w material flows t h r o u g h

the

system. A s we will see below, a s o m e w h a t different formalism is used in s u c h packages as G o l d S i m , S i m u l i n k and E x t e n d . H e r e we h a v e more, flexibility m describing what we wish t o d o in the model, a n d the model does n o t present only s t o c k s and flows. G r o u p s o f processes c a n be defined as s u b m o d e l s and e n c a p s u l a t e d i n t o special i c o n s t h a t b e c o m e part of t h e i c o n set used to put t o g e t h e r t h e diagrams. A s usual, we get more f u n c t i o n a l i t y and versatility at t h e e x p e n s e of a steeper l e a r n i n g curve a n d h i g h e r c o m p l e x i t y of design. Yet a n o t h e r o p t i o n in building c o n c e p t u a l diagrams is provided by t h e U n i v e r s a l M o d e l i n g Language ( U M L ) , w h i c h is a standardized s p e c i f i c a t i o n language for o b j e c t m o d e l i n g . It is designed as a diagrammatic, tool that c a n be used to build models as diagrams, w h i c h c a n be t h e n a u t o m a t i c a l l y c o n v e r t e d i n t o a n u m b e r o f o b j e c t o r i e n t e d languages, such as J a v a , C + + ,

P y t h o n , e t c . In this case you are a c t u a l l y

46

Systems Science and M o d e l i n g for Ecological Economics

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almost writing c o m p u t e r c o d e when d e v e l o p i n g t h e c o n c e p t u a l m o d e l . O n c e again, e v e n more universality a n d almost infinite

flexibility

is a c h i e v e d a t t h e price o f yet

greater effort s p e n t tri m a s t e r i n g t h e t o o l . Figure 2 . 1 5 presents a sample c o n c e p t u a l diagram c r e a t e d in U M L t o formulate a n a g e n t - b a s e d m o d e l o f a landscape used by s h e e p farmers, foresters a n d N a t i o n a l Park rangers, w h o are i n t e r a c t i n g o n very diff e r e n t temporal a n d spatial scales w i t h different d e v e l o p m e n t o b j e c t i v e s ( s h e e p prod u c t i o n , t i m b e r production a n d nature c o n s e r v a t i o n , r e s p e c t i v e l y ) . T h e r e are several types o f diagrams t h a t c a n be c r e a t e d using U M L . O n e of t h e m is t h e a c t i v i t y diagram, w h i c h describes t h e temporal d i m e n s i o n of your m o d e l . T h e

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V

Location

A UML class diagram of a system can be used both as a conceptua diagram and as a

way to program the model (from: http://jasss.soc.surrey.ac.Uk/6/2/2.html, reproduced with kind permission of the Journal of Artificial Societies and Social Simulation, Centre for Research on Socia Simulation, Surrey)

class diagram presented in Figure 2.1 5 in a way c o r r e s p o n d s to t h e structural d i m e n sion, but also has e l e m e n t s of t h e spatial r e p r e s e n t a t i o n such as t h a t displayed in t h e lake m o d e l in Figure 2 . 1 0 . Most software tools designed to c r e a t e U M L diagrams, such as Visual Paradigm (h 11 p: //ww w. v isua I - parad igm .com/prod uc t/vpum I/), also provide c o d e g e n e r a t o r s that will c o n v e r t your U M L diagram i n t o c o m p u t e r c o d e in a language of your c h o i c e .

48

Systems Scienrf? and M o d e l i n g fo' Ecologicst Economics More recently, there have been attempts to standardize the conceptual, diagrammatic representation o f systems using domain ontologies. A domain ontology represents a certain domain, ecosystem o r part of an ecosystem by defining t h e meaning of various terms, or names as they apply to those ecosystems. T h e idea is to define all the various c o m p o n e n t s o f ecosystems and present their interactions in a hierarchical way, st) that when you need co model some part o f t h e world you c a n pull out the appropriate set o f definitions and c o n n e c t i o n s and have your conceptual model. Several formal languages have been proposed to describe such ontologies. A m o n g them, O W L is probably the best known, and is designed to work over t h e World Wide W e b . It is yet to be seen how these oncological approaches will be accepted by the modeling community. A s with ocher attempts to streamline and automate the. modeling process, we may be compromising its most essential part - that is, t h e exploration and research o f t h e system, its elements and processes, at the level o f detail needed for a particular scudy goal. A n y attempt t o automate this part o f t h e modeling process may forfeit the exploratory part o f modeling and thus diminish the new understanding about the system that the modeling process usually offers.

To conclude... Conceptual diagrams are powerful modeling tools that help design models and c o m municate chem to stakeholders in case o f a collaborative, participatory

modeling

effort. In most cases, building a conceptual diagram is t h e first and very important step m the modeling process.

"A maxim for the mathematical resources of theory."

modeler: start simply and use to the fullest Berlinski

W h e n making decisions regarding a model's structure, its spatial and temporal resolution, we should always keep in mind that che goal o f any modeling exercise is to simplify the system and to seek the most important drivers and processes (Descartes's second principle). If the model becomes too complex to perceive and t o study, its utility drops. A s stated above, there is little g a m in substituting o n e c o m plex system that we do n o t understand with another complex system that we also do not understand. E v e n if the model is simpler than the original system, ic is useless if it is still too c o m p l e x to shed new light and to add to the understanding o f the system. S o our first rule is:

KEEP IT SIMPLE Ic is better to start wich a simplified version, even if you know ic is unrealistic, and then add c o m p o n e n t s co it. It helps a lot when you have a model thac always runs and the performance o f which you understand. T h i s is much betcer than pucting together a model that has everything in it to satisfy the most general goals and requirements. C o m p l e x models are hard to handle, they tend to go out o f control, they behave counter-intuitively and produce unreliable and uncertain results. A t every step o f model development you should try to have a running and tested version o f the model, and you can then build more into it. You will then always know

The A r t o f M o d e l i n g

49

at what point the model fails and n o longer produces something reasonable, and thus what kind of recent changes have caused rhc problem. Our second rule is:

KEEP IT RUNNING. KEEP TESTING IT Everything you know about the system is good tor the model. T h e more you know about the system, the better the model

However, that does not mean that

all the available data and information from previous or similar studies have to end up as part of the model. Modeling and data collection are iterative processes; o n e drives another. You never know which data at what stage ot the modeling study will be required, and how these will modify your interpretation and understanding o f the system. A t the same time, o n e of che most important values o f the modeling effort is that it brings together all the available information about the system in an organized and structured format. T h e model then c h e c k s that these daca are full and consistent. Even if the model turns out to be a failure and does noc produce any reliable predictions and conclusions, by bringing the daca togecher new understanding is erea red and important gaps in our knowledge may be identified. S o the third rule is:

THE DATA DRIVE THE MODEL THE MODEL DRIVES THE DATA No m i l t e r whether the goal ot the model is reached or the model fails to produce rhe expecred resulrs ; the modeling effort is always useful. W h e n building a model, a great deal of information is brought together, new understanding is creaced, and new networks and collaborations between researchers, experimenters, stakeholders, and decision-makers are emerging. T h i s clearly brings a study to a new level. W e therefore conclude that:

THE MODELING PROCESS MAY BE MORE IMPORTANT THANTHE PRODUCT

2.2

Modeling software T h e r e is a lot of software currently available char can help co build and run models. Between the qualitative conceptual model and the computer code, we could place a variety of software tools that can help to convert conceptual ideas into a running model. Usually there is a crade-off between universality and user-friendliness. At one extreme we see computer languages that can be used to translate any concepts and any knowledge into working compurer code, while at the other we find realizations of particular models thar are good only for the particular systems and conditions that they were designed for. In between, there is a variety of more or less universal tools (Figure 2.16). W e can distinguish bee ween modeling

languages,

which are computer languages

designed specifically for model development, and extendable

modeling

systems,

which

are modeling packages thac allow specific code to he added by the user if che existing methods are nor sufficient for their purposes. In contrast, there are also modeling

sys-

tems y which are completely prepackaged and do not allow any additions to the methods provided. T h e r e is a remarkable gap between closed and extendable systems in terms of their user-friendliness. T h e less power the user has to modify che system, che fancier the graphic user interface is and the easier ii is to learn the syscem. From modeling systems we go to extendable

models,

which are actually individual models that c a n be adjusted

• H

50

S y s t e m s S c i e n c e a n d M o d e l i n g f o r Ecological E c o n o m i c s Computer Code

ASSEMBLER C, C++, Java, FORTRAN BASIC

General purpose Computer Languages & Libraries

Swarm, Repast, MASON, Cormas

Environments

OpenMI, SME, SAMT

Mathematical Solvers

MATLAB, Mathematica

Modeling Languages

DYNAMO, CSMP, SYSL NetLogo, StarLogo

Spreadsheets

Extendable Modeling Systems

Modeling Systems

Extendable Models Individual Models

—

Excel, L o t u s l 2 3 , OpenOffice

Extend, GoldSim, Simulink SIMSAB, SONCHES

Simile Powersim, Madonna, Stella, Vensim, ModelMaker

OASIS CLEANER, BASINS MINLAKE WEAP Glumso, BALSECT, SimCity

Conceptual Model

Hierarchy of modeling software.

for diffeient locations and case studies. In these, the model structure is m u c h less flexible, the user c a n make c h o i c e s from a limited list o f options, and it is usually just the parameters and some spatial and temporal characteristics that c a n be c h a n g e d .

Models A n y m o d e l wc run o n a c o m p u t e r c o m e s as a p i e c e o f software. T h e r e f o r e , in s o m e cases, to solve a particular m o d e l i n g task we may try to find an appropriate m o d e l t h a t has b e e n developed previously for a similar case, a n d see if this software package, if available, c a n be adapted to the needs o f your project. T h i s c a n save you t i m e and m o n e y ; a n o t h e r benefir is that the model may h a v e already b e e n c a l i b r a t e d in a variety of l o c a t i o n s and c i r c u m s t a n c e s , and thus be more easily a c c e p t e d by a group

11

-]J

o f stakeholders

-[I-"J"

'- "• The Art of Modeling

51

S o m e models are distributed for a price, while others are available

free o f charge. T h e Register of Ecological Models ( R E M - htrp://L'co.wi;.um-kassel, de/ecohas.html) is a mera-database for models in ecology. It c a n be a good starting point if you are looking for a particular model. In some cases yon will he able t o download the executaMes from the website, in others you will have to c o n t a c t the authors. For the vast majority of models the sourcc code is unlikely to he available, and we can never he sure what actually goes on inside the processor W e can only look at the output and the d o c u m e n t a t i o n , run scenarios and analyze trends, but ultimately we have to trust the model developers that the model ;s programmed properly. W e also can make no changes to this kind of model. T h e fact that models c o m e as software black boxes may he o n e ot the reasons that model re-use is not very c o m m o n

it may take a long time to learn m i l understand

an off-the-shelf model, and it can he ^|uite frustrating if, after this investment of time and effuit, we find that the model is not q u i t e applicable to our case. It certainly helps when models are well documented, have good user guides and tutorials, ami c o m e with nice graphic usei interfaces ( G U I ) . Most o f the models that are commercially distributed have very slick G U I s that help set up these tools for particular applications. For example, the W E A P (Water Evaluation and Planning system - http://w\vw.weap21. org/index.asp) is a user-friendly software tool that helps u-ith an integrated approach to water resources planning. T h e core of the model is a water balance model that calculates t h e dynamics o f supply and demand in a river system. T o set up the model the usei is guided through a series of screens, which start with a river schematic that can be arranged on top of an Arc View map, and then takes care of data input with a series of dialogue Ixixes for water use, loss and re-use, demand management, priorities, err. T h e results arc then displayed in the same G U I in charts and tables, and on t h e schematic o f the river system. Scenarios that describe different demand and supply measures arc driving the system, and are c o n n e c t e d with the various results. These user interfaces certainly help with using the models, however, extending the model capabilities is not a straightforward task, if it i i possible at all. In particular when models are not open source, it is usually an "all or n o t h i n g " deal - you either use the model as it is, or drop it entirely if it does not have some o f the features needed tor your studyS o m e models aie deliberately designed as games, with special emphasis put on t h e graphic interface and ease of use. O n e good example is the SimO'cy computer game, which has a sophisticated s o c i o - e c o n o m i c and ecological model at its core, but n o o n e o t h e r than t h e model developers has ever seen this model and users do not know whether the model was calibrated or validated. T h e purpose here is to e n h a n c e the interactive utility of lhe program, to maximize lis user-friendliness and simplify t h e learning process

Extendable models S o m e models and modeling systems are designed in such a way that they allow additions to their structure. For example, O A S I S (Operational Analysis and Simulation o f Integrated Systems) is a software package designed to model river, reservoir and hydro-power systems ro develop operating policies and optimize water use. O A S I S has a graphical user interlace that allows easy configuration o f rhe system. You can describe how the river system looks, locate rhe inputs and withdrawals, and enter historical data sets that the system is to work with. In addition, there is an Operation C o n t r o l Language ( O C T ) - a special language used to enter rules and constraints that

52

S y s t e m s Scienrf? a n d M o d e l i n g fo' E c o l o g i c s t E c o n o m i c s

arc specific for your case study. O C L also acts as a bridge from O A S I S to o t h e r c o m puter programs. Users c a n express all o p e r a t i n g rules as operating goals or operating constraints, a n d c a n a c c o u n t for b o t h h u m a n c o n t r o l and physical c o n s t r a i n t s o n t h e system. T h i s takes care o f all sorts of " i f - t h e n " operations, which c a n go beyond just operational rules. T o m o d e l a n y system, t h e problem must simply lie approached as a set o f goals and c o n s t r a i n t s . T h e software t h e n works out t h e best means of m o v ing water through t h e system t o m e e t these goals a n d c o n s t r a i n t s . O C L allows data to b e sent a n d received b e t w e e n O A S I S a n d o t h e r programs w h i l e t h e programs are running, and e a c h program c a n t h e n react t o t h e information provided by t h e other. T h u s you are dealing with a prefabricated " c l o s e d " system, yet have some flexibility to modify it t o t h e particulai needs o t a study. T h e r e is clearly m o r e flexibility t h a n in case o f a pre-packaged m o d e l ; however, t h e user is still operating within t h e set o f assumptions and formalizations e m b e d d e d in t h e model c o r e of t h e software. T h e r e are also limitations to what O C L c a n h a n d l e as e x t e n s i o n s t o t h e O A S I S system.

Modeling systems U n l i k e p r e f a b r i c a t e d models, w h i c h are after all developed for specific systems, there a r c also g c n c r i c software tools that c a n h e l p to build m o d e l s o f any systems. T h e s e are probably most interesting to c o n s i d e r w h e n a n e w modeling task is in order. However, t h e more versatile a n d powerful t h e system gets, t h e harder it b e c o m e s t o master it and t h e more inclined modelers will b e t o stick to what they already know how t o use t h e well-known " h a m m e r - a n d - n a i l " paradox, which we will revisit in C h a p t e r 9 . H e r e , we will give a brief o v e r v i e w o f s o m e software tools that are a v a i l a b l e for m o d e l i n g , along with s o m e r e c o m m e n d a t i o n s a b o u t t h e i r applicability. It should be n o t e d t h a t t h e r e is a great deal o f d e v e l o p m e n t in progress, a n d n e w features are b e i n g added to t h e software packages q u i t e rapidly, so it is always r e c o m m e n d e d t h a t you c h e c k o u t t h e latest d e v e l o p m e n t s o n t h e respective w e b pages. N o t e t h a t neither of these tools implies a n y kind o f c o r e m o d e l ; they c a n be used to put t o g e t h e r any models

However, e a c h o n e assumes a particular m o d e l i n g paradigm a n d there-

fore has c e r t a i n l i m i t a t i o n s .

Systems

dynamics

tools

M o s t o f t h e s e appeared as a n o u t g r o w t h o f rhe systems d y n a m i c s approach o f J a y Forrester a n d his D Y N A M O language. S t e l l a was inspired by Forrester's formalism, a n d quickly gained worldwide r e c o g n i t i o n . In t h e following years a n u m b e r o f o t h e r software packages have appeared t h a t a r e b e t t e r t h a n S t e l l a in m a n y aspects, a n d are c e r t a i n l y worth investigating and c o m p a r i n g prior t o any purchase decisions.

S T E L L A - isee systems ( f o r m e r l y HPS), h t t p : / / w w w . i s e e s y s t e m s . c o m / - Free Player a n d 1 - m o n t h trial v e r s i o n - M a c / W i n

Most used in academia, and has much legacy code developed Over the past decade has been heavily prioritizing the User Interface features with nice capabilities to create game-like models, where the modeling part can be hidden from the user and only the front-end. which is similar to a Flight Simulator dashboard, is provided. Recent addition of isee MET Framework promises more integration with other tools, but is not exten sively used and tested yet.

The Art of Modeling

53

- V e n t a n a Systems, h t t p : / / w w w . v e n s i m . c o m / - F r e e V e n s i m PLE (personal learning edition) - Mac/Win VENSIM

Same basic features for stock-and-flow modeling as Stella, w i t h recent addition of some important functionality, such as calibration (will automatically adiust parameters to get the best match between model behav ior and the data), optimization (efficient Powell hill-cl mbing algorithm searches through the parameter space looking for the largest cumulative pay-off), Kalman filter, M o n t e Carlo analysis, Causal Tracing (a tree diagram s h o w s a selected variable and the variables that " c a u s e " it to change), etc. Vensim DLL is a w a y to communicate w i t h other applications such as Visual Basic, C, C + + , Excel, multimedia authoring tools, etc The DLL allows access to a Vensim model from custom-built applications; it can send data to Vensim. simulate e model, make changes to model parameters, and collect the simulation data for display

POWERSIM

- P o w e r s i m , h t t p : / / w w w . p o w e r s i m . c o m / - Free Player a n d trial v e r s i o n ,

Win This modeling tool has mostly been catering for the business community. Communicates with M S Excel. Powersim Solver is a companion product that handles calibration, optimization, risk analysis and risk management

M A D O N N A - UC Berkeley, h t t p : / / w w w . b e r k e l e y m a d o n n a . c o m / - Free R u n T i m e version, Win/Mac Runs many times faster than Stella Will do parameter calibration (curve fitting), and optimization. Has several more numeric methods to solve ordinary differential equations Stella compatible will take Stella equations almost as is and w o r k w i t h them.

M O D E L M A K E R 4 - Exeter S o f t w a r e ( f o r m e r l y C h e r w e l l ) , h t t p : / / w w w . e x e t e r s o f t w a r e . c o m / c a t / m o d e l m a k e r . h t m l - No free v e r s i o n s , W i n Same as the others in this category, plus quite extensive optimization and numeric methods, including M a r q u a r d t or Simplex methods, simulated annealing and grid search methods of initial parameter estimation, ordinary, weighted, and extended least squares methods of error scaling, comprehensive statistical report ing; M o n t e Carlo global sensitivity w i t h 14 distribution types, 5 different integration methods - Runge Kutta, Mid-Pomt, Euler, Bulirsch-Stoer and Gear Gear's is an appropriate solver lor stiff simulations w h e r e processes happen on very different timescales.

- Simulistics (formerly open-source AME, Agroforestry Modelling E n v i r o n m e n t ) , h t t p : / / w w w . s i m u l i s t i c s . c o m / - Free E v a l u a t i o n Edition, M a c / W i n / L i n u x S I M I L E

Allows

object-based

representation that handles disaggregation

and individual-based

modeling,

auto-

generates C + + model code, plug-and play modules. Supports modular modeling any p a n of a model can be extracted and used s e p a r a t e ^ Has plug-m displays, allowing field-specific graphics Also has options for spatial models w i t h some basiclinks tc GlS

54

S y s t e m s Scienrf? a n d M o d e l i n g fo' Ecologicst E c o n o m i c s

T h e basic m a t h e m a t i c a l Formalism and t h e interface c o n v e n t i o n s used in all chese packages a r e q u i t e similar, so o n c e you h a v e mastered o n e o f t h e m it should be q u i t e easy to s w i t c h 1:0 a n o t h e r if you are l o o k i n g for c e r t a i n special features. P r o s : T h e d e v e l o p m e n t o f all this m o d e l i n g software h a s c e r t a i n l y simplified t h e process o f building models, to t h e e x t e n t that p r o g r a m m i n g is n o longer needed to put t o g e t h e r models, and only very basic n u m e r i c and m a t h e m a t i c a l skills are required. S y s t e m s d y n a m i c s h a s b e c o m e widely used in a variety o f applications. C o n s : T h e r e is also a reverse side to it. M o s t of the software developers advertising their products will tell you that building a model is n o w as simple as clicking your mouse. U n f o r t u n a t e l y this is srill n o t quire so, and c a n hardly ever be so, since modeling is primarily a research process thac requires knowledge and understanding o f the system to generate more knowledge and more understanding. By simply putting together diagrams and pretending thar now you c a n run a model o f your system, you may generate false knowledge and illusions. T h e modeling systems are indeed very helpful if you k n o w how to build models; otherwise, they c a n b e c o m e deceptive distractions.

Systems

diagrams

A n o u t g r o w t h o f rhe systems d y n a m i c s a p p r o a c h is what we c a l l e d systems diagrams tools. T h e software discussed here has many m o i e icons t h a n t h e stocks, flows and parameters t h a t t h e systems d y n a m i c s tool operates with. W h o l e submodels or solvers for m a t h e m a t i c a l e q u a t i o n s , say, partial differential e q u a t i o n s , may be e m b e d ded i n t o specially designed icons chat later o n b e c o m e part o f t h e t o o l b o x for future a p p l i c a t i o n s . O n c e again we get more f u n c t i o n a l i t y and flexibility, but c e r t a i n l y at t h e e x p e n s e o f a m u c h s t e e p e r l e a r n i n g curve.

E X T E N D - Imagine-That, h t t p : / / w w w . i m a g i n e t h a t i n c . c o m / i n d e x . h t m l - Free Demo, Win/Mac As follows from the name of the product, the system is sxtendable. It encourages modularity, providing the functionality to encapsulate certain processes and subsystems into blocks that can be further reused. Extend models are constructed with library-based iconic blocks. Each block describes a calculation or a step in a process. Interprocess communication allows two applications to communicate and share data with one another. This feature allows the integration of external data and applications into and out of Extend models. Information is automatically updated between Extend and Excel, can be connected with databases (Open DataBase Connectivity), has embedded ActiveX or OLE (Object Linking and Embedding), and works with DLL (Dynamic-Link Library). Block-building is based on ModL - a language that provides high-level functions and features while having a familiar look and feel for users with experience programming in C. Also allows scripting to develop "wizards" or self-modifying models. Evolutionary Optimizer employs powerful enhanced evolutionary algorithms to determine the best model configuration.

- G o l d S i m T e c h n o l o g y Group, h t t p : / / w w w . g o l d s i m . c o m - Free Evaluation and Student v e r s i o n , W i n GOLDSIM

Uses the same approach based on an extendable library of icons ("hierarchical containers") for a variety of processes. The user controls the sequence of events, and can superimpose the occurrence and consequences

—111— The Art of Modeling

•I • 55

of discrete events onto continuously varymg systems Other features include particularly strong stochastic, Monte Carlo simulation component to treat uncertainty and risks inherent in all complex systems, embedded optimization, sensitivity analyses (e.g tornado charts, statistical measures); externa! dynamic links to programs and spreadsheets, and direct exchange of data with ODBC compliant databases. Models can be saved as player files. There are several extens on modules le.g. for Contaminant Transport using solvers for POE, financial analysis, etc.).

- The M a t h w o r k s , h t t p : / / w w w . m a t h w o r k s . c o m / p r o d u c t s / s i m u l i n k / i n d e x . h t m l - Free trial and w e b d e m o , W i n Mac/UNIX SIMULINK

Built on top of MATLAB (see beluwl Provides an interactive graphical environment and a customizable set of block libraries, which can be extended for specialized applications. More power, but harder to master. Can generate C code for your models, which can be further embedded into other applications. Based on the same concept ot expandab:e libraries of predefined blocks, with an interactive graphical editor for assembling and managing block diagrams, with functionality to interface with other simulation programs and incorporate hand written code, including MATLAB algorithms. Has full access to MATLAB for analyzing and visualizing data, developirg graphical user interfaces, and creating model data and parameters.

Pros: Power, versatility, llexiniluy. expnndnhibiy. C o n s : In a way t h e pros b e c o m e rhefr c o n s , s i n c e after investing m u c h time to fully master these systems it is most likely thac t h e y will b e c o m e your " h a m m e r " for rhe future. Besides, when b e c o m i n g wedded to proprietary 1 software there is always a risk of r u n n i n g i n t o l i m i t a t i o n s t h a t will be hard to o v e r c o m e .

Modeling languages, libraries and environments C o m p i l a t i o n s of model languages, libraries appropriate t o specific a p p l i c a t i o n s , and software e n v i r o n m e n t s aie e v e n m o r e general, rely less 011 Mime e m b e d d e d assumptions about rhe m o d e l structure and t h e logic ot c o m p u t a t i o n s , but require more prog r a m m i n g efforts.

Spreadsheets T h e w e l l - k n o w n spreadsheets are probably t h e most widely k n o w n software applic a t i o n s t h a t can also help build quire s o p h i s t i c a t e d models

Microsoft

Evcel is by

far t h e b e s t - k n o w n and widely used spreadsheet. However, there is also L o t u s

123,

w h i c h actually pioneered the spreadsheet c o n c e p t a n d is now o w n e d by I B M , o r t h e o p e n - s o u r c e O p e n - O f f i c e suite

B o t h offer very similar functionality. T h e o t h e r

o p t i o n is to use G o o g l e spreadsheets, w h i c h are found on rhe web and c a n be shared a m o n g several developers, w h o c a n t h e n access a n d update t h e d o c u m e n t from anywhere a r o u n d the world using just an I n t e r n e t browser. T h e basic f u n c t i o n a l i t y t h a t c o m e s with spreadsheets is chat formulas c a n be programmed using s o m e very s i m p l e c o n v e n t i o n s . For d y n a m i c models these formulas can be reiterated, using a T I M E c o l u m n , and using t h e results ot previous c a l c u l a t i o n s ( r o w s ) to g e n e r a t e t h e values lor t h e n e x t time step

56

S y s t e m s Scienrf? a n d M o d e l i n g fo' Ecologicst E c o n o m i c s

P r o s : T h e s e cools are free or almost free, s i n c e they c o m e as part o f M i c r o s o f t O f f i c e , w h i c h is more o r less standard these days, or c a n be d o w n l o a d e d as pare o f O p e n O f f i c e , or c a n be used over che I n t e r n e t with G o o g l e . A n o t h e r advancage is that many users already k n o w h o w to use t h e m . C o n s : S p r e a d s h e e t s c a n q u i c k l y gee very c u m b e r s o m e as m o d e l c o m p l e x i t y increases, especially if you a r e Crying to add d y n a m i c s co ic. T h e r e is n o good G U I for m o d e l i n g , so models may be hard to present a n d visualize. O n l y t h e simplest n u m e r i c m e t h o d s c a n feasibly be i m p l e m e n t e d (say. Euler for O D E ) .

Mathematical

solvers

T h e r e are several specialized m a t h e m a t i c a l packages designed to help solve m a t h e m a t i c a l problems. A s such they c a n b e useful for modeling, s i n c e , after all, m o d els a r e m a t h e m a t i c a l e n t i t i e s w h i c h n e e d to be solved. T h e s e packages are n o c very helpful in formulating models. In this regard they a r e as universal as spreadsheets, but tin I ike spreadsheets, w h i c h are q u i t e well k n o w n and intuitive to use, r h e m a t h e m a t i c a l packages have a s t e e p l e a r n i n g c u r v e and require l e a r n i n g specialized prog r a m m i n g languages. O n t h e benefit side, t h e c o m p u t i n g power and versatility o f m a t h e m a t i c a l m e t h o d s is unsurpassed.

M A T L A B - T h e M a t h W o r k s , h t t p : / / w w w . m a t h w o r k s . c o m / p r o d u c t s / m a t l a b / - Free trial version, M a c / W i n / U n i x This is a high-level technical computing language and interactive environment for algorithm development, data visualization, data analysis and numeric computation. It is faster to master MATLAB than C or Fortran, but it c e r tainly requires a major investment of time. Includes mathematical functions for linear algebra, statistics, Fourier analysis, filtering, optimization (including genetic algorithms), and numerical integration; 2D and 3D graphics functions for visualizing data; tools for building custom graphical user interfaces; and functions for integrating with external applications and languages, such as C, C + -F, Fortran, dava, COM, and Microsoft Excel. May be a great tool to analyze models, but offers little help in conceptualizing and building them. There are sister products, such as Simul/nk (see above) or Simscape that are designed to handle the modeling process.

MATHEMATICA-Wolfram

Research,

http://www.wolfram.com/products/

m a t h e m a t i c a / i n d e x . h t m l - Free w e b seminars and d e m o s , M a c / W i n / L i n u x / U n i x The software integrates numeric and, importantly, symbolic computations. It provides automation in algorithmic computation, interactive document capabilities, powerful connectivity, and rich graphical interfaces in 2D and 3D. It is based on its own advanced programming language, and it needs time and effort to master this. Has no specific tools to support modeling per se, but can be very useful to solve, run and analyze already built models. Can be very useful to study individual functions that are used in your model - for example, to test how

parameters

impact the functional

response (see, for example,

http://www.wolfram.com/products/

mathematica/newin6/content/Dynamiclnteractivity/FindSampleCodelnTheWolframDemonstrationsProject.htmll.

P r o s : M a t h e m a t i c a l power t h a t is hard co m a t c h . C o n s : S t e e p learning curve, requires a solid m a t h e m a t i c a l b a c k g r o u n d .

The Art of Modeling

57

Environments U p to 8 0 p e r c e n t of a m o d e l i n g c o d e may support various input/output f u n c t i o n a l i t y and interfaces with data a n d o t h e r programs

It makes perfect sense to build software

packages that would take c a r e of t h e s e d a t a - s h a r i n g a n d c o m m u n i c a t i o n procedures, so that modelers c a n focus o n t h e actual formalization o f processes and systems. T h e r e art- n u m e r o u s m o d e l i n g e n v i r o n m e n t s d e v e l o p e d to support m o d e l i n g and to increase m o d e l functionality.

O P E N M I - O p e n M I Association, h t t p : / / w w w . o p e n m i . o r g / o p e n m i n e w / - Open source, p l a t f o r m independent OpenMI stands for Open Modeling Interface and Environment, a standard for model linkage in the water domain OpenMI avoids the need to abandon or rewrite existing applications

Making a new component

OpenMi-compliant simplifies the process of integrating it with many olher systems. It provides a method to link models, both legacy code and new ones OpenMI standardizes lhe way dala transfer is specified and executed, it allows any model to talk to any other model (e g from a different developer) without the need for cooperation between model developers or close communication between integrators and model developers. Based on Java and NET technology, currently OpenMI has some 20+ compliant models in its library.

S M E - U V M , h t t p : / / w w w . u v m . e d u / g i e e / I D E A S / l m f . h t m l - Open source, Mac/Linux/Unix The Spatial Modeling Environment iSME) links Stella with advanced computing resources. It allows modelers to develop simulations in the Stella user-friendly, graphical interlace, and then take equations from several Stella models and automatically generate C + + code to construct modular spatial simulations and enable distributed processing over a network of parallel and serial computers. It can work with several GIS formats, and also provides a Java viewserver to present results ot spatial simulations in a vanety ot graphic formats.

S A M T - Z A L F , h t t p : / / w w w . z a l f . d e / h o m e _ s a m t - l s a / - O p e n source, Linux Spatial Analysis Modeling Tool (SAMTI is a modeling system with some GIS features, designed to help wuh spatial analysis. It is an open system that links to different models (especially fuzzy-models, neural networks, etc.). Ii can also link to a general-purpose modeling language DESIRE.

P r o s : A d d e d f u n c t i o n a l i t y to o t h e r m o d e l s and m o d e l i n g tools. C o n s : Hardly any, s i n c e m o d e l i n g e n v i r o n m e n t s mostly serv e o t h e r m o d e l i n g paradigms rather t h a n imposing any o f t h e i r own upon rhe user. In most cases, il is t h e n e x t level of m o d e l i n g that may require q u i t e good m o d e l i n g a n d systemic skills. Usually, user and developer groups are q u i t e limited a n d are very m u c h driven by e n t h u s i a s m . T h e r e f o r e , future d e v e l o p m e n t a n d support may he q u i t e u n c e r t a i n .

Agent-based

tools

A g e n t - b a s e d m o d e l i n g requires m o r e c o m p l i c a t e d formalism co describe rhe b e h a v ior and d y n a m i c s of individual a g e n t s and their spatial distribution and behavior.

58

Systems Scienrf? and Modeling fo' Ecologicst Economics Perhaps for this reason t h e r e are n o " d r a g - a n d - d r o p " a n d " c l i c k - a n d - i n n " software packages available su far. A l l software fools in this area a i e designed around s o m e p r o g r a m m i n g language. It c a n be e i t h e r versions of h i g h - e n d full fledged programm i n g languages such as C + + or J a v a , or a simplified language such as Logo

However,

it still requires s o m e p r o g r a m a i i n g ro gel t h e mode; to r u n . A l l package? h a v e links t o G 1 5 data, t h o u g h some m a k e a special effort to e m p h a s n e that

This connection

usually goes in o n e d i r e c t i o n , and is provided by routines that mi port data from raster G I S ( A r c View, A r c G I S ) a n d m a k e it a v a i l a b l e for t h e m o d e l i n g tools.

S W A R M - Swarm Development Group,

http://www.swarm.org/wiki/Swarm_main_

page - Open source, a n y p l a t f o r m This is a collection ol soliwaie libraries, written in Objective C, originally developed at the Santa Fe Institute and since then taken up as an open-source project with developers ail over the world Swarm is a software package for multi-agent simulation of complex systems. It is specifically geared toward the simulation of agent-Based nodels composed of large numbers of objects. EcoSwarm is an extension library of code that can be used for individual-based ecological models ihtip://www.humboldt edu/~ecoiriodel/iiidex htmi

R E P A S T - ROAD (Repast O r g a n i z a t i o n for A r c h i t e c t u r e and D e v e l o p m e n t ) , http://

r e p a s t . s o u r c e f o r g e . n e t / - O p e n source, a n y p l a t f o r m Repast (RFciirsive Porous Agent Simulation Toolkit! is an agent-based simulation toolkit originally developed by researchers at the University of Chicago and the Argonne National Laboratory Repast borrows many concepts Irom the Swarm toolkit. It is different in its multiple pure implementations in several languages (Java, C#, Net, Python) and its built-in adaptive features, such as genetic algorithms and regression Includes libraries for genetic algorithms, neural networks, random number generation, and specialized mathematics, has built-in systems dynamics modeling capabilities, has integrated geographical information systems fGISI support

M A S O N - G e o r g e Mason University, http://cs.gmu.edu/~eclab/projects/mason/O p e n source, a n y p l a t f o r m MASON Stands for Multi-Agent Simulator Of Neighborhoods ... or Networks ... or something . . the developers are no! sure It contains both a Java model library and an optional suite of visualization tools in 2D and 3D. It can represent continuous, discrete or hexagonal 2D, 3D or Network data, and any combination of these. Provided visualization tools can display these environments in 2D or in 3D. scaling, scrolling or rotating them as needed Documentation is limited

C O H M A S - Cirad, h t t p : / / c o r m a s . c i r a d . f r / i n d e x e n g . h t m -

Freeware

Programming environment lo model multi-agent systems, with focus on natural-resources management. It is based on VisualWorks, a programming environment which allows the development of applications in SmallTalk programming language and is freely available frottl a third party website.

The Art of Modeling

59

O P E N S T A R L O G O - MIT, h t t p : / / e d u c a t t o n . m i t . e d u / s t a r l o y o / - O p e n source,

Mac/Win A programmable modeling environment for exploring the behaviors ot decentralized systems, such as bird Mocks, traffic iams and ant colonies, and designed especially foi use by students It is an extension of the Lego programming language, which allows control over thousands of graphic individuals called "turtles" in parallel. Comes with a nice interface, making it user-friendly and ready to use Some basics of the l o g o language are simple to learn, and users can start modeling in less than an hour

N E T L O G O - Uri W i l e n s k y ( N o r t h w e s t e r n U n i v e r s i t y ) , h t t p : / / c c l . n o r t h w e s t e r n . e d u /

n e t l o g o / - Freeware, M a c / W i n / L i n u x NetLogo, a descendant of Staifogo, is a riKilti-platlorm general purpose complexity modeling and simulation environment. The design is similar to SiarLcgo: it also has a user-interface It is written in Java and includes APIs so that it can be controlled from external Java code, and users can write new commands and reporters m Java It comes with hundreds of sample models and code examples that help beg nners to get started It is very welt documented, and also has a systems dynamics component

P r o s : I hese systems oiler perhaps t h e only possible way to identify emergent properties t h a t c o m e from interaction b e t w e e n agents. Most of t h e a p p l i c a t i o n s are o p e n source, which creates infinite possibilities for linkages, e x t e n s i o n s , and i m p r o v e m e n t s C o n s : R e q u i r e p r o g r a m m i n g skills, t h e r e f o r e may t a k e a c o n s i d e r a b l e time t o learn.

Wrap-up of software It should be noted t h a t there are hundreds ami m a y b e thousands ol software p a c k ages t h a t can be related to m o d e l i n g , and by no means c a n we o v e r v i e w even a small fraction of them

My goal h e r e was t o look at some representative e x a m p l e s and try

t o put t h e m in s o m e order. C l e a r l y , for beginners, especially those with no or few q u a n t i t a t i v e and p r o g r a m m i n g skills, ir m a k e s more sense t o Stan at t h e easier end of t h e spectrum a n d e x p l o r e s o m e of t h e existing models or m o d e l i n g systems. T h e y will take c a r e of m u c h of the tedious model organization and m a k e sure t h a t t h e model is c o n s i s t e n t , they may h e l p with u n i t c o n v e r s i o n s and logic of c o m p u t a t i o n s , and t h e y will immediately offer some n u m e r i c m e t h o d s to run a s i m u l a t i o n . In some cases t h e y may actually work as is, " o f f - t h e - s h e l f , " for s o m e a p p l i c a t i o n s t h a t are repeated frequently for similai systems. A s tasks b e c o m e more c o m p l e x , t h e r e will probably be t h e n e e d to m o v e higher up t h e diagram in Figure 2 . 1 6 , and e x p l o r e s o m e of t h e more s o p h i s t i c a t e d m o d e l i n g tools and m e t h o d s . M o s t of t h e software tools, like living organisms, go through life stages after they are born, they devemp, reach maturity, and t h e n s o m e t i m e s decline and die. It is hard to predict what t h e future of many of these products will he, especially when they are corporately owned and depend upon t h e d y n a m i c s o f world markets. In this regard, well-developed o p e n - s o u r c e products promise more continuity, but e v e n they c a n fade away or be replaced with s o m e t h i n g better T h i s is what we are now seeing with Sw'jrm, w h i c h tends to morph i n t o o t h e r products, such as Repast- Similarly, S M L has been hardly developing over t h e past few years. In the proprietary world, there

76 Systems Scienrf? and Modeling fo' Ecologicst Economics does not seem to be much progress in the M o d e l M a k e r development. Similarly, Stella seems to be relying o n its previous success and has not shown much improvement over the past several years. T h e r e are also the models that are offered by federal agencies, which are free to download and use, but for which the source code is closed and proprietary. Except for the price factor, there is no big difference between these and the closed commercial products; in both cases users have very little to say about the future development o f the software and entirely depend on some obscure decision-making process and funding mechanism, either in the corporate world or by the government. T h e b o t t o m line is that we need to keep a close eye on all these systems, and be flexible enough to migrate from o n e to another. T h e "hammer—nail" syndrome should be avoided. N o modeling software is universal; there are always systems that could be better modeled using a different formalism and different m a t h e m a t i c s and software. If you confine yourself to only o n e modeling system, you may start to think that modeling is only what the software is offering. In reality, there are numerous different approaches, and all o f them may be worth considering when deciding how to model the system of interest. Models built using open-source software are most desirable, since they c a n be modified to meet particular needs of various applications. Moreover, they can be tested and fixed if errors are found. W h i l e commercial proprietary software comes as closed "black boxes", where you can never be sure what's inside, open-source models are open, and the source code can be viewed and modified. O n the other hand, commercial products tend to be better documented and supported. O n e rule o f thumb is that if a project has involved a great deal o f brainpower and enthusiasm, go for open source; if there is good funding for the project, go for commercial products. Modeling is iterative and interactive. T h e goal is frequently modified while the project evolves. It is much more a process than a product. It becomes harder to agree on t h e desired outcomes and the features o f the product. T h i s certainly does not help when choosing t h e right software package to support modeling efforts. T h e r e is also a big difference between software development and modeling, and software engineers and modelers may have different attitudes regarding software development. For a software engineer, the e x p o n e n t i a l growth o f computer performance offers unlimited resources for t h e development o f new modeling systems. Models are viewed by software engineers merely as pieces o f software that c a n be built from blocks or objects, almost automatically, and then c o n n e c t e d , perhaps even over the web, and distributed over a network ot computers. It is simply a matter o f choosing the right architecture and writing the appropriate code - the code is either correct or not; either it works or crashes, N o t so with a research model. Instead, scientists would say that a model is useful only as an eloquent simplification o f reality, and needs profound understanding o f the system to be built. A model should tell more about the system than simply the data available. Even t h e best model c a n he wrong, and yet still quite useful if it e n h a n c e s our understanding o f the system. .Moreover, it often takes a long time to develop and test a scientific model. As a result o f this difference in point o f view and approach, we tend to see much more rapid development o f new languages, software development tools, and code- and information-sharing approaches among software engineers. In some cases, new software packages appear faster than their user community develops. In contrast, we see relatively slow adoption o f these tools a n d approaches by the research modeling community. T h e applied modeling community, driven by strict deadlines and product-oriented, may be even more reluctant to explore new and untested tools, especially since such exploration always requires additional investment for acquisition,

The Art of Modeling

61

installation and learning T h e proliferation o f modeling software, as in the case of systems dynamics modeling tools, may even be considered an impediment, since if there were only o n e or two modeling tools generally accepted by all then these could be used as a c o m m o n modeling platform, as a c o m m u n i c a t i o n tool to share and distribute models W i t h so many different clones of the same basic approach we get a whole variety o f dialects, using which ir may be harder to find c o m m o n ground. In thi^ Ixiok we will mostly be using Stella for our demonstrations. Stella's success is largely due to its user-triendly graphic interface ( G U I ) and a fairly wise marketing program that mosrly rargets students and university professors. Stella helps to illustrate a lot o f modeling concepts. I do not intend ro endorse or promote S t e l l a in any way; ir is no better than the other software packages available. It is jusr a matter o f my personal experience and the legacy code that is available. T h e r e f o r e , you will need ro get at least a rrial or Player version ol Stella ro be able ro do rhe exercises and study the models that are presented in this book and can be downloaded from the book website. Doing rhis in alternative packages is an option that is only encouraged. S o m e systems dynamics tools described above offer tools to read and run Stella models. For example, M a c o n n a will take Stella equations and, with some minimal tweaking, will run rhem - actually many times faster - and offer some additional exciting features W e will see how this works in C h a p t e r 4. T h e basic mathematical formalism and rhe interface c o n v e n t i o n s used in all these packages are quite similar - so o n c e you have mastered o n e of them, it should Inquire easy to switch to another if you are looking for certain special features If you are unfamiliar with Stella, some limited instructions are available below.

mentioned

above, the G U I in Srella is extremely user-friendly and the learning curve is gradual, so it should not take long for you to be able to use it for the purposes of this course.

A very quick introduction to Stella and the like

Before you start 'earning Stella or s o m e of the other modeling packages described above, please ensure that you realize there are different modeling paradigms used m these packages and m certain respects this makes it hard to compare t h e m

n this book w e are mostly studying

dynamic models, so w e will be locking for software tha; can help us w i t h this kind of modeling. The dynamic feature means that the s y s t e m that w e study changes over time and that there ore variables that evolve. This also means that there are certain imitations and ceriain conventions There are sysiems that are not very well suited to mode mc w i t h Stella and the ike If w e simply want to use Stella for certain calculations w e may probably dc it. but this may not be the best way to go - using, say, Excel may be much simpler For example, certain standard economic systems, which are usually formulated in t e r m s cf seme equilibrium state, may be hard to define in Stella, unless w e move away f r o m the equilibrium and consider the transition processes as well. Stella has very limited capabilities for statistical analysis

f a simple empirical

model is all you need, you may be better oft w i t h a statistical software package, which would also be much better lor analyzing uncertainty and generating mere sophisticated stanst.es Keep in mind that Stella and s o m e other dynamic modeling packages a s s u m e the sc-called " s t o c k - a n d - f l c w " formalism, w h e r e the s y s t e m is to be described in t e r m s c f reservoirs. called stocks, w h i c h are connected w i t h pipes that carry Mows Stccks are therefore always m e a s u r e d in t e r m s o f certain quantities of material, energy, bicmass. population numbers, etc.. w h i l e flows are always rates of material transferred per unit of time, or energy passed per unit cf t i m e , etc So w h e n using Stella w e start w i t h identifying the state

Ml 62

S y s t e m s Science and M o d e l i n g for Ecological Economics

variables, w h i c h w i l l be called stocks, and t h e n figuring o u t w h a t m a k e s tr.ese variables increase (because t h e r e a t e Hows of material cr e n e r g y c o m i n g in), or decrease (because there a i e f l o w s that g o out). W h i l e all s o i l s o l f o r m u l a s can b e u s e d to d e f i n e the f l o w s , the stocks can be c h a n g e d only by t h e f l o w s ; they c a n n o t b e calculated in any other w a y Steils o p e n s w i t h a graphic m e n u that c o n t a i n s a n u m b e r of icons.

n=

o

•

o

=

\

STELLA ® Re s e a r c h H

©

L i , En]

CUD

R

n

A

6

§ The first four are t h e m a i n building blocks for y o g i m o d e l

Figure 2 17 d e s c r i b e s

what

they are

- STEl •

ItHol

O H I

/I 1 <

©

I

Stats

variables

Flows

Connectors

Parameters, Buxl lisry variables

The rectangles are to represent the state variables, which are also called "Stocks" in Stella. They are to describe tne elements in your system These are the quantities that change in the system thai you analyze and which you report as results of the modeling exercise For each of lhe state variable a differential (difference) equation (see Chapter 3) will De created m Stella The flows represent lhe processes in which materials (or quantities) are taken from cne slate variable anri added to another The cloud is used to show that material is exchanged with lhe oulside ol lhe system. Circles represent parameters or auxiliary variables (also called "Converters" in Stella These are ail the quantities that are either constant in the model or will oe calculated based on the values of the state variables. Lines with arrows are lo show what depends upon what. They are lo feed the value from one Stella .con into another. They cannot move material or quantities; they only convey information.

Figure 2.17 and others.

S VARIABLE

e>

77

-GO

FLOW

o

PARAM

K

now

Mair building blocks in Stella. Note :hat thay are the same in Madonna, Simile

The Art of Modeling

63

To build a m o d e l using t h e s e tools, first click o n the State variable icon a n d t h e n s o m e w h e r e in t h e w i n d o w , w h e r e this variable w i l l be located o n t h e diagram. The variable appears w i t h a " N o n a m e " n a m e . Click o n this title a n d t y p e t h e n a m e t h a t you w a n t t h i s variable t o have. EQB

' STELLA® R e s e a r c h 5.0 •

f

OH

O

S VARIABLE

Next, click on t h e Flow icon a n d c h o o s e w h e r e y o u w a n t t o d r a w t h e f l o w that w i l l g o into or f r o m the s t a t e variable. Then h o l d o n t h e m o u s e b u t t o n a n d drag t h e cursor f r o m w h e r e t h e How starts t o w h e r e it g o e s . If y o u w a n t a s t a t e variable t o receive a f l o w or t o be drained by a f l o w , m a k e sure t h a t t h e State variable icon s h i g h l i g h t e d as you put t h e cursor o n t o p of it. If it d o e s , t h e n it w i l l be attached t o t h e f l o w ; if it d o e s not, t h e n it will not be a s s o c i a t e d w i t h t h e f l o w - a n d this m i g h t create a p r o b l e m in t h e f u t u r e if y o u d o not n o t i c e that t h e r e is a c l o u d placed s o m e w h e r e on t o p of your State variable or right next t o it. You w i l l be t h i n k i n g that the f l o w
HE

STELLA® R e s e a r c h 5.0

•to\

OH

&

^

A S VARIABLE

KJ

FLOW

You m a y n o w n e e d to a d d an auxiliary variable, in a similar w a y t o above, c h o o s e the circle icon a n d place it s o m e w h e r e o n your diagram. Give it a n a m e . If y o u click a n d h o l d o n any o l the n a m e s ot t h e e l e m e n t s n the diagram, you can drag t h e n a m e a r o u n d t h e icon t h a t it b e l o n g s to. This is useful t o k e e p your d i a g r a m tidy.

•

S

•

O

\

STE LLA » Re s e a rch 5.0

I

o

•

13

m

O N

- !

A

H j

3 B

A M.

K7 S VARIABLE

-9

y

PARAM R O W

Finally, u s e t h e c o n n e c t o r a r r o w s t o link variables., f l o w s and s t a t e variables. J u s t as w i t h f l o w s , after c h o o s i n g t h e c o n n e c t o r icon y o u click t h e m o u s e o n the origin of i n f o r m a t i o n a n d t h e n drag t h e a r r o w to c o n n e c t it w i t h t h e recip e n t of t h e i n f o r m a t i o n By d r a w i n g t h i s diagram, you have already f o r m u l a t e d o n e differential (or rather difference) e q u a t i o n that w i l l g o into your m o d e l S . V A R I A B L Elt) = S _ V A R I A B L E f t - d t i + F L O V V d t

64

S y s t e m s Scienrf? a n d M o d e l i n g fo' E c o l o g i c s t E c o n o m i c s

Next, you need to specify t h e actual size of FLOW and the initial condition for S_VARIABLE Before you go any further, switch t h e model f r o m t h e so-called " M a p p i n g M o d e " to t h e " M o d e l i n g Mode." To d o this, click on t h e button on the left-hand side that s h o w s the globe You may notice that " 7 " will appear on all elements in t h e diagram that need further definition

o

. I*

Eisj&itetiS

STEllAZ Research 5.0 OlnM 1 |I®MA|

HB t a VI/ Iftl

•<7

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c i i - y i o \\

lOlHlinil

1 IffllHAl 2

SJ

If you double click o n ihe parameter icon, this dialogue w i n d o w appears: O PARAM 4 Standard • Array Required ir.pnii

Q Summer

(DCDCDQ

Hunt Ins ABS ANO i AKOAH ADHAVStLAN AKBAVSTOOEV AHRAySia.

O HAftAMft. { Place right hand side of eauMton h e r e - 1

Decoine
|| Menage- jl CanccI J

OK |

Even though it requires that you "place t h e right-hand Side of ecuation here," in reality all you need t o d o is give the value for t h e parameter that you want to use m your m o d e l

Suppose

the rate you warn to use is 0.01. Simply t y p e that value in instead of t h e highlighted words, and click o n t h e " O K " button (or ha the " E n t e r " key) if for some reason you piefer only to click, you can also use t h e numerical pad offered in the dialogue box. You can also use s o m e arithmetic expressions, like 1/100, or choose s o m e of the tjuilt-m functions that are listed mere. Next w e double click o n t h e f l o w icon and o p e n this dialogue box

The Art of Modeling

65

N o w indeed you are to "place the right-hand side of equation h e r e " The required inputs are listed above, and i' you click o n any of t h e m they will be copied :nto the equation field. Here, describe the flow as a product of the available stock in S.VARIABLE. and the rate coefficient thai has already been specified in PARAMETER. PARAMETER"S.VARIABLE goes into the equation field Click " O K " and similarly douole click on the State variable icon. This opens the dialogue w i n d o w :o specify the initial conditions

•

S.VARIABLE *

Reservoir

QConveyor

Q Oven

Q Queue

0 Non-negative • Array Allowable Inputs

©CDCD0 ( 3 ® 0 Q OGD®0 CQGDGlG

# FLOW O PARAM

i o IHR •

INITIALS. VARIABI E} • ... { Place Initial value here...)

Document

Messarje...

OK

Cancel

The list of "Allowable i n p u t s " may be s o m e w h a t confusing, because it is only rsreiy that you will need any c-f the flows or other state variables to speedy the initial c o n d i f o n s . In m o s t cases you will simply type a constant - say, 10 Or you may store this constant as a parameter and t h e n refer to it in this box. Note that there is a check-box that says "Non-negative." By default, all state variables c o m e chocked as non-negative. This can be quite misleading You want the Hows in your m o d e l to make sense and to work in such a way that lhe stocks do not get depleted and negative. By clamping t h e m w i t h this check-box, you lose track of w h a t is really happening in your model. You may w e l l be generating s o m e totally crazy behavior that is supposed to make a stock negative; however, you will not even notice that if lhai variable is clamped it is r e c o m m e n d e d that you keep this box unchecked, unless you really k n o w w h a t you are doing and have a clear understanding of the effect of various flows on the state variable N o w that all the question marks have gone, your model is ready to run. You can also check out the equations that you have generated by clicking on i h e d o w n w a r d arrow in the upper len-hand corner:

•

=

~

STELLA® R e s e a r c h 5.0

.

0 @

66

^ H H l H H H B H I H i H I H H i l ^ B H H I H i H M M H M H H H I H H M ^ H B l

S y s t e m s Science and Modeling for Ecological Economics

The e q u a t i o n here d e s c r i b e s t h e e x p o n e n t i a l g r o w t h m o d e l : I 1 S_VARIABLE((j = S VARIABLES-dtJ+(FLOWrd! — IN IT S VARIABLE ="10 INFLOWS: (-••,

FLOW •= P ARAM'S VARIABLE

o

PARAM = 0.1

Before you start running the m o d e l , s p e c i f y w h a t y o u w a n t the o u t p u t to lock like. Returning t o t h e o t h e r icons at the t o p of the w i n d o w , the next g r o u p of three is m o s t l y for convenience, t h e s e are not essentia: t o c r e a i e m o d e l s The first o n e a l l o w s you t o place b u t t o n s in t h e diagram, w h i c h may be handy for navigation or for m o d e l runs. Instead of g o i n g t h r o u g h m e n u s , you can g e t s o m e t h i n g d o n e by simply p r e s s i n g a b u t t o n m t h e d i a g r a m The next icon a l l o w s y o u t o g r o u p

LA® R e s e a r c h 5.0

certain variables a n d p r o c e s s e s into s e c t o r s , this is useful

m

t o achieve m o r e order in t h e diagram run only certain parts of the m o d e l

It also a l l o w s y o u t o

in

I \

The third icon a l l o w s

y o u t o w r a p c e r t a i n details a b o u t p r o c e s s e s and display

Create graphic

t h e m as o n e icon in t h e diagram, this also m o s t l y s e r v e s esthetic purposes.

Li

A Display value

Create table

CreatB 'a*' box

The next g r o u p of four is t h e O u t p u t Tools Click o n the Graph icon, c h o o s e a place on t h e diagram, a n d click again. A n e w w i n d o w is o p e n e d to display t h e graphs. D o u b l e click a n y w h e r e in that w i n d o w , a n d a dialogue box is o p e n e d

By d o u b l e clicking on any of t h e values in t h e left-hand side y o u add t h e m to

t h e hst of variables on t h e right-hand side These w i l l be g r a p h e d |no m o r e than five per graphic). You can check o u i t h e d i f f e r e n t o u t i o n s a n d o p t i o n s available The m o s t u s e f u l is tne scaling, w h i c h is c o n t r o l l e d by highlighting (clicking o n c e i a variable a m o n g t h o s e s e l e c t e d a n d t h e n clicking on t h e a r r o w to : h e right of it. You c a n t h e n c h a n g e t h e m i n i m a l and m a x i m a l values for this var.abie that will d e f i n e t h e scale in t h e graph.

Graph Type: •

<•) Time Series

Comparative

O Scatter

Allowable S VAHIABI. E

h

& FLOW O

Q Sketchable

s Lii

S h o w N u m b e r s On Plots

0

S h o w Grid

•

Make 5 Grid S e g m e n t s

• T h i c k Lines

Min

$

2. 3. i

0

•Benchmark

Selected 1. I - ! S_VARIABLE

~

PARAM

4. 5.

Title:

Untitled

^New

Max

Srale:

Page:

Set From

Display:

Q Bar

~ Connect Dots

1 V"

To 100

Cancel

| ||

OK

The Art of Modeling

• h a very similar way. you can define a Table using output values that variables take during the course of the simulation. Choose the Table icon, click a n y w h e r e in the diagram, and a Table w i n d o w opens, double click a n y w h e r e in that w i n d o w , and a dialogue box opens.

Allowable Blank Line S VAHIAOUi

•

Selected S VARIABLE

0

ft FLOW

| <<

O PARAM

ir>

fable type: • Comparative

Orientation: 0 Vertiral

Report: Beginning balances O Ending balances

Report Rowvalues: Instantaneous O Summed

Report interval: • Every DF Title:

Untitled Table

c->r

Page: I V Cancel

] |[

OK

• O n c e again, choose the variables that you want to display and specify s o m e of the characteristics of your Table. Similarly, you can generate output for an individual variable, in this case, you will get the value that it attains by the end of the simulation • As a result, for the m o d e l being built it is possible to design the output and run the model by choosing "Run" f r o m the Run m e n u , or pressing C o i r m a n d - R (on a Mac) or CUI-R Im W i n d o w s ) . In this case, the graphic of exponential g r o w t h is displayed

67

68

Systems Scienrf? and Modeling fo' Ecologicst Economics

Finally, look at t h e Editing Tools p r e s e n t e d by the f o u r t h group of icons •

"he navigation cursor (the hand) is t h e o n e y o u w i l l m o s t l y

ED B

be u s i n g 10 o p e n a n a close w i n d o w s a n d t o arrange elem e n t s of your d i a g r a m

|

The p a i n t b r u s h icon is :o a d d

1 t 1<

s o m e color t o t h e d i a g r a m - y o u can color individual elem e n i s or change t h e color o f youf graphics

The dyna-

m i t e ' icon is u s e d t o d e l e t e t h i n g s in t h e diagram. Be

Navigation

| Delete

n

careful - there is n o " u n d o " in Stella until version 8 if y o u b l o w s o m e t h i n g up. it is g o r e I To u s e ttvs tool, click on the d y n a m i t e icon, choose t h e e l e m e n t you w a n t t o delete, and click o n it, this will highlight w h a t is t o b e b l o w n u p D o n o t release t h e m o u s e w h a t is highlighted is really w h a t y o u w a n t t o g e t rid o f 1

button

until v o u have verified that

The " g h o s t " is v e r y u s e f u l w h e n y o u n e e d t o c o n n e c t e l e m e n t s that are v e r y far apart m tne d i a g r a m

In that case, t h e d i a g r a m g e t s t o o busy if y o u d o all t h e c o n n e c t i o n s directly

Click o n t h e g h o s t icon, t h e n o n any o f t h e e l e m e n t s in t h e d i a g r a m y o u w a n t t o g h o s t A copy of it will b e a e a t e d that can b e p u t a n y w h e r e else in t n e d i a g r a m . Once again, h e r e are t h e m a m b u i l d i n g e l e m e n t s in Stella

f information flows [ between components

fAuxliaty variables V as a graph J

This brief o v e r v i e w c o v e r s only s o m e of t h e basics o f S t e ia However, it m a y oe s u f f i c i e n t as a starter, since m o s t o f t h e dialogue b o x e s and m e n u o p t i o n s are quite self-explanatory a n d m a y be m a s t e r e d oy t h e g o o d o d tnal-and-enor m e t h o d

Please n o t e that:

• You c a n n c t c o n n e c t t w o stocks w i t h a n i n f o r m a t i o n f l o w , only material f l o w s c a n c h a n g e a stock The i n f o r m a t i o n about a stock c a n affect a maier.al f l o w or an auxiliary variable b u t not t h e i n f o r m a t i o n flow. • A maienai f l o w is a s s u m e d t o b e posnive

If it b e c o m e s negative. Stel a w i l l c l a m p ir t o

zero. A negative n f l o w is actually a n o u t f l o w ; t h e ' e f o r e , smce y o u s p e c i f y t h e d i r e c t i o n o f

your

flow, the

Sign

matters, and Ste'la m a k e s sure that all f l o w s are positive. If y o u n e e d a

f l o w that can b e c o m e negative, u s e t h e b i l l o w o p t i o n A b i f l o w can g o in b o t h d i r e c t i o n s . M a k e sure that y o u have t h e p o s i t i v e f l o w a s s o c i a t e d w i t h t h e d i r e c t i o n in w h i c h y o u first d r e w t h e b i f l o w The negative f l o w w i l l t h e n g o in t h e o p p o s i t e direction. • You c a n n o t c o n n e d auxiliary variables w i t h material f l o w s . Only appropriate in this case

n f o r m a t i o n f l o w s are

The Art of Modeling

69

A l s o please note the f o l l o w i n g rules of g o o d s t y e w h e n building your m o d e l s

• Try to keep your d i a g r a m tidy Avoid long c o n n e c t o r s and c o n f u s i n g n a m e s , a n d avoid crissc r o s s i n g f l o w s . The easier it is t o read your d i a g r a m , the less errors y o u w i l l m a k e a n d t h e m o r e appealing it w i l l be to a n y o n e else w h o n e e d s t o u n o e r s t a n d t h e m o d e . • There is n o such thing as t o o m u c h d o c u m e n t a t i o n

Every variable or How in Stella has

a d o c u m e n t option, w h i c h is u s e f u l t o r e c o r d your ideas and c o m m e n t s a b o u l w h a t y o u are m o d e l i n g and the a s s u m p t i o n s you are m a k i n g . It is e x t r e m e l y i m p o r t a n t b o t h for t h e m o d e l c e v e l o p e r and the m o d e l user.

A s m e n t i o n e d a b o v e in our review, general-purpose spreadsheet software is a simple a l t e r n a t i v e t o S t e l l a and the like

For e x a m p l e , E x c e l may be used to huild

m a n y models c o n s i d e r e d in tht.s b o o k (see Figure 2 . 1 8 ) . Yc»i c a n download

this

m o d e l ( M o d e l _ O f _ E x p o n e n t i a l _ G r o w [ h . ) from the b o o k website, and e x p e r i m e n t

r

4

File

S o

Edit

Insert

Format

- 10

FP

9 10 !L

TF. I T Ii" jI fT. t9

20 21 22 23

1 2 3 4

5 6 7 8 0 10 i; 12 13 14 15 1$ 17 14 '?

20

21 22 23 24 >i

Figure 2.18

/

U

m

11 12 I 13 31 14 641 16 I0S1 17 71561 13.467171 21 4353861 23 5734769 25 9374 246 28 53''671 3 1 384 2833 34 5227121 37.9749834 41 7724817 45.9497259 50 5447028 55 5991731 61 1590904 67.2749995 74 0024934 81 4027494 89 54 3024 3 98 4973268 inn 347nc<)

&

Window *

j z

Help u

u

| m

m m $ % . a iw formulate the value in the next cell based on the value jri in the previous one

m

r — ; — " D — dx/dt = ax exponential grov4hmodtl 0.1 grov^h rate 10 inilialcondition

Equation:

2 a= 3 x04 Time JL '

B

Data -

=BS»$B$2*35•

56

1

Tools O -

& a U

||| Helvetic*

more detail).

View

160000

140000 120000

100000 80000 60000 40000 20000

II

21

31

41 _LL 6'

n

81

91

An example of a spreadsheel model for an exponential growth system (see Chapter 5 for

70

with t h e p a r a m e t e r and the formalization o f the e q u a t i o n . I t b e c o m e s q u i t e difficult, if possible at all, to do this m o d e l i n g for n u m e r i c a l m e t h o d s o t h e r t h a n Euler. Also, it gets very c o m p l i c a t e d as the m o d e l structure b e c o m e s more c o m p l e x . C h e c k out a n o t h e r e x a m p l e ( P r e d a t o r _ P r e y _ M o d e l ) , w h i c h is a two-variable m o d e l o f predat o r - p r e y d y n a m i c s . I t is still d o a b l e , but n o t as m u c h fun as in the systems d y n a m i c s software.

2.3

Model formalization T h e model formalization stage requires t h a t e a c h o f the processes assumed in the c o n c e p t u a l model in a q u a l i t a t i v e form be described q u a n t i t a t i v e l y as a m a t h e m a t i c a l formula, logical s t a t e m e n t or graph. T h i s is what you d o in S t e l l a w h e n you double c l i c k o n any o f t h e flow icons and get the dialogue b o x that invites you to specify t h e r i g h t - h a n d side o f the equations. C h o o s i n g t h e right m a t h e m a t i c a l description t o represent your q u a l i t a t i v e ideas about a process may be q u i t e tricky and ambiguous. A t this stage, describe h o w you e n v i s i o n t h e rates o f flows b e t w e e n variables. S u p p o s e you are describing t h e growth rate o f a bacterial

various

population.

T h e variable is t h e p o p u l a t i o n size. T h e r e are two processes associated: the birth rate and t h e mortality rate. You need to decide how to describe both of these processes as f u n c t i o n s o f t h e state o f t h e system ( t h e c u r r e n t size o f t h e p o p u l a t i o n in rhis c a s e ) and the state o f t h e e n v i r o n m e n t ( t e m p e r a t u r e , a v a i l a b l e food, space, e t c . ) . Suppose t h a t it is k n o w n t h a t rhe r e p r o d u c t i o n rate is a f u n c t i o n o f temperature, s u c h t h a t at low temperatures t h e divisions are rare, and as t e m p e r a t u r e grows the n u m b e r o f divisions steadily increases until it reaches a m a x i m a l value. S u p p o s e t h a t , based o n the available data, you c a n describe this r e l a t i o n s h i p by a graphic shown

m Figure 2 . 1 9 .

In this case, m is t h e m a x i m a l growth rate

that

we know. H o w d o you input this i n f o r m a t i o n i n t o the model? O n e o p t i o n would be to use t h e S t e l l a graphing o p t i o n and redraw the graphic in the model.

Figure 2.19

One possible limiting function for temperature-dependent multiplication rate.

The Art of Modeling

71

Iri the Stella m o d e l that y o u have s t a r t e d t o build, r e n a m e the variables to reflect t h e s y s t e m that is being c o n s i d e r e d n o w

CXS

B param

Births

rv T limitation

o

-

M param

Deaths

Population

Temperaiure

N o w you have t h e stock that r e p r e s e n t s t h e p o p u l a t i o n ( m e a s u r e d here in b i o m a s s u n i t s i n s t e a d of n u m b e r s ) The size of t h e stock is c o n t r o l l e d by t w o f l o w s : t h e i n f l o w is the births, similar to b e f o r e , but n o w t h e r e is also t h e o u t f l o w , w h i c h is the d e a t h s In addition t o s i m p l y having the births p r o p o i t i o n a ' t o t h e size of t h e population, the t e m p e r a i u r e l i m i t a t i o n is introd u c e d by inserting t h e T J i m i t a t i o n f u n c t i o n in the e q u a t i o n Births = B _ p a r a m " T J i r n i t a t i o n " Population T J i m i t a t i o n s h o u l d be a f u n c t i o n of t e m p e r a t u r e , as d e s c r i b e d above. To use the graphic function in Stella t o d e f i n e it, d o u b l e ciick on t h e T J i m i t a t i o n p a r a m e t e r to o p e n t h e regular dial o g u e box that has t e m p e r a t u r e 'isted as t h e required input. N o t e t h e "To Graphical Function" b u t t o n at t h e b o t t o m left

0

T .limitation

' _ A'IAY

Requ,f«d Input*

(

l ' »

'JL'^JJ

JL 1

(jDCDCDCi) i. 7 ) ( T ) ) ^Tj

G D Q

2j

8u,ltim AFFS

AMP

ASCTAM

APRirMEAM AKRAYSTDOEV ARSAYSUM CAP

CCROOTH

COOK! I ME COS

COSWAVE 0 r .li.TiUt'OI • Craf h o* . lemparatura

T o Ctaphtcal Function

Metsagr

Cancel

If y o u click o n this button, you w i l l see a n o t h e r panel, w h i c h is d e s i g n e d s p e o f c a l i y t o input a g r a p h i c that is to d e f i n e w h a t value this T J i m i t a t i o n p a r a m e t e r is t o return, d e p e n d i n g u p o n t h e value that t h e t e m p e r a t u r e parameter w i l l teed in

72

Systems Scienrf? and Modeling fo' Ecologicst Economics

Giaphical Function j i

1.000

I

T Ii mitacio n

0.000

: •

;

— • —

:

H: 7 | : / \i

i/

0.000

temperature

T limitation

0.OOO 6.000 12.00 18 00 24.00 30.00 36 00 42.00 48 00 54 00 60 00

0.000 0.240 0.465 0.660 0.815 0 90S 0 070 0.995 1.000 1 000 1 000

60 00 Temperature

Data Points

11

Edit Output To Equation

Delete Graph

Cancel

OK

In this case, a f u n c t i o n is d e s i g n e d lhar will change b e t w e e n 0 a n d 1 (like m o s t of t h e limiting functions), w h i l e t h e t e m p e r a t u r e values are anticipated t o b e in t h e range b e t w e e n 0 a n d 60. Here t h e units for t e m p e r a t u r e a.-e d e g r e e s Celsius, while t h e o u i p u t of t h e limiting f u n c t i o n itself is d i m e n s i o n l e s s - it w i l l be a m o d i f i e r for t h e birth rate that will s l o w d o w n g r o w t h t o zero w h e n t e m p e r a t u r e s are low, a n d w i l l have o p t i m a l g r o w t h values at t e m p e r a t u r e s close t o 50°C Perhaps this is g o o d e n o u g h for bacterial g r o w t h . The actual values are t h e n either t y p e d as n u m b e r s in t h e table provided >n t h e dialogue box, or p r o d u c e d w h e n y o u d r a w t h e graphic using your cursor N o w t h e limiting f u n c t i o n s finished w i t h , b u t it is still necessary t o figure out h o w t o provide t h e data for t e m p e r a t u r e Certainly, another graphic can b e p r o d u c e d :

Graphical Function :

40.00

i ime

X

1

Tempers ture

! 0.000

io

0 000

B

«

0.000

» 573 0 0

1000.00 Time

lempetature

0 000 31 25 62 50 93 75 125 00 156 25 187 50 218 7S 250 00 2S1 25 312 SO 343 75 375.00

0.000 4.000 13.40 24.40 34 20 40 00 40.00 35.80 29.00 14.00 4.800 0.600 0.200

Data Poults

33

Edit Output

To Equation

i

Delete Graph

Cancel ) (

OK

Here, a t i m e s e n e s for t e m p e r a t u r e is p r e s e n t e d , using a built-m called T I M E t o d e s c r i b e h o w t e m p e r a t u r e is c h a n g i n g in w i t h t i m e in t h e m o d e l . S o m e real climatic data can b e c o p ied f r o m a n Excel s p r e a d s h e e t or text editor a n d t h e n pasted i n t o t h e table m t h e graphical

The Art of Modeling

73

f u n c t i o n , cr the g r a p h can again be d r a w n using t h e c u r s o r In this case, a series of annual t e m p e r a i u r e c y c l e s is d e f i n e d w h e r e t e m p e r a t u r e varies f r o m 0 to 4 0 ° C over a t i m e per.od of a b o u t 3 6 5 days There ate data for a b o u t 3 years lor 1000 days), w h i c h can be s e e n m fu'.i in t h e graphic if the "ALT" key on the keyboard is p r e s s e d a n d held:

Graphical Function

A

Time 0.000 i l 25 62 50 9?.75 125 00 156.25 187.50 218.75 250 00 281.25 312 50 343.75 375 00

0.000 4.000 13.40 24.40 34.20 40.00 40 0 0 35 80 29 00 14.00 4.800 0.600 0.200

Data Points

33

v

- 375 00 1000.00 Time

Temperature

Edit Output

To Equation

} (

Delete Graph

)

( Cancel

• (

OK

T h i s graphical f u n c t i o n is a n i c e feature, but it has o n e m a j o r d r a w b a c k . If modification o f t h e f u n c t i o n ts required to reflect s o m e newly a c q u i r e d k n o w l e d g e , it is n e c e s sary to g o i n t o t h e graphic and manually redraw it. S u p p o s e t h a t you want to c h a n g e t h e curvature, so t h a t t h e o p t i m a l birth rate is a t t a i n e d faster, or suppose t h a t t h e m a x i m a l birth rate should b e increased - in all t h e s e cases, every t i m e t h e graphic needs to be redrawn. T h i s may b e c o m e q u i t e boring. A s an a l t e r n a t i v e to t h e graphic r e p r e s e n t a t i o n of t h e data, you c a n assume a f u n c t i o n t h a t would g e n e r a t e t h e kind of response that you need. For e x a m p l e , for t h e t e m p e r a t u r e d e p e n d e n c y

shown

above the function ,,

r =

mi ts + r

c a n be used, where m is t h e m a x i m u m birth rate and

repiesents the temperature

at w h i c h t h e birth rate is h a l f t h e m a x i m a l . T h i s f u n c t i o n happens to he k n o w n as t h e M i c h a e l i s - M e n t e n f u n c t i o n , widely used to model growth k i n e t i c s and population d y n a m i c s . N o w t h e r e are two parameters that c a n easily modify t h e s h a p e of t h e f u n c t i o n . By c h a n g i n g in it is possible to raise or lower t h e asymptote, t h e m a x i m a l value to w h i c h t h e f u n c t i o n tends. By m o v i n g t„ t h e f u n c t i o n c a n

be

made to grow faster (smaller t,) or slower (larger t t ). A l l t h e s e m o d i f i c a t i o n s are made without any need to redraw any graphics, but simply by c h a n g i n g a p a r a m e t e r value.

74

Systems Scienrf? and Modeling fo' Ecologicst Economics

Similarly, the tmnesenes tor temperature can be defined as a formula instead of a graph Of course, if you are usmg real climatic data it makes perfect sense t o stick t o it and import it into the graphic function, as described above

However, if t h e timeseries is hypothetical, it

might be easier to have a formula to present it. For example, it is possible to generate the dynamics very similar to the graphic used above by usmg the following equation T e m p e r a t u r e = 2 0 * S I N ( T I M E / 3 6 5 * 2 * PI + 3 " PI/2) + 2 0 It does take s o m e effort to figure out the right amplitude and phase for the SIN function; however, even w i t h s o m e very basic knowledge of trigonometry and a f e w trial-and-e"or runs m Stella, this can be done It can even be made m o r e realistic if s o m e random fluctuations in temperature are added, usmg the R A N D O M function - another built-in in Stella (just like SIN. Pi or TIME - all these can be found if you scroll d o w n the list of Stella built-ins that appear in any parameter of f l o w dialogue box) T e m p e r a t u r e = 20 * S I N l T I M E / 3 6 5 * 2 * PI + 3 * PI/2) + 2 4 + R A N D O M i - 4 . 4 , 0 ) The output from this m o d e l looks like this.

40000 00.

7000

It may he hard quickly t o find the right function to represent the kind of response that you have in mind tor a particular process. It helps a lot if you know the behavior of some basic m a t h e m a t i c a l functions; then you c a n put togerher t h e righr response curves hy c o m b i n i n g certain functions, t h e behavior o f which you know. Figure 2 . 2 0 c o n t a i n s a c o l l e c t i o n o f some useful functions that may be used as building blocks to describe various processes. N o t e chat for t h e numerical realization of the model you will need to provide actual values for all the parameters that are used in the functions. Therefore, the fewer parameters a function uses, the easier it will he to find all the values needed. It also helps a lot if the parameters used to describe a function have an ecological meaning. In that case, you c a n always think of an experiment to measure their value. For example, in (fie temperature function considered above, m c a n be measured as the birth rate at optimal conditions, when temperature is not limiting. Similarly, t, can he estimated as the temperature value at which birth is approximately equal to half t h e maximal. B o t h these parameters c a n be measured and c a n be then used in the model. T h i s is o n e o f t h e basic differences between process-based and empirical

The linear (unction is proQably the simplest and computationally the most ellicienl one. You can combine several linear functions with "if... then" conditions to describe more complex behavior The disadvantage of sucn piecewtse linear description is the lack ol smoothness, which may sometimes result in model crashes if the time step DT is too large a - inclination of lhe line, b - the offset

Michaehs-Menten lunclion. Widely used in enzyme kinetics Also Known as Monod function in population dynamics. A very useful lunclion lo describe growth with saluration At low concentrations ol substrate x, il limits growth, growth is proportional lo lhe availability ol the suestrate. At very high concentrations of substrate growth lends to a maximum value and does not exceed it. a - maximum growth rate, oelines the saluration level, b - hall-saturation coefficient U « a/2, when x - b.

The s-shaped function is a modification of the Michaelis-Menten lunclion By increasing s y o u can make the lunclion sleeper, decreasing :he transition period from low growth rale lo salutation Also important that lhe furiciion approaches zero with a zero derivative. Makes compulations more stable in the vicinity ol zero. a - maximum growth rale, defines tne saturation level; b - half-saturation coefficient U a/2, when x = b bs+xs

Variations ol the hyperbolic function. Used to describe processes ihal are very last at low values of lhe controlling variable and Ihen rapidly decrease to a constant saturation level. This can be. say. lhe dependence of lish mortality or oxygen concentrations in water Al anoxic conditions the mortality sharply increases a - controls steepness of decline; b - odsei U - a/* + b

Figure 2.20

A list of formulas to facilitate your choice of the mathematical expressions that can

properly describe the processes in your model. Certainly there are many others that you might find mote appropriate for yoi/r particular needs. To check out how a function performs with different parameters and to choose the parameters that will best suit your particular needs, you can input the function into, say, a spreadsheet program such as f x c e l , and build graphs with various parameters. Another application that is especially useful for these purposes is the Graphing Calculator It comes as p a n of the Mac OS-X, and probably there are also versions for W i n d o w s It should be noted that in most cases using a formula with parameters is preferred, rather than inserting a graphic, to describe the process in your model. One significant advantage is that certain parameters that change the form of the curve can be easily used to study the sensitivity of your model to these sorts of changes Similarly, changing graphics is a much more tedious job and more difficult to interpret.

76

Systems Scienrf? and Modeling fo' Ecologicst E c o n o m i c s

The exponent should be used wiin caution because il always tends to grow too fast. It is useful only if lhe maximum values for x are well delined and you can be sure that thev will not be exceeded a - offset; b - steepness

Variations for the parabolic lunclion Especially useful with c < 1. Otherwise it is very much like 'he exponent - grows loo last and lends ic gel out of control When c < < 1 il seems to reach a saturation level, but actually it still continues to grow but very slowly. a - offsel at zero 0 - controls behavior at larger x values whereas s controls Denavior at lower x. U = a I bx'

Ivtev function. Also used to describe growth with saturation. The saturation level can be controlled by an additional parameter that multiplies lhe whole lunclion Michaeiis-Menten (unction does practically the same, but is simplar computationally a - controls steepness U = 1 - exp(-ax)

Steel function. A bell-shaped function that is useful lo describe processes inhibited bolh al ow and h.gh values of the controlling (actor Has been originally designed to limit phytoplanKton growth by lighi a - modifies the maximum rale as well as lhe rale ol latl at inhibiting values, b - defines the optimal values ol lhe controlling factor U - ax exp(1 - bx) c•5 c•4 c -4 c-4

{

Universal bell function A more complicated formulation lor the bell-Shaped function Olten used 10 describe temperature limitation Disadvantage - many more parameters lo define Advantage - much more flexibility and conlrol over behavior Can describe pretty much any bell luncton torm abcs-

a (1-x/C)'

Figure 2.20

a • 2 b • 0 05 a • 0 5. b - 0 2 a o 5 b - 0 002 a-3.b-04

jf

x <

c

if X 2 c

(Continued)

value at zero, value taken when x = d, optimal value, where lhe function is maximal; controls the steepness, when s > i the range ol oplimality can be made really big.

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3. Essential M a t h 3.1

Time

3.2

Space

3.3

Structure

3.4

Building blocks

The full set of dynamic equations of a given physical system presented in one of the approximate forms, along with the corresponding boundary conditions and with the algorithm for the numerical solution of these equations - inevitably containing means from a finite-difference approximation of the continuous fields describing the system - form a physico-mathematical model of the system. A.S. Monin

SUMMARY M a n y models are based on some mathematical formalism. In some cases it may be quite elaborate and complex; in many others it is straightforward enough and does not require more than some basic high-school math skills to understand it. In all cases it can help a lot if you know what the mathematics are that stand behind the model that you build or use. Most of the systems dynamics models that we use in this book are based on ordinary differential o f difference equations. S o m e basics o f those are explained in this chapter. W e look at how models can tend to equilibrium conditions, and explore how these equilibria can be tested for stability. If the spatial dimension is added, we may end up with equations in partial derivatives. W e will see how the advection and diffusion processes can be formalized. Finally, in the structural domain we may also find models that will be structurally robust and stable. S u c h models are preferable, especially when there is much uncertainty about model parameters and processes.

Keywords Discrete vs continuous, initial conditions, ordinary differential equations, state variables, difference equations, e x p o n e n t i a l growth, time-step, numerical method, Euler method, R u n g e - K u t t a

method, equilibrium, stable or unstable equilibrium,

box

models, compart mental models, continuous models, advection, diffusion, equations in partial derivatives, rigid and soft systems, structural stability, * * •*

79

80

S y s t e m s Science and M o d e l i n g for Ecological Economics Modeling

and

systems

analysis

first

appeared

as

mathematical

disciplines.

R e c i p r o c a l l y , m u c h of m o d e r n m a t h e m a t i c s has originated from models in physics. U n t i l recently, a solid m a t h e m a t i c a l b a c k g r o u n d was a prerequisite o f any m o d e l i n g effort. T h e a d v e n t o f c o m p u t e r s and user-friendly m o d e l i n g software has created the feeling that m a t h e m a t i c a l k n o w l e d g e is n o longer n e e d e d t o build realistic models, e v e n for c o m p l e x d y n a m i c systems. U n f o r t u n a t e l y , this illusion results in m a n y cases in faulty models that e i t h e r misrepresent the reality e n t i r e l y o r represent it o n l y in a very narrow d o m a i n of p a r a m e t e r s and forcing f u n c t i o n s , while t h e c o n c l u s i o n s 3nd p r e d i c t i o n s t h a t are made are most likely to be presented as b e i n g quite g e n e r a l and long lasting. T h i s does not m e a n t h a t m o d e l i n g c a n n o t be d o n e unless you h a v e a P h D in m a t h e m a t i c s or e n g i n e e r i n g . M a n y o f t h e software packages t h a t are currently availa b l e c a n indeed help a lot in t h e m o d e l i n g process. T h e y c a n c e r t a i n l y e l i m i n a t e most of t h e p r o g r a m m i n g work needed. It is i m p o r t a n t , however, for t h e m o d e l e r to k n o w and understand l h e m a j o r m a t h e m a t i c a l p r i n c i p l e s t h a t are used within t h e framework o f t h o s e packages, otherwise t h e m o d e l s will be prone to error. D a v i d Berlin-,Li offers s o m e n o t e w o r t h y e x a m p l e s of how models c a n be misused, misunderstood, and in error w h e n t h e m a t h e m a t i c s is ignored.

Let us take another look at the m o d e l that w e have d e v e l o p e d in Stella O p e n t h e m o d e i and click on the little a r r o w p o i n t i n g d o w n w a r d s in t h e uppet let: corner. (In m o r e r e c e n t v e r s i o n s of Stella t h e interface has b e e n c h a n g e d and y o u have separate t a b s o n t h e left of t h e w i n d o w for t h e interface, t h e .model a n d t h e equations.)

Stella.prlmer2.stm

O -o» O \ O

G

'— H pararr

O H

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c>

S paam

. SrJ-6

•TV T kmrialon

a i 3

i b c / A

f —

y - - Dears PopUaKyi

- a ; Temper awe

W h a t you g e t is a list of e q u a t i o n s

© O ©

Stelli.primer2.stm

W) C Population(t) = Population^ - dt) - (Birtns • Deatns)* at iNiT Population = 10 INFLOWS. A Births - B param'T limitation'Population OUTFLOWS: Deaths = M_param'Popuiation O B_param - 0 1 O M_patam = 0 05 Sjparam = 1 2 Tempeiature = 20*SlN(TIME/365-2'Pi*rPi/2)+2d*RANOOM(-d <".0) ' T_limitation = rempeialure/(S_param-Temperature)

•

- - pararr M

mmmmmmmmmmmm Essential M a t h

•MB

81

So these will be the actual equations that the model will be solving The Graphic User Interface has reaily |ust o n e ourpose

to formulate these equations and then display the

results of solving t h e m W h a t are these equations, and h o w d o they work?

As we have seen in t h e previous chapter, a system may be considered in three dimensions: temporal,

spatial and structural. In the temporal dimension we decide how t h e

system evolves in time; in the spatial dimension we research t h e spatial organization o f t h e system; and in the structural dimension we define the complexity o f the model. For each o f these facets m a t h e m a t i c s are used tn modeling.

3.1

Time Most computer models operate in discrete time. T h e time is represented as a sequence o f snapshots, or states, which change momentarily every given time interval. T h e major question to be answered when considering this temporal evolution ol a system is, what wtll be its state at time t if its state is known at t h e previous time t - l ' I f we know how the system changes state, then we c a n describe its dynamics o n c e we know the initial state o f the system. Suppose we have a population ot five cells and each cell divides into two over o n e time-step - say, l hour. T h e n after l hour we will have 10 cells, since each cell is t o be replaced by two; after 2 hours there will be 2 0 cells; after 3 hours there will be 4 0 cells, and so on T h i s ts a verbal model of a system

Let us formalize it oi describe it in math-

ematical terms. Let x ( n ) he t h e number o f cells at time-step n = 1 , 2 , . .

T h e n the

doubling process can lie described by x(n + I) = 2x<«)

(3.1)

If we provide t h e initial condition .\(0) = a, we c a n calculate t h e number of cells after any n time-steps: \{n) = a 2 " T h i s is a simple model of exponential

growth.

(3.2)

T h e n i c e thing about t h e mathematical

formalism is that it provides us with a general solution

Instead ot doing iterative

(i.e. repeating) calculations to find out the number ot cclls after, say, 1 0 0 divisions, and redoing these calculations if instead ot five initial cells we were to consider six o f them, we c a n immediately provide the result based on the general solution ( 3 2 ) . However, this model c a n only describe systems

How did we get from (3 1) to (3.2) P Certainly,

if

xfn + 1) = 2x(n),

then

similarly x(n) = 2x{n - 1) and rtn - 1) = 2Mn - 2). Substituting, we get: x(n) 2Mn - 1) = 2 • 2Mn - 2), and so on, xtn) = 2 • 2 .. 2x<0). Keeping in mind that MO) = a and that 2 - 2 .. 2(n times) = T, we get the result in (3.2)

that are very well synchronized in time, where all the cells divide simultaneously and similarly. T h i s is quite rare for real populations, where divisions occur all t h e time, and therefore rhe process is n o t so discrete. In this case it makes sense to assume a different model that we formulate in terms o f growth rate. Suppose that cach cell produces o n e new cell o n c e an hour. T h i s is more-or-less equivalent to t h e above model, but now we can remove

82

Systems Scienrf? and Modeling fo' Ecologicst Economics

the synchronization. In two hours o n e cell will produce two cells, and in half-anhour a cell will produce half o f a new cell. T h i s may n o t make sense for an individual cell, but with n o synchronization this makes perfect sense for a population of, say, 100 cells. It simply means that in half-an-hour 5 0 new cells will be produced. W e can then reformulate ( 3 . 1 ) as: x{t + A t ) = x(i) + x(t)-At

(3.3)

W e have substituted t for n to show that time no longer needs to change in integer steps. At is the time increment in t h e model. N o t e that if At = 1, model ( 3 . 3 ) is identical to model ( 3 1 ) : x(t + l ) = x ( t ) + * ( t ) - l - 2 x ( t )

(3.4)

However, if we run the same model with A; = 0 . 5 , we get a different result: x(i + 0 . 5 ) = x(t) + x ( t ) - 0 . 5 = 1.5x(t) T h e n similarly x(t + 1) = x(i -t- 0 . 5 ) + x(t + 0 . 5 ) - 0 . 5 = 1.5x(t + 0 . 5 ) . Substituting from t h e above, we get: x(i + I) = 1.5 - 1 . 5 x ( 0 = 2 . 2 5 x ( t ) , which is different from what we had for At = I in ( 3 . 4 ) . W e see that when we c h a n g e t h e time-step A;, we get quite different results (see Figure 3 . 1 ) . T h e more often we update t h e population of cells, the smaller t h e time-step in the model, the faster the population grows. S i n c e new growth is based on t h e existing number of cells, t h e more often we update the population number, the more cells we get to c o n tribute to further growth. Indeed, let us take a closer look at Figure 3.1 and zoom in on t h e first two steps (Figure 3 . 2 ) . W e start with a certain initial c o n d i t i o n - say, 2. W e decide to run the model with a certain time-step - say, At = 1. A c c o r d i n g t o ( 3 . 3 ) , rhe next value at x( I ) - x ( 0 ) + x ( 0 ) • At = 2 + 2 • 1 = 4- A t time 0 we define ( 3 . 3 ) for the first timestep, and we know that during this period o f time nothing is supposed to c h a n g e in the equation. O n l y when we get to the next point in time do we re-evaluate the variables in ( 3 . 3 ) . Now we change the equation and diverge from t h e straight bold line that we o n c e followed: x ( 2 ) = x ( l ) + x( 1) • At = 4 + 4 • 1 = 8 . If we had chosen to run the model with a time-step At = 2, then we would have stayed on the course longer (the broken line in Figure 3 . 2 ) , and then x ( 2 ) = x ( 0 ) + x ( 0 ) • At = 2 + 2 • 2 = 6. S e e the difference? Alternatively, if we had chosen t o run the model with a time-step o f At = 0 . 5 , we would have corrected the trajectory already after the first half-step taken ( t h e dashed line in Figure 3 . 2 ) , and then later o n every half-step we would have been correcting the course. A s a result, by the time we got to time 2 we would have got to a different value, a substantially different o n e compared with the case with At = 2. So the smaller the time-step we use, the more often we correct rhe course,

the less the comfyuiarional

error.

N o t e that equation ( 3 . 3 ) is similar to t h e kind o f equations that we saw previously, generated in Stella. R e m e m b e r when we were clicking t h e little triangle in

Essential Math

R F L B

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Essential Math

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to t h e model formalization: 2 . 7 2 in ( 3 . 7 ) is not t h a t m u c h larger t h a n 2 in ( 3 . 2 ) However, t h e e x p o n e n t t h a t is used in t h e m o d e l blows up this difference t r e m e n dously. T h e t i m e - s t e p used in t h e model b e c o m e s a crucial factor. T h i s is s o m e t h i n g t o r e m e m b e r : models time-step

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used.

So w h a t d o e s n i s m e a n in t e r m s of our little Stella m o d e l ? A s w e have s e e n above, t n e r e is a at in t i e e q u a t i o n s tile. S o n o w w e k n o w w h a t it is all a b o u t There is also a w a y t o change this dt. Click o n t h e "Run"' menu., a n d t h e n c h o o s e " R u n Specs." This w i l l o p e n a dialogue box that c o n t a i n s t h e t i m e specifications tor your m o d e l tun. " F r o m " w i l l specify at w h a t t i m e y o u start t h e s i m u l a t i o n , " T o " tells w h e n to end. " D T " is t n e t i m e - s t e p t o u s e in t h e simulation.

Run Specs Length of simulation: From: 0 To: 100

Unit of time:

0

3 Days

3 Cycle-time

Q Weeks Q Months

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Run Mode.

Q Hours

C Quarters

Normal

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Integration Method: @ Euler's Method Runge-Kutta 2

Sim Speed: 0

real sees - 1 unit time

Min run length: 0 sees

Q Runge-Kutta 4 Analyze Mode stores run results in memory ( 0.0 MB required ) ( Cancel i f

Let us start w i t h DT = 0 25 a n d r u n t h e m o d e l like this

OK >

The result s h o u i d 100k s o m e t h i n g

86

Systems Scienrf? and Modeling fo' Ecologicst Economics

If w e n o w change the time-step t o DT - 2 w e will g e l a similar picture, but not exactly the same

Notice that w i l h a larger t m e - s t e p w e see that the temperature changes less frequently, and. besides, the population g r o w s t o a smai'er size. See t h e scale on t h e Population axis: it has changed f r o m 4 0 0 m a x i m u m t o 3 0 0 Something w e could expect' iust as in Figure 3 2, w e see that t h e g r o w t h is slower w h e n the time-step is larger, t h e variables are updated less frequently, and therefore l h e g r o w i h base .s smaller m each time-step

Equation ( 3 . 3 ) in a more general form is x ( t ) = xU-dc)

+ f(t,x(c),,i)dt

(3.8)

where /(t, x ( i ) . a ) is the transition function that describes how the system changes at time (. It depends upon the current state of the system x(r), and a vector of parameters a — (a|,();,... . a j . These parameters do not change over time. Sometimes we assume that the parameters are hidden and write simply /(t, x). As a differential equation, this will be:

— dt

=

r,

){t,x,a)

Differential equations are very useful in formulating various dynamic models. T h e left-hand side dx/dt

is t h e instantaneous c h a n g e in t h e size o f variable x. O n t h e

right-hand side, we c a n specify what t h e processes are that c o n t r i b u t e to this c h a n g e . In t h e example

above we h a v e a very simple transition function:

f(i, x(i), a) = x(t).

In real models, the function c a n be quite complex. S o m e o f t h e more simple differential equations c a n be solved analytically. However, o n c e we start putting together realistic models o f systems, very quickly we arrive at equations that are too c o m p l e x for an analytical solution. T h e s e equations are then solved numerically, using a numerical method on a computer. T h e simplest numerical method is given by t h e approximation that is used in ( 3 . 8 ) . T h e equation in ( 3 . 8 ) is called a difference

equation,

and it is a numerical approximation of a dif-

ferential equation As we have seen above, such difference equations are discrete and can be solved o n a computer by going through all t h e time-steps starting from r h e initial condition. T h e equation in ( 3 . 8 ) is also called t h e Euler method, which is t h e

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Essential Math

89

T h e m o r e c o m p l i c a t e d a n d most o f t e n used version of t h e R u n g e - K u t t a algor i t h m uses a weighted average of several a p p r o x i m a t e d values o f /((. x) within t h e interval (t, t + dt). T h e formula k n o w n as t h e "fourth order R u n g e - K u t t a f o r m u l a " is given by

x(c + dt) = x(t) +

I-

(fc, + 2 k , + 2 k , + k j

[6

where

k, =/(t,x(0),

'<:=/

dt , v t + — , x(t) +

dt

r +f,Kr) +

?

k4 = f(t + dt.x(t)

dt i

+ dt ><})

T o run t h e s i m u l a t i o n , we also start with t h e initial c o n d i t i o n , at r^, .Xp = x(to) a n d hnd x ( = In our population model, if w e choose the R u n g e - K u t t a 4 m e t h o d w e g e t t h e population of exactly 1,484 after t h e 100 days of a run. That is a perfect m a t c h w i t h t h e analytic solution. Quite outstanding!

x(£fs + dt) using t h e formula a b o v e . T h e n we plug in X| to h n d x> = x(t| + dt) = x ( r 0 + 2dt), and so o n . O n c e again we pay a price for t h e improved a c c u r a c y o f c a l c u l a t i o n s : n o w we h a v e t o c a l c u l a t e t h e t r a n s i t i o n f u n c t i o n four times. T h e R u n g e - K u t t a a l g o r i t h m is k n o w n t o be very a c c u r a t e and b e h a v e s well for a wide range o f problems.

However, like al

numerical

methods,

it is n e v e r perfect and t h e r e are models w h e r e it fails. O n e universal rule is rhat t h e smaller t h e t i m e - s t e p , n o m a t t e r what m e t h o d we use, t h e b e t t e r t h e a c c u r a c y o f rhe s i m u l a t i o n . Tn ensure that you are getting right result

wit)\ your

numerical method, you may want

unni %'ou do not see any difference

m the results

to keep

decreasing

the

the

time-step

that you arc generating. T h e r e are s o m e

a d a p t i v e step-size a l g o r i t h m s that d o e x a c t l y t h a t automatically. O t h e r a l g o r i t h m s are also a v a i l a b l e , such as t h e A d a m s m e t h o d or B u l i r s c h - S t o e r o r p r e d i c t o r - c o r r e c tor m e t h o d s , t h a t c a n be way more ellicient for some problems, especially when very high accuracy is essential. Just r e m e m b e r t h a t there is always a price to pay for h i g h e r accuracy. T h e smaller t h e t i m e - s t e p , t h e longer it t a k e s t o run t h e m o d e l . T h e more a c c u r a t e t h e m e t h o d , t h e longer it takes t o run t h e m o d e l . However, s o m e t i m e s o n e m e t h o d is simply b e t t e r for a particular type of a m o d e l - it runs faster and gives b e t ter accuracy. S o it always makes sense t o try a few m e t h o d s o n your m o d e l a n d see w h i c h o n e works best. W e have already seen that i h e size o f t h e t i m e - s t e p c h o s e n for t h e n u m e r i c a l solution o f t h e model c a n significantly c h a n g e t h e o u t p u t produced. L e t us c o n s i d e r a n o t h e r e x a m p l e . S u p p o s e we are m o d e l i n g a s t o c k of s o m e s u b s t a n c e t h a t is accum u l a t e d due to a flow c o m i n g in and is depleted by an outflow:

Stuff pj

o In

X) Out

90

Systems Scienrf? and Modeling fo' Ecologicst Economics T h e e q u a t i o n s for this m o d e l will be-

I

I StuM(r) = Slult(: - dt)* (in - Out) * dt INIT Stuff - 0 INFLOWS: In * 1 OUTFLOWS Out ^ GPAPH(Slu«) | - (0 00. 0.09}. (0 1,0.63), (02, 106). (0.3. t 321.(0.4 1 47). (0.5 1.561,(0.6,1.621, Y (0-7 1-67). (0 8, 1.72), (0 9. 1.75). (1, 1 77)

T h e inflow is c o n s t a n t , whereas rhe outflow is ,1 f u n c t i o n of t h e s u b s t a n c e accum u l a t e d . It may be described by a simple graphic f u n c t i o n o f t h e form:

2.000

Out

Stuff 0.000 0.100 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000

0.090 0.630 1.060 1.320 1.470 1.560 1.620 1.670 1.720 1.750 1.770

Data Points:

11

0.200 Out

h 0.000

«• 1.000

0.000

Stuff

If we first run t h e m o d e ! with D T — 1 using t h e Euler m e t h o d , we get a very b u m p y rtde, and a n o s c i l l a t i n g trajectory:

Essential M a t h T h e very .same model but with a smallei time-step cif D T = 0 25

91 produces

entirely different dynamics:

I

t Stu 1< 030i

I ••• •

IV. 0

—t.

IS-

000000

—I— 300

—I— 600

900

1

1200

Time

Finally, it we switch to the R u n g e - K u t r a , fourth-order m e t h o d , we get very smooth behavior, with rhe trajectory reaching saturation level and staying there:

1 Stuff

TI M E

T h e very same model produces entirely different dynamics by simply changing the time-step assumed. Clearly you do not want to run your model with too small a DT, since ir will require more computational tune and may b e c o m e more difficult to analyse properly. However, too large a D T ts aiso inappropriate, since rhe results you produce may be entirely wrong. Here is yet another example that shows that D T matters and that ir is always important t o remember the equations that are solved to run your model Q u i t e often in models we want to do something to the entire amount stored in o n e of the variables. For example, ar certain times we need to deplete a reservoir, and then stair tilling it all over again. Or we may be looking ar an age-structured population, when

108 Systems Scienrf? and Modeling fo' Ecologicst Economics after r e a c h i n g a c e r t a i n age t h e e n t i r e population m o v e s from o n e stage (say, eggs) to a n o t h e r stage (say, c h i c k s ) . Let us c o n s i d e r a simple model of a flush tank t h a t we all use several times a day. W a t e r flows i n t o t h e t a n k at a c o n s t a n t F l o w _ R a t e = R , w h i c h is altered by a floater a t t a c h e d to a v a l v e . A s t h e water level tends to a m a x i m u m , t h e valve shuts t h e flow o f water off. K n o w i n g t h e v o l u m e o f t h e t a n k T a n k _ C a p a c i t y = V, we c a n describe t h e inflow as

1-1

F,„ = R

V

w h e r e T is t h e c u r r e n t v o l u m e o f water in t h e tank. T h e outflow is such t h a t every n o w a n d t h e n s o m e b o d y o p e n s t h e gate and all t h e available v o l u m e o f w a t e r is flushed o u t . T o describe t h e outflow, let us assume that U s e = u is a r a n d o m value b e t w e e n [0, l j , a n d let us define Flush as:

F... -

%

*

u < 0.99

J

otherwise I

If we just put these e q u a t i o n s i n t o S t e l l a , we will g e t this m o d e l :

Tank

©

X

O-0r

Flow Rate

Flush

Inflow---^^

- - o

Use

Tank Capacity

It c a n b e d o w n l o a d e d from t h e book websiteU s i n g t h e Euler m e t h o d and di=

1

1, we will g e t :

1: TV* 14.00

11 ;

\ 1

1 V

7.00

1 1

moo cnprfl 1 thtOffl

r

f

W O 00

17:0' 10/KK>«

Essential Math

93

w h i c h looks e x a c t l y h o w we wanted. Every n o w a n d t h e n t h e tank is e m p t i e d , arid t h e n it is gradually rehlled. S u p p o s e now t h a t instead o f tir -

I w e wish to get a m o r e

a c c u r a t e solution a n d M a k e cit = 0 . 5 , Now, t h e output looks quite d i f f e r e n t :

O B /

^^1

1

N o t quite as e x p e c t e d

=

li: IT lorfw

T h e tank docs n o t gee e m p t i e d any m o r e . W h a t has h a p -

pened? L e t us look at t h e S t e l l a e q u a t i o n s for this m o d e l :

i j TanKf/] = Tann{r - an + (Inltow - Flush) - dl I NIT Tank = 0 INFLOWS Inflow - Flow_Aate*(1-Tank/TanK_Capacily) OJTF.OWS:

Hush . |FUse=0.999 THEN Tank ELSE 0 O Ficw"_Rate = 0.1 c Tan* .Capacity - 12 o Use • RANDOM(O.l.H)

It is c l e a r t h a t , c o n t r a r y to what we i n t e n d e d , t h e outflow is n o t T, hut T • dt. T h a t ts why dt started to modify t h e model output so dramatically.

It should he

r e m e m b e r e d t h a t w h e n e v e r a flow is described in S t e l l a or a n o t h e r similar package, it is t h e n multiplied by dt when it is inserted i n t o t h e real e q u a t i o n s to he solved Therefore,

if it is actually t h e e n t i r e s t o c k that you want t o m o v e , you should

d e s c r i b e the flow as T/dt, T h e n w h e n it is inserted i n t o t h e e q u a t i o n s , the dl gets c a n c e l l e d nut a n d we c a n really flux the e n t i r e a m o u n t as it was i n t e n d e d T h e r e f o r e , t h e c o r r e c t S t e l l a e q u a t i o n s should b e : I

I TanKffl = Tankjt - at) + (Intlow - Flush)' d< IN IT Tank = 0

INFLOWS: Inflow = Flow_Hale't1-Tank/Tank. ..Capacity) OUTFLOWS. <7y> FkiSh = IF U$fi>0.995 THEN',TanK/rf£IELSE 0 \ O Fl0w_ Rate = 0.9 j a TMK_Cap«iiy-i8 O Use = RANDOM(Q 1 11)

Note

this

change!

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undue «nvcoinpiJt»tKinai f ^ i r o a »'
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CMK/I

Essential M a t h

95

In this case t h e left-hand side is t h e variation in your a c c o u n t , measured in $ per m o n t h . W e m a k e sure that t h e flows o n t h e r i g h t - h a n d side are presented in .similar units. F o r e x a m p l e , it is i m p o r t a n t to r e m e m b e r t h a t t h e interest rate l< is m o n t h l y , a n d should t h e r e f o r e be recalculated from t h e m o r e frequently used a n n u a l interest rate. W h e n dx/dt

~ 0 , there ts n o c h a n g e in t h e variable. If t h e inflows and outflows

are b a l a n c e d , t h e variable is a t e q u i l i b r i u m , it does n o t c h a n g e because n o t h i n g is added t o tt a n d n o t h i n g ts t a k e n away

By setting dx/dt

~ 0 , we c a n c a l c u l a t e t h e

equilibrium c o n d i t i o n s in o u r m o d e l . From ( 3 . 9 ) we get:

kx + p - q = 0 ,

(3.10) k

If you m a k e {q - p)/k

your initial c o n d i t i o n : x ( D ) = (q — p)/k,

there will

never

be a n y increase or d e c r e a s e in t h e value ot t h e v a r i a b l e ; your a c c o u n t will remain u n c h a n g e d . A n i c e guideline t o b a l a n c e your a c c o u n t ! However, w h a t will h a p p e n if your initial c o n d i t i o n is slightly larger or smaller t h a n t h e equilibrium ( 3 . 1 0 ) ? In m o d e l ( 3 . 9 ) if we a r e e v e n slightly below t h e e q u i l i b r i u m : .v < (q — p)/k t h e n dxldt

< 0 . T h e d e r i v a t i v e is n e g a t i v e w h e n t h e f u n c t i o n is decreasing. T h e r e f o r e ,

lor values less t h a n t h e equilibrium e q u a t i o n , ( 3 . 9 ) takes us further away from it and we will be g e t t i n g decreasing values for t h e a c c o u n t (Figure 3 . 3 ) . Similarly, if we start e v e n slightly a b o v e t h e e q u i l i b r i u m , t h e n x > (q — p)lk a n d dx/dt

> 0. Now

t h e d e r i v a t i v e is positive, so t h e f u n c t i o n grows, a n d t h e r e f o r e again we start m o v ing away from t h e equilibrium. T h e f a r t h e r we m o v e away from t h e e q u i l i b r i u m , t h e larger dx/dt gets, t h e f a r t h e r it takes us away from t h e equilibrium. T h i s positive feedb a c k sets us o n a p a t h o f e x p o n e n t i a l g t o w t h . T h e equilibrium state is unstable. It you t a k e o n e step away from t h e e q u i l i b r i u m , e v e n a very small o n e , you will slide further awav from it. S m a l l d e v i a t i o n s from t h e equilibrium will o n l y increase with t i m e . H e r e is a n o t h e r e x a m p l e . S u p p o s e a p o p u l a t i o n o f w o o i l e s lives o n a small island t h a t h a s e n o u g h grass to support o n l y A woorles

Figure 3.3

If t h e r e a r e more w o o : l e s t h a n

Unstable equilibrium. Small displacements from steady slate result in increasing divergence from it

96

Systems Scienrf? and Modeling fo' Ecologicst Economics 1500.

1000.

500

1.

0.

60.

30.

90.

120.

Stable equilibrium. When system is perturbed Irom steady slate A - 1000, it returns to it.

A , t h e y die o f f front hunger. W e c a n use t h e f o l l o w i n g formalism t o d e s c r i b e this system: x dt

(3.11)

A

Mere, k is t h e g r o w t h rate of woorles a n d A is t h e carrying c a p a c i t y o f t h e island. A s x a p p r o a c h e s A , t h e multiplier ( 1 •- x/A) e f f e c t i v e l y slows down t h e growth rate dxf dt., m a k i n g it zero w h e n x = A . If s o m e h o w x b e c o m e s larger t h a n A , this s a m e m u l tiplier makes t h e growth rate n e g a t i v e , providing t h a t t h e p o p u l a t i o n size decreases until it reaches t h e size x = A (Figure 3 . 4 ) . W e s e e t h a t x = A is a n equilibrium p o i n t . T h e r e is yet a n o t h e r equilibrium p o i n t , w h e t e dx/dt

= 0 . T h i s is x = 0 . From Figure 3 4 . we readily see t h a t w h e n 0 < x < A

the d e r i v a t i v e ts positive and t h e r e f o r e x grows. It x could be n e g a t i v e ( n o t t h e c a s e in our system, b u t ir. could be if t h e same formalism was used for a different system), t h e n dx/dt < 0 and t h e r e f o r e x furthet decreases, t e n d i n g t o —

T h e equilibrium x = 0 is

clearly unstable. O n t h e contrary, as we c a n s e e w h e n t h e system is perturbed from t h e equilibrium x = A , t h e sign of t h e d e r i v a t i v e is o p p o s i t e t o t h e sign o f t h e perturb a t i o n ( n e g a t i v e f e e d b a c k ) a n d t h e system is returned t o equilibrium. T h i s equilibrium is s t a b l e . A classic illustration for t h e different types o f equilibrium is t h e m o v e m e n t of a ball put: in a c o n v e x bowl or o n t h e s a m e bowl turned upside d o w n (Figute .3.5). E v e n if you m a n a g e to b a l a n c e it o n t o p of a r.urned-over bowl, t h e slightest dist u r b a n c e f i o m that state o f equilibrium will allow t h e f o t c e of gravity to move t h e ball further away. You d o n o t e v e n need t o b a l a n c e a ball inside a bowl; it will find its way t o t h e p o i n t o f equilibrium by itself. T h e third, s o - c a l l e d neutral type of e q u i l i b r i u m h a p p e n s w h e n t h e ball is p l a c e d o n a flat surface. In this case, perturb a t i o n s from t h e state o f e q u i l i b r i u m d o n o t cause a n y further m o v e m e n t o f t h e ball (Figure 3 . 6 ) . A n a l y s i s o f t h e equilibrium a n d its stability may prove t o be e x t r e m e l y import a n t for u n d e r s t a n d i n g model behavior. In some cases t h e m o d e l produces t r a j e c t o r i e s t h a t seem t o c o n v e r g e t o a c e r t a i n state, n o m a t t e r what c h a n g e s are made to m o d e l parameters. In t h a t case, c h a n c e s are t h a t r.he t r a j e c t o r y is at equilibrium a n d t h e r e

A Tfcree rvoej

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Exercise 3.3 1. Consider a population ol wizzlas v.*nch. unlike /vocz)i?s. can rr>.u-t iiiv y wiier: -r.*oeis are laig«K tr^n * car^m mimr-wi vaiv»* & it tfcarn smi^ii wiuies rim;. <> they :\«nioi find a partner for ma:irg and :he populstion dies o^'f IK.-, Cc-iymp caoaciry o* ine .si^nc v ^ i e wizales live is also a Odiously. A> a Ww* .••<>;««' vvou'd c«sc,;oq trie oasuiat'o* dynamics of vrfzztes on this isiand? 2, Afu thcic s ty cqw OiH in the

m o d e l ' Are r^ey vonl©.'

98

Systems S c e n c e and Modeling for Etiological Economics

The importance of initial conditions ... car, be easi'v seen irorv these e A S ^ i e s almost inceoenaec.t o ' f ^ e ir-'iie. c l o n e s

Ir so-n© models, -^e ctricyme o* s i r i t w n i o n 'roeec.

r e '--ode w i ' i e x p c n e r t s qrowtn.

lor example. no n a t t e r w h e - e w e s t * " w e ve-v

y e t ' - t c v w y ve-y 0
us say w e have lOl) mdrvirtuals that ric*;fii« rtc^caiino to the e*p.onftnval g i o w t h n-onfti A n * : 7?0C.'jjanv-.s m the population i l w e s ' 3 * w«i-i

25 gcnerat'c^^s. there will

10 percent m o r e indivxJuats a: -.n
J

>. ' • / csc^r.5™$

>0?C b : i i

nnc-r

wr.»cn a lior-etn.'.g w e z o s o e w w e t cea'-^cj •"> :r.- tc^n:1a m a - «•<•• 3ftr.vfr.-j abnv*

f -

tune Mywpver, -Alien yxxj ye: 1 •«;v the w r y . very I x j

numhera these 10 perceni d o x n see"'-. i c

as

c~ tn« : w o 'DJi.i's

s^i; v e

look very sim.iar you st
g r i i / ^ b e '

.

Similarly. tn the I .Jul ftxjtmplfr :!vit hrtt. .? ••ta.'ilu <}<^I'II!:M';-II ?i;r V'F )ii:i:.;.-••«, - o -v-atie-' w - e " e w e start T r * 1,1.31 conditions can b e 10 or 'COD

oH. o n w o . v w o y<>

(*-<•: s a ^ o o ^ - ' i o ' - p o " '

The s y s t e m i n the MHK a c c o v ^ e x a ^ a ' e - o w e v e ' . o.suiavs a tao.caliy d ^ e - e - v iy:;e ot behavior. In this c a s e the end re&i:l* if. e r :.rftiv o e i s f i - . r e : ! t,f the nm.111 .y d-jnx -i i: v . n sin-: w i t h a i'ttle over S50.COO. w e gn> ;!er.:i.-*KJ •o> i;»cs:)e>:ry O However. if

acccont w>n g>cw .ndeh-.iteiy

6-io -vi'/o | U S' a i»r.>e loss i r e n SbC.O^C

are a m o n g the

mc-'T

with, w e w.lI ioe«riiabiy end uu in rot^i oovettv. This IS ftOl

10 IMPFY

L1"'^'

"> J ' O i l t x t r i O i c i v systur". -s r«Ml y

model, tyjt if w e tNnk abo.»t il. so>re oi r e " v t c

-JI.:S
twi:l lny mjc!.

s:tr;f^i

0 ' w e a ' : " a x ' . ^ - . ^ & i o " .ve c-ov.v

vs-ell captu-'Od by t The ' C s i / t s w f ! y o s c v n -jee"- i c resonate w . " tts? wB«-doc«f-ev'.eo foct tha: " t i e rrJi gei richer, a n d the ;-X.\Y i»fr: i x w f t i "

i r feahiv t h « ' f t a i » cftf:anUv

. v :«o»fr

processes wr^clved that n ^ e t ^ e p.ctv'e ' " t » ? obso.»ie. crenting j m s t e ' t i M S • -y tiorts that bounce cerram md-vKi'.a's « i . r nf tnn "vd that most o ' the additonal processes w ior. Indftfr
fc.-.OS

Hfr.vf.vnf

ac.tua rv se've

tc: mi;tfr

r o w n r w - «

us

••tTengtne'iing tnis system t»*mv-

MS IW -.vill st-t- 11: •"'wptis: (J. !'us : u - . x ^

will be then used to create fr>cr«j vvoa'i--. t h . j is a n c f - e i :>cs't'-.-o feedback. -A-hiy. acos to positive

already e^-oenarto •"•(: i n e e c o ^ o ^ - C S'vSif.-^" in,>t w e u e s c - o i t o

bonk account model ICan y o . f'g j - e 3ut w i a : '.ie feedUacK s tha: ^

On t h e othor hand. I

•"<•

a-e ' J e s c n b t ^ ' i

"f this s ^'SO one o f tne r e a s o n . ,vf.v tne mainsrream econ

orrry is s o 3'M.cled to eAOCnieit.a' g r o w n as tne only Dcss tM; w c " 0" v -jo'c e.X)'iO'"-C tem peifotrrance lndce<;. m «ri uxiX'nonirtlly -jCivaiyj

:

- n u J i e a s - r : to hijre : " «

m p o n a n c e of initial conditions As 2 part.cipar-t ot ' h e e>;w»;-tii> w

.s>sicr--

Esser">t'3f Math

w i t h v®ry little KMtiai w e a l t h , as w e ba-A! see'» acove.

c . i " -i
increase this w e a l t h a n d b e c o m e as e n as l o m e o* i ~ e

i w ' i cofr"W

hv«*" it»cse w < »

tJrtQirvuny Wftll. u a y i ' f i ri;: - exactly j s i •:.!-. ::-ut v e i v

w e r e w a y afcMO ot

99

future b e c o m e s m i / ; h mr.™? urwkgly i ' | h n ncrvsrriiv

v f - v - c h . ":tts

"c't
fore t h e role o ' initial c c x x M i c n s a e c o m e s m y e n rrvvi! t?Apr.»ser: s--n n * ; v o „ « This • m e e ^ the " A m e n c a n D r e a r r " ' a m a n y - *ct<;a*v. tor m o s t

3.2

Space i l e v .".11; v c H-pivivrr * | V H V ? T h . llnvlrl

. t h e

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tpan.il

ri*! nc\i

IJ T H R

. I I ; , I ; : I . W T J;CJ I I . K L

J ' t I

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t'-IlJl-

ti>«r.< c j r tvil'-- u n J I c II*.M = t-yXtfl.Hl xto- :U-re. A i i r ln- y c c n * ••( t h e |vtraini*ririMiNft!

iVK'.X'.V aali

t h e

X - i»|.\;.

i : ;t ! i r u n i l e l

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;>

;>,'. X ;

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w u h . u i h r j y j K I I I , ,I> w f l ' W..rlj. r

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v i f u J U . - n

. i S - . c

It-: r.n h rl :!w i. .1\!.V = ? l y . . . . > , J-.

Tli<" actual - i : e . i | c h r . v . f m iv J r l - J 'i:il

tin*

. J i J i K t i o n s cli.t: ;|«CMK* :lie m t c r . ^ r % i l i r it:

=

y.

i x.ii'ipli-

n . : t «-r.(?nnh M i l i - r « - l

:iu- lAm.;

sphvif ri-pieseni .»u «vi;cU- l-i- .ly

-Klu.i!l>

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\!v:»Ji»".vs

>:.nuii

IJ:

. J^,!. rlu- jf\i-

m o k - i » i i h r 1-N--

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Compartmental models v X ' x n

i r v

I l i l!-M v r t i i c l v

.lit- l.i:j!e "*>r«i-.

HV

IOIV.P,INIV.CIIT^l

uei.K-rm

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a : i J ite 'kim
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N-j^RCMUIF E-H'h

116 Systems Scienrf? and Modeling fo' Ecologicst Economics spatially homogeneous component, also called a compartment. T h e s e are then linked together by flows of material or energy. In effect, a compartmental model is a number of box models joined together. For example, this is how we might want to present a small stratified lake, where the upper part o f the lake (epi limn ion) is separated from the deep waters ( h y p o h m n i o n ) because of the temperature gradient. T h e warmer water stays on top and gets well mixed by wind-induced currents, making this upper layer spatially uniform. However, the currents are strong enough to mix only a certain portion o f water; the rest o f the cooler water is n o t involved in the turnover and remains somewhat separated from the epilimnion. It makes sense to represent each of these spatial units as separate box models and link them by certain fluxes, such as the sedimentation process o f material across the boundary of the two compartments. Each of the. box models may be described by a system o f differential equations with initial conditions: dX i _ ~dt

-

F , ( X l ( 0 . P l ) . X 1 ( O ) = X43)

dX 2 _ ~dc

dt

= F2(X2(c),P2),X2(0) = X02

= Fn(Xn(t),Pn),Xn(0)

= XO cn

Here, o n c e again, X, is the vector o f the state variables in c o m p a r t m e n t i, P, is the vector of parameters used in the model in c o m p a r t m e n t i, X 0 j is t h e vector o f initial conditions for c o m p a r t m e n t i. A s a discrete interpretation, similar to that which Stella generates, we get a system of differential equations: X , (c) = X , (t - dt) -h F( (X, (t - dt), P, )dt

XJt)

= Xn(t-

dt) + FJXJt

-

dt),Pn)dt

T h e s e are then linked by flow equations: X,(l)

= X ; ( t - dt) + f x : X , ( c - dt) • Q , {)*> +

' J 2

X t (I - dl) • Q 0 i*<

D

; • ( X ^ t - d O - X / t - d O U

Here, we distinguish between two types o f flows; advection and diffusion. Advection describes motion caused by an external force (such as gravity, which causes sedimentation). Q defines t h e advective flux. Q,, represents the flows that flow into the ith compartment from the surrounding j t h compartments; Q ; ; are the flows that flow out o f the ith compartment. Diffusion is defined by the gradient or difference between the

1 0 1

Fssfinrial M a t h

[•main's >i evHcrtil n •••'ir'rr.' sronaS l^ciinvi* f> i• • the Hirrowi Inn; / c.«ffi{vn{iioiii' i x ; : i l . - I i i i . i l c i . s l i n . , \ ( ( ) . I:K L/lKin w:»r. ;t I f t!t\ Vds:II£. .:I»J lhe

ii-i ! hr r.ililig -n^ienal

I• 4

r:!r.|un::v:!r

Continuous models IfrhTf

v.rulillr.v

I^I:II:%II

SyUic a:!^ lh:» siiimhi! •

11.

• Is is ,»n impi«TiirW Iv.l'M r.if i h r litfr^ irv :KTII

to

I.IFC'in

:.% -|-aii.:lis

trie

H U M :

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w r h-u-c

l h » uxieixiuicii' \.>i il-ie> 'line l

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So v-rdi a tnis rouncec d an s c o u t ' Von

—

juJi got .wfif (lie c'Xidt n o w w e have i
TIIS

15 W H A T

HFTOOC-I

,j V -

Ui

—

.iv

P . —

Kt:

IHAM

*

OZ

Q , —

*

F O . X

P I

J i

•A-'^O W I I

X 1.- Un- -.TUW t-i itir «tate

realize '.hat our state vanat'e udui'i t'.vw indeiXK
!• iii:.:Llri| -js ::i

this case.

y « a 'unction t o n o '

S"C a

ir

ih.i!

iMt-i T r i u «

,\r wnr . iiiuui

iiiii; ,'.!•

r... the slrficrint.Hl Csiiinti;:iv- tluii i j - f c j r when tr>e tmiv KiCichKrv

.ft.: |rt rh.,1

we

s ^ t i A f ccoicfcnaie. i V.'hile fcelcie w e

wkij-

v.ere looking Ot instantaneous iftiinyc

• n.'dcl !n ir,i'C4<( in »-iHiiinM ( i n m r i a ! rirvliMiiinW . - \ i7:

or our functor, 93 'he t-me m c e m e n t

:!>: 1 >k '•••'•' siiili.ll t<.iivlili.mi l>: tl:c i*?

was accroaching zwo. oow w e can icofc

w.- w » r l iii |-r.v.-i.lr rhr l'in:n.Uiv : 1ie1.l1r1.%11*. m

t h e initial . n tWc sp.ice dimcii>K
temporal and spotted

Pwnpl-oui. Hm

instantaneous increment w h e n e
lift cer-vative and is shown as JX Then

ili.u iv.--

t h e iri.ci.i

ii'iulit'On- .we

than one spatial coordinate Tr.en we

•> Mj'Kt' :!! •* tin.- »|T>IMI •.•."iJnwrv :
get a A c t i o n o l Mven r-mic mdejwuxl

C.I!

em var^cies.

:ii-kifl pr«Mri>'.r i.Ni. .111.I I ' u u '.rit.}! Ml n<>liit;u;tl p.v.imf reis 11 »imi:ri. >11

W i l -s the ' ^ j n g e

time. o m j *X>b£ >3

iho cr.groe in space. It b e c o ^ s even more interesting if w e oonsiotw m c i e

IV.:C;*
»UNIL,N

II.

H EM I -

i.IC.l

i n t h e

Time and space scales ... ere somewhat rclatod m most cases w o observe that systems -.vith large- soatai scales Isizesl havo longe' terroora-' scales. I t rs interesting to note how in co3rri<: s c ^ e s l i m e and space become corr-oined Into a un-t s u d ' as a light year, w h i d t is actually the distance n a t s co.>3-'ed if trarveling v/nh the speed of light for 1 year - 1 light ye&r = 94©0528-1 x iQ m meters

T " X

102

Systems

Sr. e i c f i

and Modeling for Eco'ogic-il

Ecunoi-'cj

w h « h is Quire a distance' S c m f r , r * v j simitar is round in the subatomic world

As w e

have seen, the t i n / panicles of i ^ e m i c o w o r l d have lifebmes o ' less than a millionth of a second Tnis is extremely s n o n in the human timescale. However. their 5rze is afso very small ano ihey travel at I v g h t n ^ - i a s t velocities To make more sense of this comparison, physicists have come u p w i t h a measure called a "particle s e c o n d " - a unit of t i m e equal lo 1 0 ~ 7 2 seconds, w h i c h if. the t i m e needed for a particle to tiavei over a distance a f e w t i m e s its o w n sue Various particles have lifetimes thai vary b e t w e e n 10 B-KJ 100.000 particle seconds Such resolutions do not m s * e any sense if w e are considering geological change, rrovem e n t of c o n t e n t s ot using oi mountains However, certain s l o w processes may be abruptly .nterrupt6d by fast and violent

fluctuations

Slow geologica' change vields to an eanr-quake.

w h e n in minutes end hours w e see m o r e disturbance lhan over the thousand years before that. Modeling processes that occur o n a variety of scales is s big cha ; lenge, since it is proh i i i t i v e f / ham to represent tl'n slow processes at the scale of the raoid ones However. if w e ignore ttie singular-ties completelv w e may miss some really important c o n g e s and trans'ormations in the system W h e n sizes den't differ that much, there >s no exact relationship b e t w e e n tempora and spatift' scales- For instance, s - w i s reg^sier the« environment one© |n every 4 seconds. Even though humans are larger, they can do a t e t t e r job

For us. the world around us changes

approximately once every »/24th of a second " h i s resolution of ours is w h a t d e l o e s tne rato of char>9e o f snapshots in monies that w e watch. If w e do it >ess frequently, w e see h o w tne motion becc^nes discontinuous, figures start to move in jerks If w o do it faster, w e will not see the difference. VYe can a c t u a l insen another frame and w e will noi register it " h e y say that there is a m e t h o d of manipulating people called the " 2 5 t h frame." Th>s is w h e n a 25th frame is nserted ana the movies run at 1/25th of a second This single 26tn frame can be entirely Out of c o n t e x t »;vj humans to not consciously tegistcr i l However. apparently it affects our subconsoous and lhe information finds its w a y to ihe right pans of t h e brain, influencing our opinions and d e c i s i s This w o u l d not be possible for a fty. wtWch scans the erwvonmeiM 20 t i m e s faster th&n w e do. & fly w o u l d state at tne 25th frame for long enough t o realize that scn^ething totally out of context was being disoiayed O n the other hand, a sne

w o u ' d ^ever even see this

frame Moreover, if you move fast enough, in 4 seconds you can pe psace to another. T->cse considcroticrs arc important w h e n choosing the '.ght resolutions for your models.

Lr: j s lakf J lock a" i . i : J ] l t < ! .Icuvr. I.

lo

iitnv thi> >i:ul .••: r^ualici

C.Hi tv

Modeling advection C-XiUifcr

:L:R

H.%1 I I ! N : ! I->:I

• >! .i

,fr!,IRI

one s y M i l Jintennc^ !c: s j w u i J:v:Ji J

IsetcrvSctx*!'wc

I R U M :

I:I

re*l'.ip< - l e i * p p "

.II'iil :n:IV

" -

^UPI'FLM-

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^jvuiik* ilwi i I t who)* Ktijjtls " t rr-t? % u v l c in Iv

m : o r u t u ! M-gi;icii:». rj-.!i AA king

Tl i i ' c t K O ' : i l l l o i « .M ;l: v
iMtli ! ii:r!i l:n;r .iiiJ !fi:£l!l C'.r A). Vt'iir:i Ax .i '.nac enough w-f caii think fhciit t h . j m c - > l the li-nruiiM*' j<"N-ri!v.i a k : - . v

- w -«i*i . . v ».irv.r-it

Essential Math

C(l. X )

C|T X - J X )

103

C(L, X + J Y I

JX

W e alsu assume thai there is a certain velocity ot flow in t h e c a n a l , r, and that it is c o n s t a n t . Let us now define t h e c o n c e n t r a t i o n ol the c o n t e n t s in any given segment at rime r + A:, assuming that we know the c o n c e n t r a t i o n there at time t. S i n c e calculating c o n c e n t r a t i o n may be con I using, let us write t h e equation for t h e total amount of material in segment \ at time t + At C(< + At, x ) - A . v = C(f, x) • A * -1- C ( r . x - A x } - r - At . , I,, U" [,,

, '«»,.,

C ( t , x ) - r- At klii, ^ • I l h 11 ,,,

tvf\iiv

,, L",,,

R e a r r a n g i n g t h e terms, we get. C(t + At, x) -Ax - C(i.x)'

A \ = C(T, * - A.K)-Y-At — C O , . * ) - R- Ar

Dividing both sides by A\At: C { l + At, x) • Ax - C(t, \) • A x _ C ( t . x - A x ) - r • At — C ( t , x) • r • At A ^ At

A.* • At

Or, c a n c e l l i n g A\ on t h e left-hand 5i.de and Al on t h e right-hand side: C ( ! + At. y) - CM, x) _

CU,x - Ax) - C(t, x?

A;

Ax

N o w if" we let Ax - * 0 and At Hi! 0, we get t h e well-known a d v e c t i o n e q u a t i o n as a partial differencial e q u a t i o n : dc

_

di

dc dx

In discrete n o t a t i o n , t h e equation for c o n c e n t r a t i o n at t h e n e x t time-step is: C(t + A t , , ) = C ( i , x) -

| C ( t

-°

C(r,.v-A,)l-r-At Ax

(3.12}

Il we know the c o n c e n t r a t i o n at the previous time-step, we can calculate t h e c o n centration at the n e x t rime-step. T o he able to use this equation at any (x, <•), we still need to dehne two more conditions. First, we need to know where t o start - what was the distribution o f materia! alp rig the canal ar the beginning, at time 1 •=• 0 . T h a t will he t h e initial condition: C ( 0 , x) - c s .fjj| Besides, if you look at equation ( 3 . 1 2 ) you may n o t i c e rhar to solve it for any r we need to know' what t h e c o n c e n t r a t i o n at the left-most cell is, where x = 0 . T h a t is t h e boundary c o n d i t i o n ; CO.O) = b{t)

120 Systems Scienrf? and Modeling fo' Ecologicst Economics T h e r e may be other ways ro initialize equation ( 3 . 1 2 ) o n the boundary. For example, instead o f defining t h e value o n the boundary, we may define the flow, assuming, say, that C(t,O) = C 0 . 1 ) T h i s will be a condition o f no flow across t h e boundary, and will also be suffic i e n t to start the iterative process to solve equation ( 3 . 1 2 ) .

Modeling d i f f u s i o n Let us now consider diffusion as the driving force o f c h a n g e in the c o n c e n t r a t i o n in our system. T h e force that makes the substance move in this case is the difference between c o n c e n t r a t i o n s in adjacent segments. It is also good to remember that in this discrete approximation we ate actually dealing with points o n a c o n t i n u u m , in this case a line Ox. T h e c o n c e n t r a t i o n s that we are considering are located at these points. W e are dealing with average c o n c e n t r a t i o n s for the whole segments, and are assuming that these averages are located at these nodes. T h e r e f o r e , if there is no outside force to move the material, it would be reasonable to assume that t h e farther away the points we consider are, the less material c a n be moved between them by the c o n c e n t r a t i o n gradient.

C(t, X- AX)

C(t,X

CO. X)

+AX)

p

-

JX

Just as before, let us define the c o n c e n t r a t i o n o f material in any given segment at time t + At, assuming that we know t h e c o n c e n t r a t i o n there at time t. T h e equation for the total amount o f material in a segment at time t + At is: C(t

+ At,x)-Ax

=

C(t, x) • Ax +

In this equation,

• D • At +

Ax

C(t, x + Ax)—C{t! Ax

x)

(s

^

• D • At

Ax

empirically derived equation for

the diffusive flux between two adjacent segments. D is the diffusion coefficient that characterizes the e n v i r o n m e n t , the media; it tells us how fast diffusion c a n o c c u r in this kind o f media. After some rearranging we get: C(t, x - Ax) ~ C(t, x) _ C(c, x) - C(c, x - A x ) C(t + At, x) - C ( t , x ) Ac

=

Ax

Ax Ax

Essential Math

O n c e again, if we Let Ax —* 0 and At

105

0, we get the well-known diffusion

equation as a partial differencial equation:

—

= D

dl

In discrete notation,

—

dx:

the equation for c o n c e n t r a t i o n at t h e next

time-siep

, . , , , [Ctt. x- - Ax) - 2 C ( t . x) + C(r, ,y + A O ) D • At r { r , C ( r + At, x) = L\i. x) + Ax:

(3 13)

becomes;

If we know the c o n c e n t r a t i o n at the previous time-seep, we c a n calculate t h e c o n c e n t r a t i o n at the next time-step, lust as m t h e advection example, to calculate this equation at any (x, t) we need to define the initial condition: C ( 0 , x) = C 0 ( x ) As for che boundary conditions, in this case we will need two o f them. W e cannot. use equation (3.1 3) to calculate the value both on the left-hand side boundary C(r, 0 ) and on the right-hand side boundary C ( r , N ) , where N is t h e number of rhe max ma I segment that we consider. Therefore, we need two boundary conditions: C ( t , 0 ) = b.{r);

C(t,N)

=

(;,(;).

Similarly, there may be o t h e r types of boundary conditions, such as: C M ) = C(i.l),

C(i, N - J ) = d i . N).

T h i s will be a condition of n o flow across the boundaries.

3.3

Structure Consider a c o m m u n i t y of two c o m p e t i n g species that eliminate o n e another. W e c a n describe this system by t h e following two O l ) £ s : dr

T

=

14)

dy — = —ax dt where a and | are hunting efficiencies of .species y and x respectively T h i s model c a n be re.solved analytically: dx _ by d)-

ax

ax dx = by dy, ax:

- by

~ const

122 Systems Scienrf? and Modeling fo' Ecologicst Economics A good way t o look at system dynamics, especially in case of two variables, is to draw t h e phase portrait, which presents t h e c h a n g e in o n e variable as a function o f the o t h e r variable. Figure 3.7 presents the phase portrait for model ('3.14). It c a n be seen that the two populations eliminate each o t h e r following a hyperbola T h e initial conditions define which trajectory t h e system will follow. In any case, o n e ot t h e two species gets eaten up first, while t h e other species remains. If t h e initial condition is on t h e line equation ^'(ax) = J(by).

then t h e two populations keep e x t e r m i n a t i n g

e a c h other at infinite length, tending to c o m p l e t e mutual e x t e r m i n a t i o n . If the initial conditions are below this line, then y is e x t e r m i n a t e d and x persists. If the initial conditions are above this line, then y wins. Models like those considered above may be called rigid ( A r n o l d , 1 9 9 7 ) ; their structure is totally defined. In contrast to a rigid model ( 3 . 1 4 ) , a soft model would be formulated as: dx _ , . — - -o(x, y) y (3.15)

dt

at where a(x, y) and b(x, y) are certain functions from a certain class. It may be shown that for most functions a(x, y) and b(x, y) t h e phase portrait o f system ( 3 . 1 5 ) is qualitatively similar to t h e o n e in system ( 3 . 1 4 ) (Figure 3 . 8 ) . O n e o f t h e species is still exterminated, but t h e threshold line is n o longer straight. A n important feature of model ( 3 . 1 5 ) is its structural stability. Changes in functions a(x, y) and b(x, y) that describe some features o f the populations do not change the overall qualitative

behavior

ot the system. S i n c e in most cases our knowledge about

the objects that we model is not exact and uses a good deal o f qualitative description, soft models are more reliable for predicting the system dynamics. Unfortunately, there are very limited analytical methods t o study t h e structural stability of models. T h e only way to analyze structural stability in broader classes of models is to run extensive sensitivity analysis, varying some functions and relations in the model as well as changing parameters and initial conditions. 50.00

r

:

:

:

•

Essential Math

107

S t r u c t u r a l analysis o f models requires quite s o p h i s t i c a t e d m a t h e m a t i c s E v e n tor a simple m o d e l like t h a t a b o v e , analysis o f its structural stability lies way beyond t h e s c o p e o f this hook. In g e n e r a l , T a b l e 3 . 1 , from v o n B e r i a l a n f t v ( i 9 6 8 ) , shows t h a t t h e r e is a very small d o m a i n o f m a t h e m a t i c a l models that c a n be analyzed by analytical methods. M o s t o f t h e real-world m o d e l s turn o u t to be n o n - l i n e a r , with several or m a n y e q u a t i o n s . Besides, most o f t h e systems a r e spatially distributed, w h i c h almost precludes a n a l y t i c a l m e t h o d s o f analysis. H o w e v e r , t h e r e are n u m e r o u s e x a m p l e s of q u i t e successful a n d s t i m u l a t i n g a n a l y t i c a l studies that h a v e led t o n e w t h e o r i e s and n e w understanding. Physics especially has an a b u n d a n c e o f t h i s sort o f m o d e l . P r o b a b l y this is why most o f t h e m a t h e m a t i c s that is used in m o d e l i n g c a m e from physical a p p l i c a t i o n s .

50 00

>- 25.00

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25.00 X

Phase portrait for the soft model of mutual extermination.

Mathematical models that can be analyzed by analytical methods

Linear e q u a t i o n s Equation

Non-linear equations

One

Several

Many

equation

equations

equations

Algebraic

Trivial

Easy

Essentially impossible

Ordinary differential

Easy

Difficult

Esseniiaiiy

Partial differential

Difficult

One equation

Several

Many

eouations

equations

Very difficult

Very difficult

Impossible

Very difficult

Impossible

impossible

Impossible

Impossible

Impossible

impossible Essentia,ly impossible

Impossible

108

S y s t e m s Scienrf? and M o d e l i n g fo' Ecologicst Economics

Ecology, social s c i e n c e s and e c o n o m i c s have yet ro d e v e l o p a d e q u a t e m a t h e m a t ical m e t h o d s o f analysis. U p till now, most o f the m o d e l s tn these s c i e n c e s h a v e b e e n n u m e r i c a l , analyzed by m e a n s of c o m p u t e r simulations.

3.4

Building blocks Let us consider some o f t h e main types o f equations and formulas t h a t you c a n e n c o u n ter tn dynamic models (Figure 3.9). II you have a good ieel for how they work, y o u c a n put together quite sophisticated models using these simple formalizations as building blocks W h i l e , indeed, c o m p l e x n o n - l i n e a r models are n o t o r i o u s for springing surprises, for their u n e x p e c t e d behavior, it is always nice to have s o m e level o f c o n t r o l regarding what is going o n in t h e model

Knowing some o f t h e m a t h behind t h e equations and

formulas in a modeling software package such as S t e l l a will add s o m e predictability to how your m o d e l m a y behave. Knowing h o w some o f t h e very simple formalizations perform as s t a n d - a l o n e modules will help you to c o n s t r u c t models that will be better behaved a n d easier to calibrate. Certainly, interaction o f these processes will create new a n d uncertain behavior, w h i c h it will be hard or impossible to predict in .some cases. However, in many o t h e r cases you will be able to h a v e a pretty good e x p e c t a t i o n of what t h e output will be w h e n you put together t h e building blocks.

(A) Constant growth dx/dt = a

X

where a = const

Solulion: * = c + at. c - initial condition for x

• , 1

"1

.

1 j 1

o-V a

co^

^

.

;»

-.on

t1 oo

There is a constant flow of material into the stock. If there is also a constant outflow, then consider a as the net rate of flow, a =• in •- out (B) Exponential growth •00 00 dx/dt - ax

i

1

i

X

where a = const

L ^ J

Solulion * - c e a '

XO i 00 1

in wm

1»36

n*>

The added positive feedback creates exponential growth. Dynamics can easily get out ol control because ol the very fast growth Keep a small, especially at first when you are only testing the model

• Figure i i ' i n3.9 ^ H

Growth (A| Constant growth; IB) Exponential growth, (C) Growth with saturation;

ID) Growth with peaking; IE) Delayed response

uiy

11111

^^^^^wwpwtwwwwwii^^wwwww^^^^^^WBWipiwiiiuu » n i - • Essential Math 109

(C) Growth with saturation

dx/dt = ax - bx2

> 90

where a and b = const

*

3C&•

Solution * =

(p&lSSo —' •

In

Ojt

b

a + bc(e" - 1) Otf*

The exponential growth is now d a m p e n e d by exponential decline

SOO

toe

900

12»

A) smaller populations the linear

function (ax) dominates As numbers increase the parabola (bx2) overwhelms a n d shuts down growth The solution is the so-called logistic equation Note that the model is identical to the model with bx2 = ax(i - bx/a) The carrying capacity in this model is then a/b.

carrying capacity a x

W h e n * = a/b the growth is zero a n d the model saturates

(D) G r o w t h with p e a k i n g

rop

dx/dt - ax - bxsl where a, b, and s = consl

o•

O

A 00

n

v l7 5C

Mil

MM

A simple way to make the model peak a n d Ihen decline is to have a variable exponent in the ou'.tlow part a n d make this outflow grow with time. In this case again at lirst the outflow is very small and the system grows Later on the outflow b e c o m e s dominant a n d gradually reverses the dynamics eventually getting the system down to zero. Used less often than the first three blocks but still may b e handy. (E) Delayed response

dx/dt = ax - bx2(t

At).

where a a n d b = const At - is time delay

0 00

«OT

»0«

A powerful way to get pretty c o n l u s n g results. In this model of saturated growth (see above) w e a s s u m e d that mortality is controlled by the population size several lime-steps ago This may be if w e assume lhat mortality is due l o a disease a n d the disease has an incubation period of At. If At = 1 we still have a saturation If a r = 2 w e suddenly run into oscillations as s h o w n in graph With At > 2 we have a population peak and collapse somewhat similar to the dynamics in the previous bloc*. The delay lunclion should be always used with caution, since it can easily destabilize your model

Figure 3.9

(Continued)

110

Systems Scienrf? and Modeling fo' Ecologicst Economics

Further reading If you feel

that your

be more

than enough.

Calculus

and Analytic

just type "differential nations

to choose

Berlmski,

D

math

critical

however

O n Systems Analysis

tations.

I. Arnold His classic

why mathematics

of some

expla-

most of his criticism

A n E.s\sa;y Concerning

the Limitations

and Biological

M I T Press - This does a

Sciences.

can be quite important

of the classic

Ivis been stressing book:

some analytical

mal
of

for building good models

crecirises, inducing booh

modeling

h is important

Berlinski

of Bertalanffy

to remember

Some

and

chat mod-

cases they may be useful even with

flawed

(he difference

between

soft and rigid mo dels in his 1997

presen-

A r n o l d , V. 1. ( 1 9 9 2 ) . Ordinary differential

for those who want

to get a better

understanding

equations. of modeling

Springer-Verlag with O D E ' s and

techniques.

von Bertalanffy, L. ( 1 9 6 8 ) . General mathematics

days you car. also find a lot on the web.

mathematics.

Cart be recommended master

Political,

than mathematical objects,

or inadequate Vladimir

will

from.

really good job explaining

els are more

in calculus

arid you ufli get quite a few links tut/i j;rert;v good

mto Google

M e t h o d m the Social,

may be overly

it Any textbook

T h o m a s , G . B , Finney, R . L . ( 1 9 8 9 ) . Elements of

A d d i s o n - W e s l e y These

Geometry equations"

(1978).

Mathematical

Meadows,

you may uwnt to refresh

is coo flaky

Try this one for example-.

and ide.as about

System

Theory.

the building blocks

in

G e o r g e Braziller - Contains some modeling.

important

4. M o d e l Analysis 4.1

Sensitivity analysis

4.2

Model calibration

4.3

Model testing

4.4

Conclusions

SUMMARY T h e r e are many ways in which a model can be analyzed and tested, and some o f them have become more-or-less standard for the trade T h e r e may be many unknowns or assumptions that go into the model. Sensitivity analysis is a way to figure out how important these assumptions are and what effect they may have on the model performance. Sensitivity can be tested by disturbing a model c o m p o n e n t that is not known for certain (a parameter, a function, a link), and then seeing how this disturbance propagates through the model structure and how different the results that c o m e from the disturbed model are. A second standard analysis is performed to see how closely the model c a n be made t o reproduce the experimental data (qualitative and quantitative). T h i s is model calibration. T h e mixlel parameters are modified to minimize the difference between model output and the available data. Finally, other tests can be conducted to validate the model and verify its performance. T h i s analysis includes different methods, ranging from diligent debugging o f software code and mathematical formalizations to comparisons with independent data sets, and extensive scenario runs

Keywords U n c e r t a i n t i e s , parameters, initial conditions, critical parameters, inverse problem, data model, error model, T h e i l ' s index, R1 index, weighted average, empirical model, trendline, process-based modeling, o b j e c t i v e function, minimization, trial and error, optimization, M a d o n n a software, curve fitting, open systems, C L I M B E R model, validation, verification, scenario, credibility. *

*

*

C h o o s i n g variables and c o n n e c t i n g them with flows and processes is not enough t o build a model. Actually, this is just the beginning o f t h e modeling process. By identifying t h e variables and formalizing i h e processes that c o n n e c t them, in Stella or in any o t h e r modeling tool, only o n e possible description o f t h e system is created. W e still need to make sure that this description really describes the system, and then try to use the model in a meaningful way to generate additional knowledge about the system. W h y else model at all? l his stage o f testing and working with the preliminary model built is called model analysis.

Il the model is a mathematical formalization - say, a system of ordi-

nary differential equations - we may try to solve the equations. If this is possible, we get a functional representation for all model variables and c a n pretty much say what

111

112

Systems Scienrf? and Modeling fo' Ecologicst Economics

they will he at any time or place, and see clearly how different parameters affect them However, as previously indicated, the c h a n c e s are quite slim that we will get an analytical solution. W e may still try to analyze t h e phase plane o f the model variables and derive some general understanding of the model behavior - perhaps by testing for equilibrium conditions, or trying to identify when variables grow and when they decline T h e more results we c a n derive from this analytical analysis the better, because all the analytical information we obtain is general and it describes the system behavior lor al kinds of parameter values that we may insert into the model - not just the single set of parameters that we use when we run the model numerically o n the computer

4.1

Sensitivity analysis If no analytical analysis is possible, we have to turn to numerical methods. Using Stella, in order to see how t h e model performs we need to " R u n " it. By doing this, we numerically solve t h e system of difference equations that S t e l l a has put together based on the diagram and process formalizations that we have formulated. A numerical solution o f a model requires that all parameters take on certain values, and as a result ts dependent on the specified parameter values. T h e result o f a model run is dependent on the equations we choose, and t h e initial conditions and parameters that are specified. S o m e parameters do not matter much; we c a n vary them quite significantly, but will not see any large changes in t h e model dynamics. However, other parameters may have a very obvious effect 011 t h e model performance. Even small c h a n g e s in their values result in dramatically different solutions. Analyzing model performance under various c o n d i t i o n s is called sensitivity analysis. If we start modifying a parameter and keep re-running the model, instead o f a single trajectory we will generate a b u n c h o f trajectories. Similarly, we c a n start changing t h e initial c o n d i t i o n s or even some o f the formalizations i n t h e process descriptions. By comparing t h e model Output, we get an idea o f the most essential parameters or factors in t h e model. W e also get a better feeling of the role of individual parameters and processes in how t h e model output is formed, what parameters affect what variables, and within, which ranges the parameters may be allowed to vary. T h i s is very important because, in contrast t o an analytical solution where we could find an equation relating model output to the input parameters, with numerical models we do not have any other way to learn what the c o n n e c t i o n is between the various parameters arid the model output, e x c e p t by rerunning rhe model with different, parameter values. Whereas in t h e analytical solution we c a n use a formula that clearly shows how a parameter allecis the output, in case o f numeric runs we know nothing about what t o expect from the output when a parameter changes.

In Stella, there is a method of making estimates for model sensitivity Choose "Sensi Specs

J

allow you to set up your sensitiviTy test

S e c i o ' Specs... Sensi Specs...

XY

R u n Specs... X X R Range Specs...

1. Double click on the parameter that you want to t e s t for model sensitivity. It will b e moved to the right pane

XR

I sinp

The following s t e p s will b e required:

2. Highlight the paiametet in the light pane

Help

S-Run

" in t h e Run menu A window will open that will

.

Check Units

Model Analysis

113

3. S e t t h e n u m b e r of p a r a m e t e r values that you w i s h to t e s t for. 4. C h o o s e h o w you want t n e p a r a m e t e r to c h a n g e its value. 5. M a k e s u r e you click t h e S e t button to lill in the table on the right, w h e r e p a r a m e t e r values will b e automatically calculated t o run t h e model. Sensitivity Specs Allowable

Selected (Value) e.param (0.1)

CUFcpJlatio.i

O M.patatr O S'.param

Ru"» Value "I oToi

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2 3

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0.5

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If you n o w click "OK." t h e model will run several t i m e s in a r o w for t h e different values of the parameter c h o s e n . B e f o r e you d o that, you n e e d to prepare your output. M a k e sure you c r e a t e a "Comparative" graph t o s e e t h e difference in t h e output that you will b e generating. For e x a m p l e , in t h e modei that, w e w e r e building ebove, if w e start changing t h e Birth Rate p a i a m e t e r w e will produce a family c f curves, which s h o w that the model is quite sensitive to c h a n g e s in this parameter Define Craph CraphType:

© T i m e Series

G Scatter

O Bar

t onnect Dots

55 Comparative . Allowable \<* \-C* jO jO O O O

Births Deaths B.param M.param S.pa'am Temperature T.limitation

TU.e

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114

Systems Scienrf? and Modeling fo' Ecologicst Economics

If you modify another parameter, say the o n e that is related to t h e effect of temperature, you will g e t another bunch of trajectories. You may already notice that apparently the change in this parameter has a less prominent effect. While you can s e e s o m e considerable variation, you do not g e t the curve to decline to zero - at least not lor the values ol the parameter chosen for this experiment

Sensitivity analysis explores t h e parameter space a n d can help us identify some o f t h e critical parameter away to infinity model output

values, where t h e model might, for example, crash or run

Every c o m b i n a t i o n of parameter values translates into a specific

It is like testing the landscape for hidden surprises and trying to cap-

ture trends in model behav ior in response to the changing c o m b i n a t i o n s of parameter values, figuring out how t o make certain variables grow, or decline and at what time. Later on in the modeling process, when we collect evidence of the model actually representing the system, and have sufficient confidence in the model performance, we c a n perform turthei sensitivity analysis to t h e point where we make conclusions about

Model Analysis

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Qualitative comparison may become difficult as we close on our target, getting the model output almost identical to data. W e may be still improving the results somewhat, but we c a n no longer distinguish t h e gains by simply staring at the graphics. A n o t h e r case is when we get a better match between output and data m one time range for o n e set o f parameters, but achieve a better m a t c h with a different time range for another set o f parameters. W h i c h parameters do we c h o o s e then? In these cases, visual comparisons can fail. Q u a n t i t a t i v e mathematical formulas can then b e c o m e useful. O n e simple formula for the error model is: (4.1

where x, are the data points and y, are t h e values in the model output that coirespond in time or space to the data points. N o t e that this formula tracks the relative proximity o f t h e two models - that is, for larger values we allow larger errors. T h e smaller the error, E, the better t h e model calibration. T h i s index is quite similar to T h e T s measure o f forecast quality: 1/2 (4.2)

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1=1

Eo

1=1

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Very often, the metric used to compare the models is the Pearson m o m e n t product correlation coefficient,

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ii

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the peaks. N o t e that unlike the above error models, where t h e best fit c a m e with the minimal value o f E, here the best fit is achieved when r 2 = I. T h e s e formulas become more cumbersome if we calibrate for several variables at once. In the simplest case, we can always take an average o f error models for individual state variables: k

E E" =

E

J

——

118

Systems Scienrf? and Modeling fo' Ecologicst Economics

500

• •

450 400 350 300 250 200 150 100 - | 50

0

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1

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1

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1

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Experimental model ol the microbial system.

w h e r e E. are t h e individual errors calculated using e q u a t i o n s {4.1 ) - ( 4 . 3 ) or similar metrics. In some cases, calibration with regard t o o n e variable may be more important t h a n rhe til for t h e o t h e r ones. For e x a m p l e , when we luive t h e best data model for a c e r t a i n variable but very a p p r o x i m a t e information about t h e o t h e r s , we would want t o make sure that we calibrate more to t h e reliable information, while t h e i m p o r t a n c e o f o t h e r data sets may be secondary. In this case, we may want t o introduce c e r t a i n weights i n t o t h e formula s o that particular variables ger mote a t t e n t i o n in the comparison:

w h e r e u-, are rhe weights associated with l< different variables,

5>, = i ;=l

T h e error model is t h e n affected most of all by t h e variable that h a s t h e h i g h e r w e i g h t . T h i s m e a n s tt is m o r e efficient t o get t h e error d o w n for thar variable as far as possible, s i n c e t h e total error t h e n gets reduced t h e most Let us consider a n e x a m p l e . Suppose that we h a v e b e e n running an e x p e r i m e n t in r h e lab measuring t h e growth o f a batch of microorganisms over a period o f 1 0 0 hours, t a k i n g a sample every > hours. W e then use a spreadsheet program t o store t h e results a n d t o present t h e m in a graphic format (Figure 4 1 ) A l s o suppose that we are measuring a certain limiting factor - say, temperature, or substrate availability - that describes h o w suitable t h e lab e n v i r o n m e n t is tor t h e grow th o f the organisms that we are observing (Figure 4 - 2 ) . W e are normalizing this measured value t o bring it within a range o l |0,1| T h i s c a n be d o n e it we divide all t h e data by t h e m a x i m u m observed value. Let us build a m o d e l ol t h e system. S u p p o s e we are n o t interested in t h e structure o f t h e system a n d w a n t to build an e m p i r i c a l , ' b l a c k - b o x " model.

Empirical model T h e output thai we have consists o f t h e data about t h e n u m b e r of organisms. T h e input is time, and t h e i n f o r m a t i o n about t h e t e m p e r a t u r e in t h e e n v i r o n m e n t O n e s i m p l e empirical m o d e l c a n be created i m m e d i a t e l y in a spreadsheet program. F o r e x a m p l e , in E x c e l it is c a l l e d "adding a u e n d l i n e t o t h e g r a p h . "

Model Analysis

119

i 0.9 0.8-

©

«

0.7-

©

0.6 0.5 0.4 0.3

0.2 0.1

i 0

i

i 10

i

i 20

i

i—i—i—i—i—i 30

40

50

i—i—i—i—i—i—r 60

70

Figure 4.2

Experimental mode 1 of temperature in the microbial system.

Figure 4.3

A trencline es a black-hox model that uses time as input.

80

90

In this case, t h e o n l y input i n f o r m a t i o n that is used is time. T h e model is t h e e q u a t i o n of t h e line, w h i c h is a p o l y n o m i a l o f order 2: y = -0.0978x2 + 14-554*-81.443 A s we c a n see in Figure 4 - 3 , t h e trendline does a pretty good j o b of representing t h e model results, t h o u g h there is obviously a difference between t h e model output and t h e data points available. N o t e that E x c e l labels t h e i n d e p e n d e n t variable x, while in our case it should rather be t for time. B y adjusting s o m e o f t h e parameters in t h e model, we may m a k e t h e m o d e l output closer t o o r further away from t h e data points measured in t h e e x p e r i m e n t . Actually, this is e x a c t l y h o w E x c e l c a m e up with this e q u a t i o n . It took a general form o f a s e c o n d order p o l y n o m i a l a n d started t o tweak t h e three coefficients. W e c a n see h o w this works if. instead of " A d d i n g t h e t r e n d l i n e " in t h e C h a r t m e n u , we set up a general form of polynomial and use t h e " S o l v e r " option in t h e " T o o l s " m e n u . W e will t h e n be able actually to see h o w t h e values o f t h e three coefficients will be modified while E x c e l will be optimizing s o m e t h i n g t o get t h e t w o curves t o m a t c h as closely as possible. T h i s process o f t w e a k i n g t h e model p a r a m e t e r s in an a t t e m p t t o get a b e t t e r r e p r e s e n t a t i o n o f t h e data a v a i l a b l e is t h e c a l i b r a t i o n o f t h e m o d e l . I n our c a s e , t h e c o e f f i c i e n t s o f t h e polynomial a r e t h e u n k n o w n m o d e l parameters t h a t h a v e b e e n varied in a n a t t e m p t to get t h e p o l y n o m i a l t r e n d l i n e as c l o s e as possible t o t h e data points.

120

Systems Scienrf? and Modeling fo' Ecologicst Economics

y - -1E

0 8 * 6 * I E - 07x'< + 0 0004*'* - 0 . 0 5 2 7 * 3 + 2.2704x ? - 23 969x + 52.391 R 2 - 0 9775

T h e R-squared value for t h e model described a b o v e is R2 =

0.9232.

R e c a l l that

tliis error model is such that t h e fit is getting b e t t e r as R~ is a p p r o a c h i n g I . If we use a n o t h e r m o d e l , a sixth-order p o l y n o m i a l , we c a n improve t h e R* value a n d raise it to R ' =

(Figure 4 4 )

0.9773

In this case we will have t o guess t h e best values for

seven parameters instead ol three. Even though we g e t very high R " values from these models, they have t h e problem of generating n e g a t i v e output at c e r t a i n times. T h i s should b e prohibited clue t o t h e natute o f t h e modeled process - t h e population numbers c a n n o t be negative. T h e simplest way to avoid this is t o c l a m p t h e model with an " i f s t a t e m e n t :

=

'

j 0,

if - C.097S.X- + 1 4 . 5 5 4 x - 8 1 . 4 4 3 < C

[ - 0 . 0 9 7 8 X -

+ 14.554.x - 8 1 . 4 4 3 ,

otherwise

T h i s would be t h e n our e m p i r i c a l model, where t h e n u m e r i c c o e f f i c i e n t s are t h e calibrated values. T h e r e are other statistical tools that are available in Excel (such as the Solver or t h e G o a l Seek tools) or in other packages that may be furthei used foi a refinement of our calibration. W e may also try to bring in the other available data set - that is, temperature - and run multiple regression for time and temperature t o try t o improve further our empirical model; however, this will require more sophisticated statistical tools than Excel, unless we formulate our own equation and use the Solvei to minimize the erroi model. In any case, what ts important is that, when building these empirical models, we entirely rely o n t h e information that we have in t h e data sets. W e c o m e up with some type of equation a n d t h e n quite m e c h a n i c a l l y adjust t h e parameters in an attempt t o reproduce the data as well as possible. A l l t h e information we know about the system is in t h e data. It may b e somewhat risky t o use t h e same model in different c o n d i t i o n s for example, w h e n t h e temperature is consistently 5° lower. Temperature has n o t been included in this model at all, and clearly t h e results will be totally off if it c h a n g e s

Process-based model Instead o f further exploring t h e e m p i r i c a l model, let us try t o build a process-based model for t h e m i c r o b i a l system that we are studying

W e will draw on some o f our

understanding o f population growth, c o n s i d e r s o m e o f t h e processes t h a t may b e involved, and describe t h e m in t h e model. T h i s brings up a whole different paradigm

Model Analysis

121

of modeling, where, in addition to the information c o n t a i n e d in t h e data sets, we bring in other information available from similar studies c o n d u c t e d before o n similar systems, or from general ecological theory, or from mass-conservation laws, or simply from c o m m o n sense

For the microbial system that w e are considering, just a s for any other population, the proce s s e s of growth and death are most likely playing an important role.

Perhaps w e can try

to d e s c r i b e t h e life of the whole population in terms of t h e s e t w o p r o c e s s e s . The simplest m o d e of population growth can b e then presented by l h e following Steila equations: Population^ = Population!! - dt) + (Growth - Mortality) ' dt INIT Population = 10 INFLOWS: Growth - GrowthRate*Lim_facior*Population*(l -Population/C_Capac'Uy) OUTFLOWS: Mortality = MortalityRate* Population C_Capacity = 5 0 0 GrowthRate = 0 6 MonalityRale = 0 . 1 5 . W e can alsc. insert the values for the limiting temperature factor a s a graphic: Lim. factor = GRAPH(TIME) ( 0 . 0 0 , 0.3051. 110.0, 0.47), (20.0, 0.65), (30.0. 0 815», (40.0, 0 7), (50.0, 0.505), (60 0. 0.7451. (70.0. 0.93), (80.0. 0 86), (90.0. 0.71). (100. 0 0 0 ) By looking at t h e cata points w e s e e that, after t h e initial period of rapid growth, t h e population size s e e m s to saturate at a certain level As w e have s e e n above, there is a simple way to control growth in t h e model by introducing the Carrying Capacity, which r e p r e s e n t s the maximum number ol organisms that can survive in t h e lab environment With the parame t e r s listed above, the model produces t h e following dynamics:

Hours

122

Systems Scienrf? and Modeling fo' Ecologicst Economics

Curve HI on t h e graph r e p r e s e n t s t h e experimental values that w e -iave been observing. while curve (21 is the simulated behavior

Here, too, w e s e e that there is 3 certain

error or distance b e t w e e n t h e two m o d e l s T h e size of this error d e p e n d s on t h e parameter values used in the model. Let us run sensitivity analysis for the three parameters in this model

ui:i:lcd M Population. I - 2 - I • I

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T h e s e graphics s h o w how the model reacts to c h a n g e s in GrowthRate (from 0 3 to 0.8), C_Capacuy (Irom 3 0 0 to 7001 and MortalityRate (from 0.1 to 0.3) W e may notice that c h a n g e s in growth rate and mortality have a r alher similar eflect. mostly altering how t h e population c h a n g e s during lhe initial growth period As might b e expected, the carrying capacity value defines where t h e population saturates later on W e may already start to make s o m e m e a n ngful c h a n g e s to the p a r a m e ' e r s . trying to make the output closer to t h e data

Model Analysis

r~\ r\ r-1 fi

123

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To k e e p track of o u ' gains and l o s s e s , w e can put together an error model. Described in terms of Stella equations, tne error model might b e a s follows Error(t) = E r r o r l t - d t ) + \Er_ln) ' dt I NIT Error =

0

INFLOWS: E r j n = Population - DATA)*2/DATAA2 This formula reproduces t h e metrics described above a s t h e s u m of s q u a r e s E in >4 U Notice that at each time-step w e add another error term, whicn m a k e s it equivalent to (he summation that w e s e e in (4.1). Keeping in mind t h e results of sensitivity analysts, w e can now start to tweak s o m e o l t h e model p a r a m e t e r and s e e h o w this changes the o i s t a n c e b e t w e e n the data ana poouiahon tnat is also m e a s u r e d by t h e erroi variable. M o s t ukelv the GrowthRate will need to g o down a little to make the nopulation grow slower, but the C _ C a c a o t y snould probably g o up to make it saturate at a higher level. That should bring the moael output s o m e w h a t closer io t h e Data

T H I S IS

an iterative tnal-and-error p r o c e s s that

may or may not g e t us to *.ne oerfec': match

You may have noticed that there is a difference m calibrating empirical and process-based models. In empirical models, we rely entirely o n the information that we have in the dara sets. W c c o m e up with some rype of equation, and rhen quite mechanically adjust t h e parameters in an attempt to reproduce the data as well as possible. A l l t h e information we know about the system is in the data, and the parameters usually c a n take any values as long as the error mode! is minimal. In process-based models calibration is different, since we are restricted by t h e ecological, physical or c h e m i c a l meaning o f t h e parameters that we change. Besides, there ate usually some estimates for the size o f the parameters: they are rarely precisely measured, but at least t h e order of magnitude or a range is usually known. Moreover, there are o t h e r factors that may play a role, such as confidence in the

124

Systems Scienrf? and Modeling fo' Ecologicst Economics available estimates for the parameter, sensitivity o f the model to a parameter, e t c , T h e s e are important considerations in the calibration process. A c t h e bottom of any calibration we have an optimization problem. W e will learn more about optimization in C h a p t e r 8, but here we just want to note that optimization in this case is about seeking a minimum for t h e error model. W e have certain parameters for which values are known and others that are only estimated within a certain domain o f change. W e call the latter ones "free" parameters. T h e s e are t h e ones to change in the model in order to minimize the size o f the error. T o perform optimization, we first formulate a goal function

(also called an objective

function).

T h e n we try to make this function as little (or as large) as we c a n by changing different parameters that are involved. In case o f calibration, the goal function is the error model E — /(P, C , R ) , described as a function o f the parameter vector P , the vector of initial conditions C and the vector of restrictions R . S o we search for a minimum: min £ over the space o f the free parameters P and initial conditions C , making sure that the restrictions R (such as a requirement that all state variables are positive) hold. It is rare that there is a real system model that will allow this task to be solved analytically. Ir is usually a numerical procedure chat requires the employment of certain fairly complicated software. T h e r e are different ways to solve this problem. O n e approach is to do it m a n ually, as we did above with the so-called crial-and-error method or educated-guess approach. T h e model is run, then a parameter is changed, then the model is rerun, the output is compared, the same or a n o t h e r parameter is changed, and so o n . It may seem quite tiresome and boring, but actually this process is extremely useful in understanding how the system works. By playing with che parameters we learn bow they affect output (as in the sensitivity analysis stage), but we also understand t h e synergetic effects that parameters may have. In some cases we get quite unexpected behavior, and it takes some thought and analysis to explain how and why t b e specific c h a n g e in parameters had this effect. I f no reasonable e x p l a n a t i o n can be found, c h a n c e s are there is a bug in the model. A closer look at the equations may solve t h e problem: something may have been missed, or entered with a wrong sign, or some effect may n o t have been accounted for. In addition co the educated-guess approach, there are also formal m a t h e m a t i c a l methods that are available for calibration. T h e y are based on numerical algorithms that solve the optimization problem. S o m e modeling systems have t h e functionality to solve che optimization problem and do the curve fitting for models. O n e such package is M a d o n n a . O n e big advantage of M a d o n n a is that it can also take Stella equations almost as is and run t h e m within its own shell. M a d o n n a also has a nice graphic user interface of its own so it is as well for us to start putting the model together directly in M a d o n n a , if we e x p e c t some optimization co be needed.

To d o the parameter calibration for our Stella model in Madonna w e will have to: • Go t o the Stella equations • Save t h e m as a text file (File - > Save As Text) • Open t h e file f r o m Madonna, using t h e Open c o m m a n d in t h e File m e n u

Model Analysis

125

« (Alternatively you can " c h o o s e all" and " c o p y " the equations from Stella, and then " p a s t e " them directly into an Equations window in Madonna: however, in this c a s e you will have to remove al! t h e "INFLOW:" and "OUTFLOW:" s t a t e m e n t s in t h e equations by hand) • Define the control s p e c s such as t h e STARTTIME, STOPTIME. and DT The model is now ready t o run in Madonna. Fiunmng the s a m e population model, built now in Madonna., w e g e t the following output, which is - not surprisingly - identical to the Stella output:

500 • 450

c "00

1

350 SIARTTIM6 -0 STOPTIME -!00 DT -0 02 OTMIN .10-6 OTOJT -0 -0 001 HOOrrOL -0 IMT prior INIT Population-10 C Capacity -soo Growlnftale -oe Morualilyflale -0 15

o. 3 0 0 |

250

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100 50

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40

50 Time

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As w e start running t h e model, t h e first thing w e notice is that Madonna runs much faster than Stella. That is b e c a u s e in contrast to Stella, which interprets the equations on t h e fiv, Madonna has a built-in compiler that first compiles our model and only then runs it. On s o m e models, t h e difference is quite significant, up to orders of magnitude. This is especially essential f c optimization, since all optimization algorithms require numerous model runs t o be performed. The next thing w e need lo do to calibrate our model is input t h e data into Madonna. This is done as pail of the optimization dialogue, which in this c a s e is called Curve Fitting In t h e " P a r a m e t e r s " menu, w e c h o o s e "Curve Fit...." A dialogue box w>n open:

Available: INIT Crror INIT Population i r.,i .viK (".-ov-ttiiii',.. MGi1.iliiyH.if.:

C_Capaclty 6rowtllRatC MortalityRate | t, Remove « |

: Minimum: Guess«1: . I

111 Variable: I Population ,i| 10 Dataset: (*catlt>data r) | Import Dataset- |

Guess *?; Maximum:

• Multiple fit*: Add 11 1 Remove
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126

Systems Scienrf? and Modeling fo' Ecologicst Economics

Here, w e need to specify four items 1. Choose the free parameters that can be changed for model calibration 2, For each parameter, identify the maximal and minimal allowed values, and t w o "guesses" values in the domain of change that will be used to initialize the optimization process 3. Choose the state variable that w e are calibrating - "Population" in this case 4, The data set to which w e w i s h to calibrate the model - "#calibdata" in this case The data set should be in a file, one value on a r o w which can be generated, say, f r o m Excel if t h e data are saved as Text O n clicking the " I m p o r t Data set. ." button, w e will be given the opportunity t o choose the file w i t h t h e data. Now, if w e press the " O K " button, s o m e number crunching will begin; after 144 model runs w e will get a n e w set of parameters that provides a much closer fit b e t w e e n t h e data and the simulation model, The n e w values for t h e model parameters are: C_Capacity = 5773, GrowthRate - 0,42061, MortalityRate - 0.0760 512

T h e calibration problem may n o t have a unique solution. T h e r e may be several parameter vectors P that produce almost similar output or deliver the same or almost the same minima to t h e optimization task. In that case, it may be unclear what

parameters

to choose

for the model.

Other

considerations

and restric-

tions may be used to make the decision, For instance, with C _ C a p a c i t y = 6 0 0 , G r o w t h R a t e = 0 . 5 , M o r t a l i t y R a t e = 0 . 1 , we get a fit almost as good as that achieved with Madonna. W h i c h o f the two parameter sets should we choose for the model? Normally this decision is made based o n the other information about the system that is available. For example, there may be some experimental data that would either identify a value for o n e o f the rate coefficients, or at least put a range o n them. T h e n we c a n see which o f the calibrated values is in better agreement with these restrictions. In some cases this information may n o t be available, and there may be some uncertainty about the system. T h i s c a n further drive our experiments with the system, or tell us more about the system behavior. Suppose we have done our best when finding t h e values for all the parameters in the simulation model and yet still the error is inappropriately large T h i s means that something is wrong in o n e o f the models that we are comparing. Either t h e c o n c e p tual model needs to be revised ( t h e structure changed or the equations modified), or the chosen scales were incorrect and we need to reconsider the spatial or temporal resolution. Alternatively, the data are wrong - which happens quite often, and c a n never be dismissed as a possibility. T o conclude, there are different ways to describe systems by means o f models. T h e r e are different models that may be built. The process match the output from another

model is called calibration.

of adjustment

of one model to

T h i s is probably the most gen-

eral definition, In most cases we would speak o f calibration as t h e process o f fitting the model output to the available data points, or "curve fitting." In this case, it is the data model that is used to calibrate the m a t h e m a t i c a l model. Note that there is hardly any reason always to give preference to the data model. T h e uncertainty in the data model may be as high as the uncertainty in the simulation

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130

Systems Scienrf? and Modeling fo' Ecologicst Economics T h e solution is to approximate the parameter values based on che data we have abouc che dynamics of state variables, or flows. T h a t is the model calibration procedure. W e are solving an inverse problem: finding t h e parameters based o n t h e dynamics o f the unknowns. T h i s would be fine if we could really solve that problem and find t h e exact values for the parameters. However, in most cases this is also impossible and, instead, we are finding approximate solutions that c o m e from model fitting. But then how is this different from the fitting we do when we deal with empirical models? In that case, we also have a curve equation with unknown coefficients, which we determine empirically by finding che best c o m b i n a t i o n o f parameters that make the model output as close as possible to the data. T h e only difference is that instead ot some kind of generic equation in the empirical models (say, a polynomial of some form), in process-based models we have particular equations that have some ecological meaning. T h e s e equations display certain behavior by themselves, no matter what parameters are inserted. A polynomial c a n generate pretty much arbitrary dynamics as long as the right coefficients are chosen. However, an equation of exponential growth will always produce an exponent, and, say, a classic predator-prey system (considered in the next chapter) will always produce oscillations, n o matter what coefficients we insert. O t course, for some parameters they may crash even before generating any meaningful output, but otherwise the dynamics will be determined by the type o f equations used, at least tor a large enough range of coefficients. S o we may conclude that, to a large extent, we are building a good model as long as we chose the right dynamic equations to describe our system. O n top o f the basic dynamic equations we overlay t h e many o t h e r descriptions for the processes that need to be included in the model. T h e s e may be the limiting factors, describing che modifying effect o f temperature, light or other external conditions. T h e r e may be some other details that we wish to add to the system. However, if these processes are n o t studied experimentally, and if the related coefficients are not measured, their role in the model is n o different from that o f the coefficients that we have in an empirical model. Ir\ b o t h cases we figure out their values based on a time-series o f model output; in both cases the values are approximate and uncertain. T h e y are only as good as they are the best ones found; we c a n never be sure that a better parameter set does n o t exist. S o the bottom line is that there is a good deal o f empiricism in most processbased models, and the more parameters we have estimated in the calibration process, the more empiricism is involved, t h e less applicable the model will be in situations outside the existing data range. How c a n we make sure that we have really captured the essence o f the system dynamics, and can reproduce t h e system behavior beyond the domain that we have already studied? T o answer these questions, the model needs to undergo a process o f vigorous testing. T h e r e is not (and probably never will b e ) a definite procedure for model testing and comparisons. T h e obvious reason is that models are built for various purposes; their goals may be very different. Moreover, these goals may easily change when the projecc is already underway. T h e r e is no reason why goal-setting should be left out of the iterative modeling process. A s we start generating new knowledge and understanding with a model, its goals may very well change. W e may start asking new questions and need to modify the model even before it has been brought to perfection. Besides, ecological and s o c i o - e c o n o m i c systems are open, which makes their modcling like shooting at a moving target. W h i l e we are studying the system and building a model o f it, it is already evolving. It evolves even more when we start administering control, when we try t o manage the ecosystem. A s a result, models can very well

Model Analysis

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132

S y s t e m s S c i e n c e and M o d e l i n g fo< Ecological Cconorrvcs

.ir«™ ten-peratu'e snrfu. including internal oscillations .1 tne o r n a t e system, and exieinai f o r c n g . PCA1&/ from the There a>0 i w i W f l O W f i * V J ^ H ^ o n i x i r c c J SOlflf CytfOS 0< 8 7 t r d 2»0 y m the so-ra&ed O e V n e s and Gteissberg s d a r cycles, but no o r e so far has oetected a 1.470-yeei soiar eveta • t t - H n u r t & i <-r».gh, w h e n «o»oed by pottx&c freshwater i n c u t into t h e N o t h Atlantic Ocrsc in rydes o*. 8 7 a n d 210 years CLIMBER simulated rapid climate shifts similar to the D O events * n 3 Of 1 . 4 * 0 vesrs T n a robust 1.47&y»ar response time car- be r

attributed to tne s t j p e r p o s t o n o» t h e t w o snorter cycles, together w i t h strong y nonlinear dynamics and the Jong character.stc t m e s c e i e of the thermohaline circulation For Holocene c o n a t o n s . w m a » events d a r e t o c a * R axJd therefore be concluded that the glacial 1 470v«ar d - a > e cycles c o u U nave oeen t u g g e d Cy s o a r k / O ^ g , despite mo absence o1 4 1 4 'C-year *~iat c * d o A fteouencv o< 1.470 years « " o t k x n d m (he forcing, i t « found onSr m the mooe< response. The s u o y is r w a i m e d at suggesting • certain m e d r i m for solar r f l u e n c e o n f r e s N v t l e r r u * e s r v s s n t x i d t * S O K J ^ O w i t h r r o r e detailed and highe*resolution models The simplified approach tmpfees that tne k n o w n solar frco..enc«n$ are present 01 the h y d r o i o ^ t a * Cyc»e. »nd that transfctes n t o tne D - 0 cycles by the complex and non-linear nature of t n e s y s t e m Whenever t h e modof can reproduce s o m w h n g mat is oesserved but was not used t o build the model, it speaks m favor of m o d e l vak±tv Moleover. >n 2001 m o d e l fer-. -s wenp u b shed show-ng thai A'c&c and A n t r c t c t c - n p e r a a r e s c y d e out of ohase ~ » v e w e r e t o m o records f r o m the South Atlantic Ocean and p a r s of Antarctic* that s h e w that the cokl events in the N o n h Atlantic w e r e a s s o c i t e o w i t h unusual w a r n i n g there rcha "cipalar seesaw effect*) However, it w a s only 2006 that the EPtCA l e a m r h e E u r o p e * - Proje-v: toice Coring in Antarctica) published t h e * data tn»t connect n e A m w « < up» and d o w n * 01 cfcmelo w i t h the much greater ones of Greenland Once aga
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Model Analysis

A n o t h e r important step in model analysis is verification.

133

A model is Verified

when it is scrupulously c l u c k e d for all sort of internal inconsistencies, errors a n d bugs. T h e s e a m he tn the equations chosen, in the units used, or in links and c o n nections. T h e r e may simply be p ^ g f i i n m i n g buys in the code that is used to solve the model on rhe computet, or there may be conceptual errors, when wrong d a n sets are used to drive the model. O n c e again, there is hardly a prescribed method t o weed these out. just c h e c k at id retdieck. R u n the model and rerun ii. Test ir and test again. T h e r e is n o agreed procedure for model verification, especially when, models b e c o m e c o m p l e x and difficult t o parameterize and analyse. W e just keep studying its behavior under all sorts of conditions. O n e efficient method of model testing is t o run rhe model with extreme values o f forcing functions and parameters. T h e r e are always certain ranges where che forcing functions can vary. Suppose we are talking about tempera lure. W e make the t e m perature' as high as it possibly c a n be in a particular system, or as low as it c a n be, and see what happens t o the model. Will it still perform reasonably well? Will the output stay w i t h i n certain plausible values, or will t b e model crash? if so, we need co try co figure ouc why. Is it something that can be explained? If so, t h e n probably die model can be still salvaged and we may simply need to remember that die forcing function should stay within certain allowed limits. If t h e behavior cannot be explained, we need to keep digging - most likeiy, there i.s something wrong. just as when we are testing a new car, the best way to find out how it performs is to force it. S t e p o n the pedal, and let it tun as fast as ic c a n . S e e if something goes wrong, and where it might fail. T h e beauty of testing tbe model is that it is n o t wrecked when >t goes wrong! If we force the car t o o hard, we will ruin it. W i t h t h e model, we c a n do whatever we want t o it - change all the parameters a,s much as we wish. If che compucer does not o v e r h e a t , we can always go back t o previous parameter values, and the model will run again like new. However, we will collect some valuable information about what to expect from it, where the bugs and the features are, what we c a n let users do t o it, and where we should add some limits co make sure chey do not have surprises that we c a n n o t explain. A n o t h e r important c h e c k is based on first principles, such as mass and energy conservation. Ic is important to make sure that there is a mass balance in the model, so that nothing gets created from nothing and nothing is lost. R u n n i n g scenarios is another great way t o test a model. T h i s step may already be considered as model use rather than just testing. A scenario in this c o n t e x t is a story about what c a n happen to the system. T o define a scenario, we need t o formulate all t b e forcing functions (say, patterns o f climate, or pollution loading, or land use patcerns) and all the control parameters (say, management rules, or external global variables). In a way, we are modeling what the external forcings are to which the system wil' be reacting. For example, if we arc considering a model o f landuse c h a n g e for an urban area, w^e c a n formulate a so-called "business as usual" scenario that will assume thac all the existing development trends c o n t i n u e inco che future, che population, the economy, the investments, etc, will c o n t i n u e io grew at t h e same rate, there will be n o additional controls or limits incroduced, or climatic perturbations, etc. 'I hese we feed into die landuse model and run it to generate patterns of landuse under this scenario. W e may then figure out a different scenario - perhaps a sustainable development plan. W e will need Co formulate this in terms o f the model. T h i s means that we translate the sustainable d e v e l o p m e n t plan into t b e parameter values and forcing functions that will most closely describe that. In a way, we model what we think will

134

Systems Scienrf? and Modeling fo' Ecologicst Economics be a sustainable future. In our case we may assume that there is a control over population growth, so that certain birth-rate reductions are introduced. Furthermore, we will tie e c o n o m i c growth to the natural resources that are available in the area, and make the growth rate slow down as natural capital gets depleted W e c a n also include some rules for investments that would stimulate t h e green economy. As a result, we will get a different set o f parameters that control the model, and the model run will now produce some different pattern o f landuse as a result of this scenario. Yet another scenario can be put together for devastating climatic conditions - say, a storm that will flood the area and destroy property and population. W e will need to formulate some climatic conditions describing this storm. O n c e again, we are modeling certain conditions ot forcings for the system. N o t e that scenarios are also models, coherent and feasible models of external conditions that will then drive the model of the system that we are studying. N o t e that scenario runs are also powerful tools o f model testing. In this case, we are likely to explore the unknown domains of model parameter values. W e do not have the data about the model behavior that we might expect, but we do want the model to produce something qualitatively reasonable. If that does n o t happen, we may question t h e model validity and have some clues where to look for errors. For example, if a model o f Sustainable growth results in patterns o f further urban sprawl, this would be a warning indicating that something is not working right in the model. W e should take a closer look at the formalism we used, or perhaps at the parameter values that we calibrated. T h e bottom line regarding all this testing is that there is no perfect model. It is hardly possible to get a perfect calibration, and the validation results will likely be even worse. N o matter how long you spend debugging the model and the code, there will always be another hug, a n o t h e r imperfection. Does this mean that this is all futile? By no means! As long as we reach new understanding o f the system, as long as the model helps to c o m m u n i c a t e understanding to others and to manage and control the system, we are o n the right path and our efforts will be fruitful. Any model that is useful ts a good

4.4

model.

Conclusions O n e obvious conclusion from all the above is that putting the model together is not just about establishing variables and c o n n e c t i o n s and writing the. equations for them. W e also need to do a lot of number crunching, running the model many times. If t h e model is c o m p l e x and requires a great deal o f computer power to run it, we will be limited in the e x t e n t o f testing and improving that can be done with the model. W e will have to be prepared to do che j o b o n our slow desktop (and spend more time), or we will need t o find a more powerful super-computer (and spend more $ $ ) , or we will have to limit ourselves in the amount o f testing and calibrating that we c a n do (and get a poorer model and less well-understood system). Yet a n o t h e r option is to go back to the model design ^tage and try t o simplify the model. T h e r e is a potential C a t c h - 2 2 in this process. O n the o n e hand, t h e more information about the system we cnn use in our model, the more processes we c a n include and the more detail about these processes we c a n formalize, t h e better our model should be and the more it should be able to tell us about the real system. O n the other hand, the more complexity there is built into the model, t h e longer it will take

Mrtrtol Arifllysis

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136

Systems Scienrf? and Modeling fo' Ecologicst Economics the decision t o be made. T h i s is probably a good way coi frame it. Here, we include b o t h die model goaI and t h e model users in the evaluation process: Indeed, there is no use talking about some overall .universal model v a l i d l y ; rhe model in valid only with respect to the goals that it is pursuing, and only the users o f die model c a n define whether it suits their needs or n o t . T h e r e is a good deal ot concern about the uncertainties that are inherent in almost any modeling effort. Prettv much any stage o f the modeling process is full ol uncertainties. W e shirt from the goals of the study and immediately we realize that there are different expectations that various users may have for a model. T h e goals are communicated in some linguistic form, i:i words, and this m itself is a model of a collection of thoughts or ideas about what we want. S u c h models already may he fumy, and may change as the mind, knowledge and ideas of people evolve- Especially when we are dealing with socio-economic processes that include people, their opinions, and priorities, we immediately enter a realm of huge uncertainty and much guesswork. Very much like in quantum physics, where rhe mere o c c u r r e n c e of i h c experim e n t influences its results, so it is 1:1 social work, where, lor example, hy polling people and asking t h e m a question we immediately hias the o u t c o m e hy how we ask the question and by the simple fact ol the question, which already can make people think differently from how they might have done without being exposed to the question. "How do you value that forest?" Well, c h a n c e s are the respondents even noticed the forest and could not care less about us existence that they are asked about it, they may start thinking

never

However, now

" S o why would they ask m e ?

Actually yes, there is that lorest. A n d 1 remember going there as n kid. O n c e . A n d it was pretty cool. A n d how am 1 going t o look if 1 say that I don't care about this forest? No, probably 1 should say that I value it at least somewhat- And maybe actually there is value in it, or why would they ask o t h e r w i s e ' " W e see that the response already different from what is was supposed to be at first T h e person quickly built a mental model, analysed it and produced an answer, which in fact is still lull of uncertainties, especially since we will never know what the real chain ot thought was anil what intermediate evolution t h e person's mind had gone through. It does n o t get any better as we step up to the next stages o f model building. As we have already seen, we hypothesize alt sorts ot things about a system when we model it. Besides, we need to simplify it, introducing even more uncertainties. A n d then of course there is all ot the calibration process, when looking at the sensitivity test should be enough t o realize that different parameters c a n result in a dramatically different model output, A model that does not have much sensitivity to its parameters, thai is quite robust, will be adding less t o the overall uncertainty t h a n will a model that 15 very sensitive to certain parameters. Sensitive parameters then need t o be measured with especially high accuracy, which may not be possible in some cases. Obviously, as models b e c o m e more c o m p l e x , overall uncertainty also grows very fast. In some cases, greater complexity c a n make the model more robust to variations m parameters; however, this normally comes at the expense ot overall model c o n trollability, when the c o m p l e x model starts to operate as an entity in itself, and we approach t h e Bonnini paradox situation - that is, we replace the real-life complex system by another complex system - t h e rnodelStill, we will model. T h e r e is simply n o o t h e r better way to perform analysis and to produce synthesis. W e have to find a way to simplify a complex system if we want to understand it. As long as we are ready to go back, to try again, to reiterate and test, test, test, we will eventually e n d up with a useful product. A n d if it is useful, it means that the model we have built is a good one.

M o d e l Analysis

137

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. Simple M o d e l , C o m p l e x Behavior Classic predator-prey model Modifications of the classic model Trophic chains Spatial model of a predator-prey system Conclusions

SUMMARY N o n - l i n e a r systems are chose that can generate the most unusual and hard to predict behavior. A system o f two species where o n e eats the o t h e r is a classic e x a m p l e o f such non-linear interactions. T h e predator-prey model has been well studied analytically and numerically, and produces some very exciting dynamics. T h i s simple twovariable model can be further generalized to explore systems o f many species that are linked into trophic c h a i n s . Further complexity is added when these populations are considered spatially as so-called metapopulations.

Keywords Lotka-Vol terra model, non-linear systems, trophic function, equilibrium, phase plane, carrying capacity, M o n o d function, Kolmogorov theorem, periwinkle snail, e v e n and odd trophic levels, Yellowstone wolves, Stella arrays, Simile, Spatial Modeling Environment ( S M E ) . W ** Two-state-variable systems have been honored with t h e most attention from mathematical modelers. T h i s may be readily explained by t h e dramatically increasing complexity of mathematical analysis as the number of variables grows. A s seen previously, it is only the simplest models chac can be treated analytically. O n the other hand, two state variables produce m u c h more interesting dynamics than one variable, especially if there is some non-linearity included. Mathematically, such systems are more challenging and certainly moie rewarding. All sorts of exciting mathematical results have c o m e from analysis of these systems

In addition to advancing mathematics, analy-

sis of these simplest two-state-variable systems has provided a wealth ol results thai may have important ecological implications and are certainly interesting in the art o f modeling even in more general and complicated cases. O n e ot the first and also best-studied communities is the so-called " p r e d a t o r prey" system, where organisms o f o n e population serve as food for those of the other. V u o Volterra studied tish populations, and in 1926 formulated a model that turned

139

140

Systems Science and Modeling for Ecological Economics out to be very insightful regarding t h e understanding of p o p u l a t i o n d y n a m i c s . Alfred L o t k a proposed t h e same m o d e l in 1 9 2 5 , so t h e m o d e l is s o m e t i m e s k n o w n as t h e L o t k a - V o l r e r r a m o d e l , or jusr the V o i t e i r a model, s i n c e ir was he w h o did most o f t h e m a t h e m a t i c a l analysis.

5.1

Classic predator-prey model S u p p o s e we are c o n s i d e r i n g a p r e d a t o r - p r e y system, w h e r e rabhits are the preys and wolves are t h e predators. T h e c o n c e p t u a l model for this system c a n be presented hy t h e simple diagram in Figure .5.1. In this case we are not c o n c e r n e d with t h e effects ol t h e e n v i r o n m e n t upon t h e c o m m u n i t y , and focus only o n t h e i n t e r a c t i o n s b e t w e e n t h e two species. L e t x ( t ) be t h e n u m b e r o f rabbits arid i ( t ) be t h e n u m b e r of w : olves at time t. S u p p o s e t h a t t h e prey p o p u l a t i o n is limited o n l y by t h e piedator and, in t h e a b s e n c e ot wolves, rabbits multiply e x p o n e n t i a l l y . T h i s c a n be described hy t h e e q u a t i o n :

W h e n t h e wolves are brought i n t o play, they start to c o n s u m e rabbits at a rate o f V = V ( x ) , w h e r e V(A) is che n u m b e r o f rabbits chac e a c h wolf c a n find and eac o v e r a unit t i m e . N a t u r a l l y this a m o u n t depends on t h e n u m b e r o t rabbits a v a i l a b l e , x, because when there are just a few rabbits it will be harder for t h e wolves to find chetu t h a n when t h e prey are everywhere. T h e form of t h e f u n c t i o n for V ( x ) may be differe n t , but we may safely assume that it is m o n o t o n e and increasing T h e n t h e e q u a t i o n tor rabbits will be — dt

= ax -

V(x)y

(5.2)

T h e growth ot t h e wolf population is determined by the success ot t h e wolves' h u n t i n g activities. It makes sense t o assume that only a c e r t a i n part of t h e biomass ( e n e r g y ) c o n s u m e d is assimilated, while some part ol it is lost. T o a c c o u n t lor that, we describe t h e growth of t h e wolf population as kV(x);y, where 0
(5.3)

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5.1

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A simple conceptual model for a predator-prey system: wolves eat rabbits

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Iri [he absence of rabbits, the wolf population exponentially decreapSi V{.x} is called rhe trophic function, and it describes the rare of predarion as a function of the prey -abundance- T h e form of the trophic function is species-specific, and may also depend upon environmental conditions. Usually it grows steadily when the prey population is sparse, but then tends to saturation when the prey becomes abundant. Holliri" has identified three main types ot trophic functions, as shown in Figure 5.2. T h e hrst two types o f the trophic functions { A , B) are essentially the same, except that in case B the function has a well pronounced saturation threshold, l he third type of trophic function behaves differently for small values ot prey densities. It tends to ~ero with a lero derivative, which means that near zero the trophic function decreases taster than the prey density. T h i s behavior is found in populations that can learn and find retuge trom the predator. For such populations there is a better c h a n c e to persist, because the predator cannot drive the prey to total extinction. Volterra considered the simplest case, when the trophic function is linear. This corresponds to function B below the saturation threshold- T h e wolves are assumed to l^e always hungry, never allowing the rabbits to reach saturation densities. T h e n we can think that the trophic function is linear: V = fix. T h e classical Volterra predator-prey

model

is

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formulated as: c/x— = xx dt ^ dt

ix (5-4)

k(Sxy-m

It can easily be seen that this system has two equilibria. T h e first is the so-called trivial one, which is when both the wolves and the rabbits are driven to extinction, x = 0, y — 0. T h e r e is also a non-trivial equilibrium when \ v — [i/hf},

y* = Dcjji.

Obviously, it the community is at an equilibrium state, it stays there. However, the chances that the initial conditions will exactly hit the non-trivial equilibrium are null. Therefore, it is important to find out whether the equilibria are stable or not. For a simple model like this, some qualitative study of the phase plane may precede further analytical or numerical analysis of the model. In fact, we may note that when there are more rabbits than at equilibrium (x > x^). the population of wolves decreases (dyldt < 0). T h e opposite is true when .v < x * . Similarly, when there are more wolves than at equilibrium (y > )'*}, the population ol rabbirs declines (dx/di < 0); it grows

Figure 5.2

Different types of lhe Irophic function, according to Hailing

142

Systems Scienrf? and Modeling fo' Ecologicst Economics w h e n y < y*. W e may therefore break t h e phase plane into four areas and in e a c h o f t h e m show t h e d i r e c t i o n o f t h e trajectory o f t h e m o d e l solution (Figure 5 . 3 ) . T h i s qualitative analysis already shows t h a t there appears to he s o m e cyclic m o v e m e n t around t h e equilibrium point

T h e trajectories are likely to wind around this

point. T h e r e is still a c h a n c e that t h e point is stable, in w h i c h case we start circling around t h e equilibrium, gradually m o v i n g hack into t h e c e n t e r . However, rhis qualitative analysis only indicates that t h e model trajectories will loop around t h e n o n - t r i v i a l equilibrium, but it is n o t clear w h e t h e r these loops form a spiral converging towards t h e equilibrium ( p o i n t stable) o r w h e t h e r t h e spiral will he heading away from t h e c e n t e r ( p o i n t u n s t a b l e ) . In a n y case, we may e x p e c t oscillations in populations o f rabbit a n d wolf. Let us see what a simple S t e l l a model c a n tell us about the d y n a m i c s in t h e predat o r - p r e y system (Figure 5 . 4 ) . You c a n e i t h e r put t o g e t h e r a model yourself for further analysis, o r d o w n l o a d it from t h e book website. T h e phase portrait very well m a t c h e s our e x p e c t a t i o n s . W e d o get t h e loop t h a t b e h a v e s e x a c t l y as out q u a l i t a t i v e analysis predicted. A s e x p e c t e d , the model produces c y c l i c b e h a v i o r , w h e r e an e x p l o s i o n in t h e rabbit

population

is followed by a peak in t h e wolf p o p u l a t i o n . T h e rabbits are t h e n wiped o u t , after

Figure 5.3

The direction of change on the phase plane for the Volterra model

In I, both x a n d /decline; in II, xdeclines as / g r o w s , in III, x g r o w s and / f a l l s , in IV, both x a n d / g r o w

Figure 5.4

The Stella diagram for the predator prey model

Simple Model, Complex Behavior

143

w h i c h t h e wolves die from s t a r v a t i o n , a l m o s t co e x t i n c t i o n . W h e n t h e r e a r e very fewwolves left, t h e rabbits start t o multiply again a n d t h e p a t t e r n recurs (Figure 5 . 5 ) . If we run t h e m o d e l with the Euler m e t h o d , we s e e t h a t t h e r e is n o trend towards t h e e q u i l i b r i u m m t h e c e n t e r , a n d t h e a m p l i t u d e o f t h e o s c i l l a t i o n s gradually increases until t h e system crashes. However, if we switch t o t h e R u n g e - K u t t a

fourth-order

m e t h o d , we find t h a t actually we get a closed loop in t h e phase plain. P o p u l a t i o n s of b o t h wolf and rabbit follow t h e s a m e identical trajectory, going through t h e s a m e p a t t e r n o f o s c i l l a t i o n s (Figure 5 . 6 ) . T h e r e is n o c o n v e r g e n c e towards t h e equilibrium in t h e c e n t e r , a n d n e i t h e r is t h e r e a r u n - a w a y from it, w h i c h we erroneously susp e c t e d at first w h e n r u n n i n g t h e m o d e l with t h e Euler m e t h o d . H o w e v e r , unless we find a n a n a l y t i c a l solution we c a n n o t be really sure t h a t this will b e t h e kind o f b e h a v i o r chat we g e t u n d e r all c o n d i t i o n s a n d c o m b i n a t i o n s o f parameters. Luckily, in che cime o f V i t o Volterra t h e r e were n o c o m p u t e r s a n d h e studied t h e m o d e l q u i t e rigorously, a n a l y t i c a l l y proving thac t h e model crajcccories always loop a r o u n d t h e equilibrium p o i n t . 1: may be n o t e d thac t h e inicial c o n d i t i o n s Curn ouc to b e very important for t h e o v e r a l l a m p l i t u d e o f t h e c y c l e . N o t e chac if all che paramecers stay c h e s a m e hue t h e initial c o n d i t i o n s are modified t h e system still produces a c y c l e , alchough its form may c h a n g e quire dramacically. T h i s is a s o m e w h a c u n e x p e c t e d result, showing t h a t t h e current state o f che svscem depends very m u c h upon che scace o f t h e system a c o n s i d e r a b l e length o f t i m e ago, w h e n t h e initial c o n d i t i o n s were established to start up t h e process. T h e c h a n g e s in t h e p a r a m e t e r values also d o n o t c h a n g e t h e overall form o f t h e t r a j e c t o r i e s , w h i c h are still looping a r o u n d t h e n o n - t i i v i a l equilibria. H o w e v e r , they d o m o v e t h e loops o n che phase plane (Figure 5 . 7 ) . Seel la is unlikely Co get che loops using any o t h e r m e t h o d o f integration t h a n fourth-order R u n g e - K u c c a . T h e Euler m c c h o d quickly resulcs in increasing oscillations

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The dynamics of prey and predator in the Volterra model as solved by the Runge-Kutta

fourth-order method. A Graphs for the Wolves and Rabbits B Phase portrait for the Volterra model. that eventually e x p l o d e t h e system. S e c o n d - o r d e r R u n g e - K u t t a persists for longer, but eventually also tends to fall apart. T h i s is a n o t h e r illustration o f t h e i m p o r t a n c e o f careful c h o i c e of d i e time-step and rigorous analysis of t h e influence of t h e timestep upon t h e simulation results. If there were n o analytical solution available for t h e Volterra model a n d we had been running ir with t h e Euler m e t h o d in Stella, we would have b e e n getting qualitatively different results, and would n o t even be suspecting that the true d y n a m i c s o f t h e system are rorally different. T h e m a j o r result thar c o m e s from t h e Volterra m o d e l is t h a t p o p u l a t i o n c y c l e s o l t e n registered in held studies may be e x p l a i n e d by s o m e i n t e r n a l d y n a m i c features o f t h e system. T h e y d o n o t necessarily s t e m from some e n v i r o n m e n t a l forcings, such as t h e seasonal v a r i a t i o n s in c l i m a t i c (actors. C y c l e s may o c c u r simply as a result o f i n t e r a c t i o n b e t w e e n t h e two species.

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5.00 - i 1 Rabbits 2: Wolves

25.00 Figure 5.8

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Dynamics of Rabbits and Wolves with carrying capacity introduced for Rabbits.

3.00 Rabbits v Wolves

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Phase portrait for the Volterra model with prey saturation run with different initial

conditions.

the system dynamics does not depend upon t h e initial c o n d i t i o n s . T h e c o e x i s t e n c e state appears t o IK? stable, and the oscillatory behavior is o n l y iransieni (Figure 5 . 9 ) . A s might be e x p e c t e d , t h e m o d e l also h c c o m e s more robust with respect t o t h e n u m e r i c a l m e t h o d for its s o l u t i o n . W e c a n safely run t h e m o d e l with Euler m e t h o d a n d m u c h larger time-steps, y e t still arrive at t h e Mine steady s t a t e (Figure 5 . 1 0 ) . Let us c o n s i d e r s o m e further a d j u s t m e n t s for t h e Volterra model. A s n o t e d a b o v e , a n o t h e r simplification in t h e m o d e l , t h a t was hardly realistic, was t h e assumption regarding t h e linear t r o p h i c f u n c t i o n . T h e wolves r e m a i n e d equally hungry, n o m a t t e r h o w many rabbits r h e y had already e a t e n

T h i s s e e m s unlikely. Let us n o w

148

Systems Scienrf? and Modeling fo' Ecologicst Economics

3 00 - i Y

R a b b i t s v. W o l v e s

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X

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Phase portrait for lhe Volterra model with prey saturation run using Runge-Kutta (blue) and Euler (red) methods c h o o s e a H o l l i n g type II f u n c t i o n a l response, assuming a M o n o d t r o p h i c f u n c t i o n , t o describe how wolves e a t rabbits:

H e r e , /> is t h e m a x i m a l growth rate a n d K is t h e half-saturation c o e f f i c i e n t . T h e f u n c t i o n m a k e s sure t h a t t h e process ( p r e d a t i o n , in this c a s e ) o c c u r s with s a t u r a t i o n at p, a n d it r e a c h e s P I 2 w h e n t h e prey p o p u l a t i o n is equal t o K ( t h i s e x p l a i n s t h e " h a l f ' in t h e n a m e )

T h e f u n c t i o n is identical co che M i c h a e l i s - M e n t e n

function

t h a t we e n c o u n t e r e d a b o v e : for s o m e reason in p o p u l a t i o n d y n a m i c s it is k n o w n as the M o n o d f u n c t i o n , w h i l e in c h e m i c a l k i n e t i c s it is k n o w n as M i c h a e l i s - M e n t e n . T h e d y n a m i c s in this model are s o m e w h a t similar t o those in t h e classic m o d e l . W e get n o n - d a m p i n g o s c i l l a t i o n s for t h e variable, or a c y c l e in t h e phase plane. However, t h e r e is a m a j o r d i f f e r e n c e : now, different initi.il c o n d i t i o n s result in l h e s a m e limit c y c l e . N o m a t t e r where we start, we e n d up looping a l o n g t h e s a m e trail in t h e phase plane. T h i s is c a l l e d a limit cycle,

a n d it is stable (Figure 5.1 1). T h e r e are

m a t h e m a t i c a l m e t h o d s t o prove that t h e c y c l e in this case is indeed stable; h o w e v e r , chis is a bic coo c o m p l e x t o describe h e r e . A5 in t h e previous c a s e , when prey growth was stabilized by carrying capacicy, h e r e again t h e model c a n be solved by t h e Euler m e t h o d as well as by R u n g e - K u t t a . W h e n e v e r you h a v e a " s t a b l e " situation t h a t a t t r a c t s t h e t r a j e c t o r i e s , Euler works too. T h e c y c l e it g e n e r a t e s will be slightly different from t h a t w h i c h t h e R u n g e K u t t a m e t h o d derives, but qualitatively t h e b e h a v i o r o f t h e system will be i d e n t i c a l .

K o l m o g o r o v ( 1 9 3 6 ) considered a vers' general system t h a t c o v e r s all t h e cases studied a b o v e . H e analyzed a system o f two ordinary differential e q u a t i o n s : ~ dt

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Simple Model, Complex Behavior

149

1 - 2 : Rabbits v W o l v e s

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W e c a n sec t h a t Volterra's system is a special c a s e of t h i s system; however, t h e r e are m a n y o t h e r systems t h a t c a n be also described by t h e s e e q u a t i o n s - t h e Volterra system is j List o n e of t h e m . T h e f u n c t i o n s y.(x), V(.v) and

c a n be any, a l t h o u g h

as l o n g as we are describing p o p u l a t i o n d y n a m i c s they h a v e t o c o m p l y with c e r t a i n o b v i o u s restrictions: 1. dzldx < 0 ; « ( 0 ) > 0 >

- this is to say that t h e prey birth rate is decreasing

as t h e prey population grows ( t h e d e r i v a t i v e o f o v e r x is less t h a n 0 ) , g o i n g from p o s i t i v e t o n e g a t i v e values. T h i s is s o m e t h i n g we were yetting w i t h t h e carrying c a p a c i t y f u n c t i o n in ( 5 . 5 ) , w h i c h is q u i t e a natural assumption for p o p u l a t i o n s with i n t r a s p e c i f k c o m p e t i t i o n a n d a limited resource. W i t h t h i s a s s u m p t i o n , e v e n with n o predator t o c o n t r o l it t h e prey p o p u l a t i o n grows, but it is t h e n stabtli:ed at a c e r t a i n value given by t h e e q u a t i o n a ( x ° ) = 0 . 2 . dK/dx

> 0 ; X ( 0 ) < 0 < K(«>) - this is t o m a k e sure t h a t t h e p r e d a t o r birth rate

increases w i t h t h e prey population. It starts with a n e g a t i v e value, w h e n there is n o food a v a i l a b l e , a n d t h e n increases to positive values. 3 . V ( x ) > 0 for x > 0 ; and V ( 0 ) = 0 - this is to make sure that t h e t r o p h i c f u n c t i o n ts positive for all positive values of t h e prey p o p u l a t i o n . It also equals zero w h e n t h e r e are n o prey. U n d e r t h e s e c o n d i t i o n s , system ( 5 . 6 ) has e i t h e r two o r t h i e e positive equilibria: 1. T h e trivial e q u i l i b r i a x = 0 ; -y = 0 2 . x = x° ( w h e r e x° is t h e solution to flt(x) = 0 ; y = 0 3 . Point ( x * , y*), w h i c h is t h e solution t o

a(x*)x* - V(x*)y* = 0 k'(x*) = 0 a t ct(x*) > 0 , t h a t is w h e n x * < x ° .

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Simple Model, Complex Behavior

153

those tines in the even levels. By further increasing the inflow into the system we do hot change the values in the even levels, whereas the odd levels gradually continue to increase then equilibrium hioinass. To c h e c k whether this is just a c o i n c i d c n c e that might go away if parameter values are modified, or whether it is something real regarding the system dynamics, we may take a look at the equations a n d figure out t h e equilibria. In the most general form, the equations for the model are. T; = N - u T|TT, = u,T,T, - u , T T j

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which allows us to calculate back, starting from T ^ . j , all. the eqgjlibria for odd ( e v e n ) trophic levels if k is even (odd)

N o t e thac ail of them are constant and independ-

ent o f N . From t h e first equation, we have either T| = N / ( u ( T ; ) , or T ; — N./{u|T,). Therefore, if we know ail the equilibria for odd trophic levels, we c a n calculate the value for T ; , and then use T H , =

to calculate all t h e remaining equilibria.

Similarly, if k is odd and we know all the even equilibria, we c a n calculate T ] . and then build up t h e equilibrium values for all the remaining even trophic levels. W h a t is important is that we get every o t h e r trophic level constant and independent of t h e amount of flow into the system, whereas material accumulates only o n the remaining trophic levels W e have an alternating pattern of equilibria, where every other trophic level simply passes material through to the next trophic level. T h e analytic treatment confirms some o f "he assumptions that we made from watching the dynamics of the system in Madonna. Moreover, ic confirms t Ilk this is really the way the. system behaves beyond the simulation period and pifemeter values chosen. T h e overall dynamics look quite similar to the second case discussed above, when we introduced carrying capacity for the prey population (equation 5 , 5 ) . T h i s mighi well be expected, if we realize that at carrying capacity we have ;i constant flow of external resources into the system, which is exactly the formulation wc are considering now N = const S o the fact that t h e system equilibrates and the equilihnum appears to be stable is quite consistent with what we observed in the simple two-species system W h a t is somewhat surprising is the distinctly different l*'haviof observed in the odd and even trophic levels.

154

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S y s t e m s Science and Modeling for Ecological Economics

5.14

A Stella model of a five-level trophic chain with mortality

In the model a b o v e we assumed t h a t natural m o r t a l i t y is negligible c o m p a r e d w i t h t h e predator uptake. S u p p o s e this is n o t so. Let us c o n s i d e r a t r o p h i c c h a i n , w h i c h has a c e r t a i n f r a c t i o n of biomass r e m o v e d from e a c h t r o p h i c level due t o m o r t a l i t y (Figure .5.14), and see how t h e model d y n a m i c s is influenced hy c h a n g e s in t h e a m o u n t of resources N provided to t h e system T h e apparently subtle c h a n g e in t h e m o d e l f o r m u l a t i o n results in quite substantial differences in t h e system dynamics. O n c e again, we c a n easily put t h e model t o g e t h e r in S t e l l a or, e v e n better, in M a d o n n a

Ii we l o o k at h o w the system reacts

to c h a n g e s in t h e flow o f t h e e x t e r n a l resource N , we may see t h a t now, for substantially h i g h flow i n t o t h e system, all t h e hve t r o p h i c levels c a n c o e x i s t a n d e q u i l i b r a t e a t c e r t a i n values t h a t appear t o be stable. It we start to d e c r e a s e t h e e x t e r n a l flow N , t h e species e q u i l i b r a t e t o lower and lower values, until t h e last, fifth, t r o p h i c level b e c o m e s e x t i n c t . T h e fourth level t h e n follows a n d so o n , until all species b e c o m e e x t i n c t w h e n t h e r e are n o e x t e r n a l sources of e n e r g y or material ( N = 0 ) (Figure 5 15). T h i s result may h a v e an interesting e c o l o g i c a l i n t e r p r e t a t i o n . T h e more resources flow i n t o a trophic c h a i n , t h e longer t h e trophic c h a i n that c a n be sustained. N o r o n l y do the equilibrium values increase; also, entirely n e w t r o p h i c levels spring up.

Simple Model, Complex Behavior

9

e

155

TrChain2 - Ran 1: T i , T2, T 3 . T4. TS vs. TIMF

a

2s

10

80

80

O

100

120

110

160

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10*

Figure 5.15

180

T r C h a j n 2 - Sliders

Dynamics of five trophic levels in a trophic chain with mortality.

The length of the trophic chain is defined by the amount of resource flowing into the system.

T i n s kind of p h e n o m e n o n has been observed in real-life systems. In agriculture, it has b e e n n o t i c e d that when larger a m o u n t s o f fertilizers are applied new pests appear, w h i c h effectively e x t e n d s t h e existing t r o p h i c c h a i n , adding a new level t o !t. A t this time, however, we still c a n m a k e qualitative c o n c l u s i o n s only about t h e system we have analyzed, and only lor t h e parameter values that we have used. W i t h respect to parameter values, t h e system seems t o he quite robust. W e may start modifying t h e coefficients in a fairly wide range (as long as rhey stay ecologically feasible that is, positive and perhaps less than I for most of t h e rate coefficients, like mortality). T h e system behavior seems to be t h e same. However, if we want to consider a trophic c h a i n with more species involved, we may need to put together a n o t h e r model and repeat t h e analysis. It is most likely that, qualitatively, t h e dynamics will be the same, but still we c a n n e v e r be 1 0 0 percent sure unless we pe.rtoi m s o m e analytical treatment. A full a n a l y t i c a l s o l u t i o n t o this problem c a n b e found in S v i r e : h e v and L o g o f e t ( 1 9 8 3 ) . Here, let us take a q u i c k g l a n c e at what t h e equilibria c a n look like, a n d w h a t makes species fall o u t o f t h e system. T h e system of a l g e b r a i c e q u a t i o n s t h a t defines t h e equilibria in tins model is q u i t e simple:

AUvnys diexk N - d,T, - u,T(T2 = 0 u,T| - u . T , - d 2 = 0

T&meiJutujyou atudytkjil idutimt

U

,-I

T

~

-

U4T, -

US - D ,

,-I

d, =

0

= 0

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ti

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fee if you out

YET MUM ARN-ftriUKts OK FWAAHETM or try to fujure cwuUtim/,

out

ejuUibruuv.

Systems Science and Modeling for Ecological Economics From the last equation, we immediately get: +

T

T4 = —

- const

Playing with t h e odd a n d even numbers, as we did above, we c a n now c a l c u l a t e t h e o t h e r equilibria:

1,

_

+ d, Ui

where we c a n substitute the value for T^ from die above and see that T 2 = constKnowing T j , we c a n calculate T , -

N

'

U|Tj

d) +

N o t e that this time the equilibrium is dependent on the external flow N S o far, all the equilibria have been positive af any time. Rased o n t h e second equation, we c a n now calculate _ T, =

UITI

~ d2 U;

F o r this equilibrium we need to make sure that T | > dj/iij, otherwise the equilibrium is negative and makes n o ecological sense. T h i s condition translates immediately into a requirement for N : t h e flow ol external resource has to be larger than a certain value. Similarly, f o r T ; , to be non-negative we need T j > d^/uj or, substituting for T - „ J2ui

Tr

- —' + , > d

T h i s explains why, with decreasing N , the equilibria for T|, T j and T 5 are getting smaller and smaller, and eventually t h e species ceases to exist as the equilibria become negative. However, this does not explain t h e fate o f the other two trophic levels, T i and T ; , which are supposedly constant and independent o f N. S o what is going on.1 Let us take a closer look ac the model dynamics in t h e animation above. N o t e that actually at first, when we start c u t t i n g the input o f N , t h e equilibria for T i and T , are indeed fixed and do n o t c h a n g e . It is the o t h e r three equilibria chat show a downward trend. It is only after

hits zero that T

;

and T 4 start t o change. But

note: when T 5 becomes e x t i n c t , we n o longer have the same five-level trophic 1 hatn Instead we have only four trophic levels, and the equations that we are to solve now change. Now, for four trophic levels, we have T ( and T i c o n s t a n t and independent o f N . whereas T j and T 4 are defined by N and decrease with N. Indeed, chis is what we see in che a n i m a t i o n . N o w T , and T-i stay fixed until T 4 hies zero, when o n c e again the system and t h e equations are redefined. Again t h e system has an odd number of" levels, and now T j becomes fixed while T| and T^ start to fall Now that we have figured out what goes on in t h e system, we can with far greater confidence describe the system behavior with an arbitrary number o f trophic levels. T h e r e 15 strong evidence that the equilibria are stable, and we have understood how the odd and even trophic levels are alternating their behavior as the flow of resource

" •

"

• •

•

Jl

•"•

—

Simple Model, Complex Behavior

157

i n t o t h e system c h a n g e s . W e also k n o w t h a t the parameters o f t h e model define t h e intervals in rhe N c o n t i n u u m rhat correspond t o t h e particular n u m b e r s ot rrophic levels in t h e system. L e t us look at h o w t h e system evolves in t h e o t h e r d i r e c t i o n , w h e n we start with N - 0 , a n d then start increasing N . O n c e N > 0 , t h e r e is a resource that c a n support o n e species. A s N increases, t h e p o p u l a t i o n in this t r o p h i c level keeps growing until N passes a threshold, after w h i c h a n o t h e r species in t h e n e x t t r o p h i c level appears A r rhis point t h e first trophic level stabilizes, a n d from now o n all t h e resource is transmitted to r h e new trophic level, r h e population of w h i c h starts to grow. N e x t , after N passes a n o t h e r threshold, a n o t h e r , third t r o p h i c level appears. N o w t h e s e c o n d t r o p h i c level freezes, while t h e first a n d t h e third ( o d d ) t r o p h i c levels start t o grow. T h e n , at some point, as N passes a n o t h e r threshold, a fourth ( e v e n ) trophic level b e c o m e s established. From n o w o n , odd levels b e c o m e frozen, a n d e v e n levels start t o grow biomass. A n d so o n . X

*

In b o t h rhe t r o p h i c c h a i n s considered a b o v e , we had t h e input o f e x t e r n a l resource, i n d e p e n d e n t o f t h e biomass in t h e first t r o p h i c level. W e assumed that it was t h e resource rliat was always limiting growth, a n d t h e r e were as many organisms in t h a t t r o p h i c level as were needed t o uptake all t h e resource that was made a v a i l a b l e . T h i s is d i f f e r e n t from what we had in t h e classic m o d e l . W h a t will t h e t r o p h i c c h a i n look like it ( h e resource is n o t l i m i t i n g . ' T h i s may appear t o be a fairly subtle c h a n g e in t h e system; h o wever, t h e d y n a m i c s will be quite different. Let us put t o g e t h e r a simplified version with o n l y three t r o p h i c levels:

T-(t! = Ti(t - dt) + IN -

R,)*dt

INITT, = 1 N = u0*T) R. = u ) < T 1 * T 2 T2 = T 3 lt - dtl + IB, INIT T 2 = 2 R, = u , * V T

R2l'dt

2

R, = UJ*T?-T3

T a (t) = T3|t - dtl - |R2 -

R3l*dt

INITT 3 = 1 Rj

=

l^^TT^TT

R3 = U3T3 u0 = 0 1 u, = 0.1 u2 = 0 1 113 ~ 0 . 1

N o t e t h a t in this model N is n o t c o n s t a n t ; instead, it is a linear f u n c t i o n of T , . N o w t h e model looks e x a c t l y t h e same as t h e " c l a s s i c " model but with o n e additional t r o p h i c level. W e c a n import theses e q u a t i o n s i n t o M a d o n n a , o r q u i c k l y a s s e m b l e the model in S t e l l a or o n e o f t h e o t h e r packages to d o some preliminary qualitative analysis. W i t h t h e model "as is," we get t h e familiar o s c i l l a t i o n s (Figure 5 1 6 ) . However, it we c h a n g e t h e c o e f f i c i e n t s u, e v e n slightly, we get a d r a m a t i c a l l y differe n t picture: e i t h e r t h e species b e c o m e e x t i n c t , o r they start t o grow e x p o n e n t i a l l y (Figure 5 . 1 7 ) .

158

Systems Scienrf? and Modeling fo' Ecologicst Economics

100

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Tina

Figure 5.17

Dynamics in a three trophic leve model with no resource limitation with unequal rate

coefficient. The system either dies off or species produce infinite growth.

It Uf or to are e v e n slightly increased, trophic levels T ( a n d T-, grow e x p o n e n tially while T ; keeps o s c i l l a t i n g a p p r o a c h i n g a positive e q u i l i b r i u m . A similar trend is produced w h e n u, oi in are decreased. If u<> or u? a r e e v e n slightly decreased, t r o p h i c levels T | and T , g o e x t i n c t w h i l e T 2 keeps o s c i l l a t i n g a p p r o a c h i n g a positive equilibrium. A similar trend is produced w h e n ii| or u.t a r e increased. A quick a n a l y t i c a l look at t h e equilibria gives us o n l y a very general idea about the u n d e r p i n n i n g s o f t h e s e trends. First, we find that there are t w o e q u a t i o n s for equilibrium ul t h e s e c o n d t r o p h i c level: T

:

= uju\, a n d T 2 = u^/uj. S e c o n d , we see

t h a t for t h e equilibria in t h e first and third t r o p h i c levels we h a v e ^ T j = u 2 T v T h e equilibrium in t h e s e c o n d t r o p h i c level is t h e r e f o r e feasible only if U|Uj = u 0 u 2 . T h e s e c a l c u l a t i o n s e x p l a i n some o f t h e q u a l i t a t i v e d y n a m i c s we observed a b o v e . If U,U) = UfUj, w e get stable oscillations; if U|U5 >

u 0 u 2 , we h a v e t h e downward

t r e n d that leads t o species e x t i n c t i o n s . O t h e r w i s e , we h a v e o s c i l l a t i o n s following an

Simple Model, Complex Behavior

159

exponential growth trend. W e could have been expecting this fiom what we saw in the model; however, it might have been hard t o guess the exact relationship between the parameters thac defines t h e course o f the traiectories. W e also see thac there is a relationship between T , and T-,, which makes them behave in a similar way - something we also observed from t h e model output However, this is probably all we c a n say about t h e system, based on this primitive analysis. W e do not know what makes T , and T i grow to infinity o r vanish from t h e system, when the parameters are chosen tn some specific way- U n l i k e t h e "classic" model, which produced the loop in the phase plane for any c o m b i n a t i o n of parameters, n o w a loop is possible only for specific values. Moreover, it would be hard to imagine in real lite an exact equality of t h e kind U|Ui = UpUj. T h e r e f o r e , we may conclude that a three- or more trophic level system ol the predator-prey type is unstable and unlikely to exist in reality. W h a t will happen if, instead o f three, we have four trophic levels' Will t h e results be t h e s a m e ' T h e answer is a definite N O . T o our surprise, the system always persists, even though it goes through some dramatic oscillations which in many cases appear to resemble chaos. O n c e again, it is strongly recommended that you reproduce the model in o n e o f the modeling packages. Below are t h e equations that you can simply paste into M a d o n n a and e n j o y t h e model performance yourself:

T,(t> = T , ( t - d t ) + iN - R,)*dt INITT, = 1 N = uO'T, R, = c , * T , * T 2 T 2 (t) = T 2 (t - dt) + (R, - R 2.i* at INITT 2 = 2 R, =

L,*T,*T2

R2 =

UJ'T2*T3

T3M

= T 3 (t - dt) + (R 2 -

R 3 )*di

INITT) = 1 R2 = U2*T2*T3

R3 = ^3* Td(t) = Td(t - dt) + IR 3 -

Rj'ai

INITT., = 1 R3 = UJ*TJ*TA R* -

U4"T4

U3 = 0.1 •J, -

0.1

u2 = 0 1 U3 -

0 1

u„ = 0.1

T h e variety o f designs that the trajectories produce when we start modifying che parameters is truly remarkable. A few examples appear in Figure 5.18. In the left-hand column we are looking at the regular graphs o f state variables vs time; in the righc-hand column we have the scaccer graphs, where T ( and T> are displayed as functions o f T ,

100

ISO Time

200

250

04

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u0 - 0.1

06

08

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14

16

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— 0 1 u2 = 0.1 u., = 0.1 22 •

50

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2SO

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04

u0 = 0 13 U,

06

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0.1 U2 = 0 1 u, - 0 1 4

50

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ISO Time

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OS

5

u0 = 0 13 u, = 0 06 u ? =0.1 a, - 0.1

200

250

300

0

u0 - 0.13 u, - 0.06 — r, . Tfe! w

2

= 0.19 u;l - 0.1

3 T3

T 4 4 1

_

3

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0 5 Jr 0 u0 = 0 15 u, - 0 06

B i l T H ' U m

I

02

C< 06

oe

12

14

16

I(

- 0.17 u3 - 0 2

Adding another trophic level (fourth) stabilizes the system and makes it persist, even

though some of the oscillations secni to be chaotic. The left-hand column shows the dynamics of the four populations; the right-hand column graphs are phase dynamics of populations ol the first two trophic levels as functions of the third trophic level population These show how irregular the oscillations may become

Simo'fc MocJo'. C o m p l e x B e ^ a v i o r

161

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Systems Scienrf? and Modeling fo' Ecologicst Economics

5.4

Spatial model of a predator-prey system T h e models we have looked at so far have been local - that is, spatially they had no resolution, assuming that the whole area that we were modeling was uniform, and that t h e same populations with the same parameters o f growth and death were distributed across t h e area. W e did not know or care about any spatial differences. But what if that is n o t t h e ease? Suppose we do care about spatial differences. Suppose that the populations have different numbers across the landscape. How c a n we model the system in this case? First, let us decide o n how to represent space. In C h a p t e r 2, we saw several ways to make space discrete so that we c a n put the spatial dimension into a model. W e need t o decide o n the form and size o f the spatial segments that we wish to use. In doing that, as always in modeling, we will be looking at the goal o f the study and the spatial resolution o f the data that are available. T h e n we will select modeling software for these spatial simulations. Stella may not be the best tool for this. Theoretically, we could replicate our model several times and have several stocks for prey and several stocks for predator, representing their numbers in different spatial locations. W e could also add some ailes o f transition between these stocks, representing spatial movement between different places. T h e Stella model would look like Figure 5 . 1 9 (see page 1 6 5 ) . In this case, we assume that organisms migrate to the compartment where the existing population size is lower T h i s could probably work for two, three or four locations - maybe even ten - but then the Stella model would become almost incomprehensible. W e could use the array functionality in Stella, which would make it a little bit easier to handle. If you are unfamiliar with arrays in Stella, read the pages o f the Help File. It does a really good j o b o f explaining how to set up arrays in Stella. For example, t h e model above on a 3 X 3 grid o f 9 cells can be presented with a diagram that looks quite simple (Figure 5.20; see page 165); however, the equations are not simple at all:

RabbitslcoM,rowl ](t) = RabbitslcoM ,row1 ) ! t - d t ) + (R_births[col1 ,row1 | - P r e d a t i o n I c o M . r o w l ] Rjmigration(coM ,row1 ]) * dt INIT Rabbits[col1 ,row1 ] - 1 RabbitslcoM,row2]
>

dt

INIT RabbitslcoM,row3] - 3 Rabbits|col2,row1 ](t) = Rabbits[col2,row1|(t~dt) + (R_births|col2,rowl | - P r e d a t i o n [ c o l 2 . r o w 1 | R_migration|col2,row1 ]) * dt INIT Rabbits(col2,row1] = 3 Rabbits[col2,row2](t) - Rabbits!col2,row2)(t-dt) + (R_births|col2,row2)-Predation[col2,row2lR_migration(coi2 / row2)) * dt INIT Rabbitslcol2,row2] = 2 Rabbits[col2,row3)(ti - Rabbits|col2,row3|(t-dt) + (R_births(col2,row3]-Pfedationlcol2,row3)R_migration[col2,row3]) * dt INIT Rabbitslcol2,row3| - 1

Simple Model, Complex Behavior

163

Rabbits|coi3,row1 |(t) = Rabbits|col3,row! | ( t - d l ) + (R_births|col3,row1 |-Predation|co(3,rowl ] R_migration(col3,row11) * dt I NIT Rabbits[col3,row1 ] - 1 Rabbits|col3,row2|(t) - Rabbits|col3,row2|(t-dl) + (R_births[col3,row2|-Predation[col3,row2]R_migration[col3,row2]) * dt I NIT Rabbits[col3,row2] - 2 Rabbits[col3,row3](t| = Rabbits|col3,row3l(t-dt) + (R_births[col3,row3l-Predation[col3,row3|R_migrat!on!coi3,row3]l * dt INIT Rabbits!col3,row3] - 3 INFLOWS: R_births|column,row] = alpha* Rabbits|column,row| OUTFLOWS: Predation(column,row| = beta*Rabbits|column,row| * W o l v e s [ c o l u m n , r o w | R_migration[col1 ,row11 •= gannma*(Rabbits!col1 ,row1 |-Rabbits(col2,row1]) + gamma*(Rabbits|col1 ,row1 ]-Rabbiis[col1 ,row2]) R_nmgration(col1,row2] = gamma*((Rabbits|col1 , r o w 2 | - R a b b i t s l c o l l , r o w l |) -r (Rabbitslcoll, r o w 2 ) - R a b b i t s [ c o l 2 , r o w 2 | ) + (Rabbitslcoll , r o w 2 ] - R a b b i t s ( c o l 1 ,row3)J) R _ m i g r a t i o n [ c o h , r o w 3 l = gamma*((Rabb'is|col1 , r o w 3 | - R a b b i t s [ c o l 1 , r o w 2 ] ) + (Rabbitslcoll, row2]-Rabbits|coi2,row3|}) R..migration|col2,row1] = g a m m a * ((Rabbits|col2,rowl ] —Rabbitslcoll ,row11) + (Rabbits|col2, r o w l ] - R a b b i t s [ c o l 2 , r o w 2 ] ) + (Rabbits[col2,row1 ) - R a b b i t s l c o l 3 , r o w 1 ] ) ) R_migration|col2,row2| = gamma*((Rabbits|col2,row2]-Rabbits[col2,row11) + (Rabbits[col2, row2]-Rabbits|col2,row3|) + {Rabbits|col2,row2j-Rabbitslcoll ( row2I) + (Rabb!ts[col2,row2]Rabbits|col3, row2D) R_migration|col2,row3) = g a m m a * ( ( R a b b i t s ( c o l 2 , r o w 3 | - R a b b i t s ( c o l 1 ,row3|) + (Rabbits(col2, r o w 3 l -Rabbits|col2,row2|) + (Rabbits|col2,row3]-Rabbitslcol3,row3|)) R.. nnigration|col3,row1) = gamma*((Rabbits|col3,row1 ) - R a b b i t s [ c o l 2 , r o w 1 ]) -+ (Rabbits|col3, r o w l ] - Rabbits|col3,row2])) migration [col3,row2] = gamma*((Rabbits[col3/ow2|-Rabbits|col3,row11} + (Rabbits|col3, row2] - Rabbitslco'2, row2)) + (Rabbits(col3,row2] - Rabbits|col3, row3D) R_migration[col3,row3| = gamma *{(Rabbits|col3,row3]-Rabbits(col3,row2]) + (Rabbits|col3, row3] - Rabbits[col2 ; row3D) Wolveslcoll ,rowl|(t) - W o l v e s l c o l l , r o w l l ( t - d t ) + ( U p t a k e | c o l l . r o w l ] W_mortalitylcol1 ,row1 ] - W _ m i g r a t i o n I c o l 1 ,row1 ]) * dt INIT W o l v e s l c o l l , r o w l ) = 1 Wolveslcoll.row2)(t) = Wolveslcoll , r o w 2 ) ( t - d t ) + (Uptake(col1 , r o w 2 | W_mortalitvlcol1 , r o w 2 | - W _ m i g r a t i o n | c o l 1 , r o w 2 l ) * dt INIT W o l v e s l c o l l , r o w 2 ) - 2 W o l v e s l c o l l , row3](t) = W o l v e s l c o l l , r o w 3 l ( t - d t ) + (Uptakeicoll , r o w 3 | W _ m o r t a l i t y | c o l 1 , r o w 3 ] - W _ m i g r a t i o n | c o l 1 , r o w 3 ] ) * dt INIT W o l v e s l c o l l , r o w 3 | - 3 Wolves|col2.row1 ){t) = Wolves[col2,rowl | ( t - d t ) + (Uptake|col2 ; row11W _ n n o r t a l i t y [ c o l 2 , r o w 1 ] - W _ m i g r a t i o n | c o l 2 , r o w l | ) * dt INIT Wolves|col2,row1 ] = 3 Wolves|col2,row2](t) - W o l v e s | c o l 2 , r o w 2 | ( t - d t ) + (Uptake[col2,row2)W_mortality|col2,row2J-W_migration(col2,row2|) * dt INIT Wolves|col2,row2] - 2

164

Systems Scienrf? and Modeling fo' Ecologicst Economics

Wolves[col2,row3](t) = W o l v e s [ c o l 2 , r o w 3 ] ( t - d t ) + ( U p t a k e [ c o l 2 , r o w 3 | W_mortality[col2,row3]-W_nnigration[col2,row3|i * dt INIT W o l v e s ( c o l 2 , r o w 3 ] = 1 W o l v e s | c o l 3 , r o w 1 ] W _ m i g r a t i o n | c o l 2 , r o w 3 | = d e l t a * { ( W o l v e s | c o i 2 , r o w 3 ] - W o l v e s [ c o l 1 , r o w 3 ] } + (Wolves[col2 row3|-Wolves[col2,row2|) + (Wolveslcol2,row3]-Wolves|col3,row3])) W _ r n i g r a t i o n ( c o l 3 , r o w 1 ] = d e l t a * ( ( W o l v e s | c o l 3 , r o w 1 ] - W o l v e s [ c o l 2 , r o w 1 ! ) f (Wolves[col3, rowl |-Wolves|col3,row2|)) W _ m i g r a i i o n ! c o l 3 , r o w 2 ] - d e l t a * ( ( W o l v e s | c o l 3 , r o w 2 l - W o l v e s ! c o l 3 , r o w 1 ]) + (Wolves|col3, row21 - W o Ives I col 2, r o w 2 ]) + ( W o l v e s [ c o l 3 . r o w 2 ] - W o l v e s [ c o l 3 J r o w 3 D ) W _ m i g r a t i o n [ c o l 3 , r o w 3 | = d e l t 3 * f ( W o l v e s [ c o l 3 , r o w 3 ] - W o l v e s | c o l 3 , r o w 2 ] ) + (Wolves|col3, r o w 3 | - W o l v e s | c o l 2 . row3])) alpha = 1 beta = 1 delta = 0 . 0 2 g a m m a = 0.01 k = 0.1 mu = 0 1

In particular, it is a real h e a d a c h e to define t h e e q u a t i o n s o f m o v e m e n t , migration. W e assume t h a t our cells are arranged as in Figure 5 . 2 1 , and both wolves a n d rabbits c a n m o v e to t h e n e x t c e l l if t h e population si2e t h e r e is lower t h a n in t h e c u r r e n t cell. T h e r e will be lots o f c l i c k i n g o n t h e S t e l l a diagram t o define all t h e c o n n e c t i o n s . A s t h e n u m b e r o f spatial cells grows, t h e m o d e l description

quickly

Simple Model, Complex Behavior

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b e c o m e s very c u m b e r s o m e ; it b e c o m e s especially hard t o input t h e data, visualize r h e outpur, or define various scenarios thac i n v o l v e spatial d y n a m i c s . I m a g i n e defining a m o d e l with a hundred or m o r e array e l e m e n t s ! T h e r e has to be a b e t t e r way t o d o this. Let us t a k e a look at s o m e o t h e r software tools t h a t may be m o r e suited t o these tasks t h a n S t e l l a . O n e p o t e n t i a l l y powerful tool for spatial m o d e l i n g is S i m i l e , and we will e x p l o r e an e x a m p l e in t h a t m o d e l i n g system.

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Figure 5.23

Output from the Simile predator-prey model using the Plotter helper to create a

time-dependent graph This is identical to what we were generating in Stella

(Figure 5 . 2 2 ) . H e r e , we slightly modified t h e m o d e l , describing G r a s s as prey a n d R a b b i t s as predator. T h a t would be o n e t r o p h i c level below what we were c o n s i d ering a b o v e , but t h e r e is really n o need for much c h a n g e in h o w we formulate t h e m o d e l . W h e r e a s in its systems d y n a m i c s S i m i l e follows S t e l l a ' s formalism

quite

closely, it also goes way beyond Stella's f u n c t i o n a l i t y in a lot o f ways. A s you may n o t i c e , in Figure 5 2 2 , there a r e quite a few more i c o n s or building blocks in S i m i l e . W e will not go i n t o m u c h detail describing all o f t h e m - that c a n always be d o n e by d o w n l o a d i n g t h e tree trial version o f t h e package a n d e x p l o r i n g t h e different e x a m ples a n d c o n t r i b u t e d models. T h e H e l p tile and t h e T u t o r i a l for S i m i l e is n o w h e r e nearly as f o o l p r o o f as in S t e l l a , s o be prepared t o spend q u i t e s o m e t i m e it you decide to e x p l o r e t h e more a d v a n c e d features o f t h e software. A m o n g t h e s e leatures let us m e n t i o n t h e following. • Modularity. In S i m i l e , you c a n c r e a t e a " s u b m o d e l " t h a t c a n be t h e n used in o t h e r models. T h i s is handy for disaggregation o f models, for c r e a t i n g spatial m o d e l s or for substituting o n e model c o m p o n e n t for a n o t h e r . • C + - * - axle. S i m i l e g e n e r a t e s C + + c o d e , w h i c h c a n be used w i t h i n t h e framework o f o t h e r systems, interfaces o r e n v i r o n m e n t s . It c a n be ported to different c o m p i l ers producing optimized c o m p u t e r c o d e . • ErtendcWr interfaces.

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Figure 5.25

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A two-dimensional graphic visualization that the Spatial Grid display helper generates.

The intensity of the color corresponds to the population numbers ol Rabbits in different cells. T h e r e is actually an easier (but also n o t very w e l l - d o c u m c n t e d ) way t o d o this it" you define t h e array as b e i n g 2 D . You d o t h i s by d o u b l e c l i c k i n g o n d i e b a c k g r o u n d o f your stack o f cells, w h i c h o p e n s a dialogue b o x : O

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H e r e we c a n input t h e d i m e n s i o n s o f t h e array, m a k i n g it t w o - d i m e n s i o n a l . Let us specify t h e d i m e n s i o n s 1 0 , 1 0 . N o w t h e array will be treated as rows a n d c o l u m n s , a n d we will n o t need t o worry a b o u t t h e c o n v e r s i o n o f a linear array i n t o a matrix.

170

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Using the Plotter helper to output an ensemble of tiaiectones fot all the cells in the model

The initial conditions are generated in random in the 11,2! interval, and each cell then develops on its own

T o view t h e results ot our spatial runs, we c a n c h o o s e t h e helper called "grid display." W h e n defining t h e grid display we will he requested to " c l i c k o n the variable c o n t a i n i n g t h e positions ot IDs ot t h e c o l u m n s " - c l i c k o n t h e " c o l " variable. T h e n we will he asked t o c h o o s e t h e variable t o display, a n d will c l i c k o n R a b b i t s . If wc now run the m o d e l , we c a n observe how R a b b i t populations vary in all t h e cells (Figure 5 . 2 6 ) . N o t e that in this case t h e graphic display produces an e n s e m b l e ol 1 0 0 curves, w h i c h o r i g i n a t e s o m e w h e r e in t h e interval [1,2) a n d t h e n o s c i l l a t e like in the p r e d a t o r - p r e y m o d e l considered before. S o lar, che c e l l s h a v e b e e n working i n d e p e n d e n t l y . T h e r e has b e e n n o i n t e r a c t i o n b e t w e e n v a r i a b l e s in different cells. T h a t is n o t particularly interesting. Lev us now m a k e t h e R a b b i t s m o v e horizontally. Suppose t h a t , as in the S t e l l a m o d e l wc considered a b o v e , we want to m a k e R a b b i t s m o v e from cells with h i g h e r density to c e l l s w h e r e t h e r e arc less R a b b i t s . T h i s is similar t o t h e diffusion process. For e a c h cell we add t h e migration flow (Figure 5 . 2 7 ) . w h i c h c a l c u l a t e s t h e m o v e m e n t o f R a b b i t s in e a c h o f t h e four d i r e c t i o n s : f r o n t , b a c k , left and right. First, we define an array of R a b b i t s in all c e l l s - R _ A . T h e n Migration = delta col -

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Simple Model, Complex Behavior

171

Spatial predator-prey model in Simile with migration added lor Rabbits. The R_A variable stores the values for Rabbits in all cells as an array. The decision for migration is based on the number of Rabbits in adjacent cells. Rabbits jump to the neighboring cell if the population there is less than in the current cell.

T h i s was p r e t t y clumsy, b u t s t r a i g h t f o r w a r d .

F o r e a c h c e l l , we c o m p a r e t h e

n u m b e r o f R a b b i t s w i t h t h e n u m b e r s in t h e four a d j a c e n t c e l l s . I f t h e d i f f e r e n c e is p o s i t i v e , we g e t a p o s i t i v e flow from t h e c e l l to t h e n e i g h b o r i n g c e l l . If ir is n e g a t i v e , we g e t a flow from t h e n e i g h b o r i n g c e l l i n t o t h e c e n t c r c c l l . H e r e , we used t h e e l e m e n t b u i l t - i n f u n c t i o n e l e m e n t ( [ A ] , i ) , w h i c h returns t h e icK e l e m e n t o f array A . N o t e t h a t h e r e we are t r a n s l a t i n g t h e 2L) d e f i n i t i o n in t e r m s o f ( r o w , c e l l ) b a c k i n t o t h e I D definition. T o test h o w t h i s works, we will initialize t h e m o d e l differently. L e t us m a k e t h e R a b b i t s biomass e q u a l , say, t h r e e o n l y in o n e c e l l (e.g. i = 2 5 ) , a n d m a k e t h e b i o m a s s e q u a l o n e in all o t h e r cells. L e t us also s w i t c h o f f all t h e e c o l o g i c a l

predator-prey

d y n a m i c s by s e t t i n g t h e g r o w t h , d e a t h a n d p r e d a t i o n rates to zero. If t e s t i n g a p a r t i c ular process, h o r i z o n t a l d i s p e r s i o n in this c a s e , it is i m p o r t a n t to e n s u r e t h a t n o t h i n g is i n t e r f e r i n g with it. I f we r u n t h e m o d e l , we will s e e h o w rabbits gradually disperse across t h e a r e a ( F i g u r e 5 . 2 8 ) . N o t e t h a t we h a v e also added a v a r i a b l e , s u m _ R , t o t h e d i a g r a m . T h i s v a r i a b l e is equal to s u m ( [ R _ A ] ) , a n o t h e r b u i l t - i n f u n c t i o n w h i c h returns t h e sum o f e l e m e n t s o f a n array. T h i s is useful t o c h e c k t h a t we are n o t losing o r g a i n i n g rabbits; it works as a mass c o n s e r v a t i o n c h e c k . A s long as s u m J R does n o t c h a n g e , we are O K . W h a t is also n i c e a b o u t S i m i l e is t h a t we c a n c h a n g e t h e size o f t h e area a n d t h e n u m b e r o f c e l l s just by c h a n g i n g t h e " s i z e " v a r i a b l e a n d t h e n u m b e r o f i n s t a n c e s o f t h e " C e l l " array. T h i s c a n b e d o n e by d o u b l e c l i c k i n g o n t h e C e l l s u b m o d e l a n d t h e n s p e c i f y i n g t h e d i m e n s i o n s . F o r e x a m p l e , we c a n s w i t c h from t h e I 0 X 1 0 grid t h a t we w e r e e x p l o r i n g a b o v e t o a 1 0 0 X 1 0 0 grid in just a m o m e n t , a n d start g e n e r a t i n g s i m i l a r d i s p e r s i o n p a t t e r n s o n a m u c h finer grid o f c e l l s ( F i g u r e 5 . 2 9 ) . I m a g i n e building a s i m i l a r m o d e l o n a 1 0 0 X 1 0 0 grid in S t e l l a !

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Systems Scienrf? and Modeling fo' Ecologicst Economics

Simile can also save equations; however, here it is done using a p r o g r a m m i n g language, Prolog, which makes II a bit harder to read for s o m e b o d y unfamiliar w i t h t h e conventions of that language - especially w h e n the model b e c o m e s more complex

For simple models like

the one w e are studying, it is still quite easy t o understand w h a t the s t a t e m e n t s are about. B e l o w is the Grass-Rabbits model as described tn Simile M o d e l R__G_array10000 Enumerated types: null Variable R_A R_A =

I Rabbits]

Where: [Rabbits! = Cell/Rabbits Variable sum_R sum_R =

sum(iR_A|)

Submodel Cell Submodel Cell is a f i x e d _ m e m b e r s h i p s u b m o d e l w i t h dimensions [10000], Enumerated types. [ Compartment

Grass

Initial value = 2 Rate of change = Compartment

Growth - Grazing

Rabbits

Initial value = if (index(1) = = 2550) then 300 else 1 Rate of change = + Uptake - Mortality -

Migration

Comments: For random initialization rand_const(1,2) Flow Grazing Grazing =

beta*Grass*Rabbits

Flow Growth Growth =

alpha*Grass

Flow Migration Migration =

delta* ((if col > 1 t h e n Rabbits-element([R_A],(row - 1)

size + c o l - 1 ) else 0) + (if c o l < s i z e then Rabbits-element((R_A],(row-1 )*size + col + 1) else 0) + (if r o w > 1 t h e n Rabbits-element([R_A],(row-2)*size + col) else 0) + (if r o w < s i z e t h e n Rabbits-element(|R_A],row*size + col) else 0)) Where. [R„A] - ,/R_A Flow Mortality Mortality =

mu*Rabbits

Flow Uptake Uptake =

k*Grazing

Variable alpha alpha = Variable beta beta = Variable col col -

fmod(index(1) - 1 .size) + 1

Variable delta delta =

0.1

Simple Model, Complex Behavior

173

Variable k k =

0.1

Variable m u mu =

0.1

Variable r o w row -

f l o o r l t i n d e x ! 1} -

U/sizel + 1

Variable si?e size =

100

10 Figure 5.28

Spatial output lor the model with migration.

First we use a simplified initial condition to make sure that we can generate a pattern of dispersion, as w e might expect to see in a model that is similar to the diffusion process.

Rabbits (100x100. time = 100.0)

Figure 5.29

The same model but with 10,000 cells active.

Switching Irom one model dimension to anutliei is easy, it itujuites only ulidiiymy one parameter and the definition ol the array size N o w that we a r e c o n f i d e n t about h o w rabbits m o v e horizontally, we c a n switch t h e e c o l o g i c a l piocesses b a c k o n and s e e h o w t h e system performs in space. O n c e again initializing R a b b i t s a n d G r a s s r a n d o m l y o v e r t h e landscape, we c a n s e e how, due to dispersion, t h e p a t c h e s b e c o m e blurred; every n o w a n d t h e n , w h e n G r a s s is

174

Systems Scienrf? and Modeling fo' Ecologicst Economics

Figure 5.30

Spatial output for the model with randomly generated initial conditions. The diffusion

creates blurred patterns of distribution of Rabbits

d e p l e t e d , t h e o v e r a l l p o p u l a t i o n falls t o a l o w t h e n , f o l l o w i n g g e n e r a l p r e d a t o r - p r e y d y n a m i c s , R a h h i t s reappear (Figure 5 . 3 C ) . \X;e c a n also o u t p u t t h e results as t i m e g r a p h i c s for e a c h c e l l

Figure 5 . 3 1 p r e s e n t s e n s e m b l e s o f I C.CCC c u r v e s f o r R a b b i t s

a n d G r a s s in e a c h o f t h e IC.CCC cells. T h i s g r a p h i c a n d t h e q u a n t i t y o f c o m p u t a t i o n s t h a t stand b e h i n d it s h o u l d really b e a p p r e c i a t e d . I n t e r e s t i n g l y , in s p i t e o f all this spatial variability, t b e totals for R a b b i t s a n d G r a s s follow e x a c t l y t h e classic p r e d a t o r prev p a t t e r n t h a t we h a v e s e e n b e f o r e ( F i g u r e 5 . 3 2 ) . W e l l , a l m o s t e x a c t l y , as we c a n s e e from t h e sea trer-plot X Y diagram in Figure 5 . 3 3 . W h e r e a s previously for just t w o v a r i a b l e s in o n e c e l l t h e R u n g e - K u t t a m e t h o d p r o d u c e d a n e x a c t e l l i p s o i d , w i n d i n g o v e r a n d o v e r i t s e l f again a n d a g a i n , w i t h 1 0 , 0 0 0 i n s t a n c e s o f t h e s a m e m o d e l t h e b e h a v i o r b e c o m e s q u i t e d i f f e r e n t . T h e r e is c e r t a i n l y far m o r e r e a s o n t o e x p e c t t h a t it is t h e error t h a t a c c u m u l a t e s a n d t a k e s us slowly o f f t r a c k

L e t us c h e c k : is it t h e e r r o r

t h a t causes tins, o r s o m e t h i n g e l s e ' T h e first r e m e d y t o d e c r e a s e c o m p u t a t i o n e r r o r is t o s w i t c h t o h i g h e r - o r d e r n u m e r i c a l m e t h o d s o r t o d e c r e a s e t h e t i m e - s t e p . T h e r e is n o t h i n g b e t t e r in S i m i l e t h a n R u n g e - K u t t a , s o h i g h e r - o r d e r m e t h o d s 3 r e n o t a n o p t i o n . I l o w e v e r , we c a n easily d e c r e a s e t h e t i m e - s t e p . A b o v e , we h a d D T = C. I. Let us m a k e it D T = O.Cl N o w it will take us almost IO t i m e s longer t o run t h e m o d e l , y e t u n f o r t u n a t e l y we are n o t g e t t i n g a n y d i f f e r e n t o u t p u t S t i l l t h e t r a j e c t o r y k e e p s w i n d i n g towards t h e c e n t e r . S o w h a t else c o u l d be c a u s i n g i t '

Simple Model, Complex Behavior

175

Time

Figure 5.31

Using lhe plotter, w e can view dynamics for all the 10,000 cells.

L e t us g o hack t o t h e o r i g i n a l m o d e l . In order t o get t h e r e , we will the

horizontal

fluxes

(Migration = 0

if d e l t a = 0 ) , a n d initialize

rhe s a m e . N o w we are s i m p l y r u n n i n g a b u n c h taneously. T o m a k e t h e m o d e l

o f predator-prey

remove

all t h e c e l l s models

run faster, w e c a n also m a k e t h e spatial

simuldimen-

s i o n s s m a l l e r : let us s e t t h e size equal t o 2 , a n d t h e d i m e n s i o n o f c e l l s equal t o 4 If w e n o w run t h e m o d e l , we will

finally

get r h e e x p e c t e d ellipse ( F i g u r e

5.34A).

N e x t , let us initialize t h e four c e l l s t h a t we h a v e r a n d o m l y s e l e c t e d . T h e result is s o m e w h a t u n e x p e c t e d ( F i g u r e 5 . 3 4 B ) , a n d answers o u r d i l e m m a : it is t h e r a n d o m n u m b e r s in r h e initial c o n d i t i o n s t h a t m a k e t h e t o t a l p o p u l a t i o n d y n a m i c s s o diff e r e n t . I f we i n c r e a s e t h e n u m b e i o f cells (size = 1 0 ) , t h e p o p u l a t i o n s t e n d t o be less c h a o t i c a n d t e n d towards a limit c y c l e ( F i g u r e 5 . 3 4 C ) . T h e g r a p h i c in Figure 5 . 3 4 D is p r o d u c e d by t h e s a m e

1 0 , 0 0 0 c e l l s with

horizontal

migration

switched

o n ( d e l t a = 0 . 1 ) , as w e h a d in Figure 5 . 3 3 , b u t a f t e r s o m e 1 , 5 0 0 t i m e - s t e p s . W e see t h a t h e r e also t h e r e is a c l e a r t r e n d t o t h e c e n t e r , w h e r e t h e p o p u l a t i o n a l m o s t equilibrates. T h i s is q u i t e r e m a r k a b l e , s i n c e , as you m a y r e c a l l , o n e o f t h e m a j o r c r i t i q u e s o f t h e classic L o t k a - V o l t e i r n m o d e l was t h a t it d e p e n d s s o m u c h u p o n t h e i n i t i a l

176

Systems Scienrf? and Modeling fo' Ecologicst Economics

D

+

-

iff

—sum R. run 2 17000 15000 p \ 13000|- \ 11000 9000 7000 5000

Idi+JhMI _sum_G. run2 40000

Figure 5.32

Ouipui lor the lotal numbers ol Rabbits and Grass in all the 10,000 cells.

The totals seem to follow ilie classic p r e d a t o r - p r e y oscillations observed before, w h e n dealing w i t h a spatially aggregated modei

sum_G 40000 —

30000

20000

10000-

• Fa i g i u . rme ^ 5 f .t 3k 3l

A scatter graph lor XY graph) w h e r e the numbers for Grass are displayed as a f u n c t i o n

of the number for Rabbits It s h o w s that the oscillations are damping

Simple Model, Complex Behavior

177

w '

-

I

: i

to

Figure 5.34

•

js

/

f

it—' j *

'—rfe

is-

Resolving the mystery of dampened oscillations

A. W h e n w e have no spatial heterogeneity, the population is spatially uniform, and w e have an ideal predatorprey ellipse as in the classic model B. W i t h the population randomly initialized in just four cells w e get a chaotic behavior that fills the w h o l e interior of the ellipsoid. C W i t h 100 cells randomly initialized, the area of chaotic dynamics shrinks to a smaller domain D. W i t h 10.000 cells there is no more chaos a n d the trajectories t e n d to a small limited c y c l e , a r o u n d w h i c h they keep oscillating. This behavior no longer d e p e n d s upon the initial conditions, as long as the cells aie initialized w i t h difletent values

c o n d i t i o n s . T h e c l a s s i c m o d e l d e s c r i b e s a p o p u l a t i o n o v e r a c e r t a i n a r e a , w h e r e spat i a l h e t e r o g e n e i t i e s a r e i g n o r e d a n d all t h e o r g a n i s m s a r e l u m p e d i n t o o n e r e p r e s e n t i n g t b e t o t a l p o p u l a t i o n . H o w e v e r , in reality t h e y are c e r t a i n l y

number unevenly

d i s t r i b u t e d o v e r s p a c e . I f we split t h e s p a c e i n t o j u s t a few r e g i o n s a n d p r e s e n t

the

d y n a m i c s in t h i s s p a t i a l c o n t e x t , we g e t r e s u l t s t h a t a r e s i g n i f i c a n t l y d i f f e r e n t f r o m t h e c l a s s i c m o d e l . A c t u a l l y , it t u r n s o u t t h a t t h e s t a b l e o s c i l l a t i o n s are a n of t h e a v e r a g i n g o v e r s p a c e .

With

s e v e r a l s p a t i a l e n t i t i e s we h a v e a

artifact

converging

d y n a m i c , which also n o longei depends upon the initial c o n d i t i o n s ll we l a k e a c l o s e r l o o k at t h e s p a t i a l d i s t r i b u t i o n s t h a t c o r r e s p o n d t o t h i s q u a s i e q u i l i h r i u m s t a t e , we m a y f i n d s o m e weird s p a t i a l p a t t e r n s ( F i g u r e 5 . 3 5 ) . from the randomly distributed

initial c o n d i t i o n s

Starting

(Figure 5 . 3 5 A ) , after s o m e

1,000

i t e r a t i o n s , as t h e t r a j e c t o r y o n t h e p h a s e p l a n e c o n v e r g e s t o w a r d r h e c e n t e r o f t h e e l l i p s o i d a s p a t i a l p a t t e r n e m e r g e s t h a t , w h i l e c h a n g i n g t o a d e g r e e , s t i l l persists, as can

be seen from t h e series o f snapshots taken a p p r o x i m a t e l y every

50

iterations

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Systems Scienrf? and Modeling fo' Ecologicst Economics

Figure 5.35

The spatial distribution in the 10,000-cell model w i t h migration

Starting with random initial conditions (Al. after some 1.000 iterations a pattern is formed, w h i c h then persists IB-I). Thus there is a pattern that emerges both in time and space.

( F i g u r e 5 . 3 5 B — I ) . Ic is n o t c l e a r h o w and why this p a t t e r n emerges, hut it is i n t e r e s t i n g to register that e m e r g e n t p a t t e r n s c a n result from this k i n d ol n o n - l i n e a r d y n a m i c s , Using

the so-called

association

submodel

concept

in S i m i l e

w e c o u l d put

t o g e t h e r m u c h m o r e e l e g a n t s o l u t i o n s for this m o d e l ; h o w e v e r , t h e s e m o d e l s also b e c o m e far m o r e difficult t o build a n d c o m p r e h e n d L e t us put t o g e t h e r

an association

submodel

called

NextToCell.

It will b e

d e f i n e d by t w o r e l a t i o n s h i p s : " s e l f a n d " n e i g h b o r . " T h e s e a r e c e l l a t t r i b u t e s t h a t a r e

Simple Model, Complex Behavior

179

provided by rhe s t a c k o f cells with t h e submodel in each o f t h e m . T h e e x i s t e n c e o f N e x t T o C e l l s u b m o d e l is defined by t h e c o n d i t i o n c o n d l .

C fill

condl =

1

(coLself = = col_neighbor and row_self

= = row_neighbor) and

a b s ( c o l _ s e l f - c o l _ n e i g h b o r ) < 1.5 a n d a b s ( r o w _ s e l f - r o w _ n e i g h b o r ) < 1 5

T h i s c o n d i t i o n is true only if t h e c o o r d i n a t e s ( c o l , row) o f t h e two cells are adjac e n t t o each o t h e r - thac is, rhe difference b e t w e e n the c o l and row c o o r d i n a t e s is less t h a n 1.5 and the cell is n o t itself. I n this way we c a n describe all eight cells in t h e vicinity o f a given cell. For e a c h o f these n e i g h b o r cells we define a variable called

migration = Rabbits_neighbor - Rabbits_self

T h i s is t h e difference b e t w e e n t h e n u m b e r o f R a b b i t s m t h e c e l l and t h e neighboring c e l l . T h i s value is t h e n fed back i n t o the model and is used to define t h e flow called

In = d e l t a * s u m ( { m i g r a t i o n _ s e l f } )

H e r e we a r e s u m m i n g all t h e m i g r a t i o n s for t h e eight n e i g h b o r i n g cells and, with t h e diffusion rate o f delta, using this sum to update t h e n u m b e r o f R a b b i t s in t h e current c e l l . N o t e t h a t w h e n

R a b b i t s _ n e i g h b o r > Rabbits__self

t h e flow is positive, and it is n e g a t i v e o t h e r w i s e . T h i s should be sufficient to describe the diffusion process o f R a b b i t s in o u r system. Indeed, i f we run t h e mode! we get s o m e very plausible distribution t h a t looks very similar to w h a t we h a v e b e e n genera t i n g a b o v e - but, we have to agree, this f o r m u l a t i o n is way more e l e g a n t .

" • IPSWPPI Systems Science and Modeling for Ecological Economics

180

D

H

5b,

*

e

x

B C D

+ |Oji_«l|jQ.

(V)

f Cxecute (or

292.6

Current time

7.4

Display Interval

o.i

E

f Page 1 |

( Run control I Run settings

aj,

i l j Kb

H

:

P a 9 * 2 |[ P a g e 3

El ' <

«

' •

s |

'' P a g e an

Add a variable

^rdCen Grass •ED K aboil s •••Growth -*» Gra2lng Uptake

-$• Mortality 0 aisha -8 beta ® m u « k

• delta 8 col — % row

lf1

RoDDitS (10x10 Umo » 7 31

NeitToCen — ? COnflt 0 migration

SME

model

Let us e x p l o r e yet a n o t h e r way t o build and run spatial m o d e l s . T h e S p a t i a l M o d e l i n g E n v i r o n m e n t ( S M E ) is n o t q u i t e a m o d e l i n g system, s i n c e it does not require a language or formalism o f its o w n It c a n take t h e e q u a t i o n s from your S t e l l a model a n d t r a n s l a t e t h e m into an i n t e r m e d i a t e M o d u l a r M o d e l i n g L a n g u a g e ( M M L ) , w h i c h is t h e n translated into C + + c o d e . A t t h e s a m e time, S M E will link ycur model t o spatial data if needed L e t us first put t h e s a m e G r a s s - R a b b i t s model i n t o S t e l l a and m a k e sure t h a t it runs properly

A s a result, we will e n d up with t h e f o l l o w i n g syscem of S t e l l a

equations: Grass(t) = Grasslt - dt) + ( G _ g r o w t h

Grazmg)"di

INIT Grass = 2 INFLOWS. G.growth = alpha'Grass OUTFLOWS Grazing = b e l a * G r a s s * R a b b i t s Rabbitslt) = Rabbitsft - d t l + (Uptake INIT Rabbits - 1 INFLOWS Uptake -

k*Grazing

OUTFLOWS. R.morlality = mu'Rabbits alpha = 1 beta = 1 k = 0.1 rnj = 0 1

R_mortality>*dt

Simple Model, Complex Behavior

181

For these S t e l l a e q u a t i o n s , we d o < E d i t - > S e l e c t A i l > a n d t h e n < E d i t - > Copv>. N e x t , we open a T e x t Editor on our c o m p u t e r ( o n a M a c i n t o s h it will be B B E d i t . or T e x t E d i t ; in W i n d o w s ic is probably t h e N o t e p a d ) a n d paste t h e e q u a t i o n s inro the file, t h e n save t h e file using t h e .eqns e x t e n s i o n and n a m i n g it R _ G L e q n s . W e n o w need to get S M E r u n n i n g . S M E is o p e n source and is available for d o w n l o a d from S o u r c e Forge, t h e main repository o f o p e n - s o u r c e projects. T h e U R L is http://souiceforge.net/projects/smodeuv. S M E is available for L i n u x and M a c O S X o p e r a t i n g systems; t h e r e is n o W i n d o w s version so far. O n c e we h a v e d o w n l o a d e d and installed S M E , we n e e d to s e t up t h e S M E project. Having

chosen

a name

for our p r o j e c t

-

let us say R _ C , representing

R a b b i t s & G r a s s - we o p e n t h e T e r m i n a l window and e n t e r t h e c o m m a n d : > S M E project R_G I f t h e installation h a s b e e n d o n e properly, chis sets up t h e p r o j e c t directory. N o w we c a n put t h e e q u a t i o n s file that we c r e a t e d in S t e l l a into t h e directory Models. W e will call t h e m o d e l R _ G l and perform t h e S M E c o m m a n d : > S M E m o d e l R_G1 N o w we get: Current proiect d i r e c t o r y is / D o c u m e n t s / S M E / P r o j e c t s / Current project is R_G Current m o d e l is x x x Current scenario is x x x Current m o d e l s e t to R_G1 Current project s e t to R_G It is n o t i m p o r t a n t at this time, but let us also c h o o s e a s c e n a r i o n a m e . W e will see what t h a t is later o n . U s i n g t h e c o m m a n d > S M E scenario S1 we get: Current project d i r e c t o r y is / D o c u m e n t s / S M E / P r o j e c t s / Current proiect is R_G Current m o d e l is R_G1 Current scenario is S1 N o w we c a n import and configure che m o d e l : > S M E import T h i s will take rhe e q u a t i o n file a n d translate ic into t h e M M L ( m o d u l a r m o d e l i n g language) specification. T h e r e will probably n e v e r b e a n y need to see t h e result, hut for t h e sake o f curiosity it is possible to l o o k at t h e file M o d e l s / R _ G 1 . M M L for t h e M M L specification and t h e n look at M o d e l s / R _ G l / R _ G l _ m o d u l e . x m l , w h i c h is t h e same file in a n i n t e r m e d i a t e X M L s p e c i f i c a t i o n . A t t h e s a m e tune t h e first config file h a s b e e n g e n e r a t e d in C o n f i g / R _ G l . M M L . config. T h i s file srill c o n t a i n s just a list of all variables a n d parameters o f the m o d e l . Let us d o t h e build c o m m a n d n o w : > S M E build

182

Systems Scienrf? and Modeling fo' Ecologicst Economics S o m e t h i n g is processed, i h e r e are s o m e messages, and at t h e e n d it c a n be seen t h a t s o m e C + + c o d e h a s already b e e n c o m p i l e d . T h i s is n o t i m p o r t a n t a t this time, s i n c e we will probably still need to do s o m e m o r e configuring before we get s o m e t h i n g m e a n i n g f u l . W h a t is i m p o r t a n t is t h a t a c o u p l e o f m o r e config files are generaied. S e e w h a t is in t h e C o n f i g directory n o w : R.Gl.biflows, R_Gl.conf, R_Gl.Sl, and R_Gl.Sl.conf.out T h e most i m p o r t a n t file is R _ G l . c o n f . T h i s will he t h e config file t h a t we will be working with most o f the time. A t this time it has t h e following list o f parameters: # global DS(1 0,0} n(1)s(4332) ngi(O) op(0> OT<1,0.20) d(0J UTM(0,0.0,0.0) UTM(1.1.0,1.0) $ R_G1_imodule • ALPHA 4

pm(1) pm(1)

BETA

* GRASS

s(1) sC(C)

* GRAZING

ft(u)

" G.GROWTH

ft(u)

4

K

pm(0 100000)

* MU

pm(0 100000)

* RABBITS

s i l ) sC(C)

* R_MORTALITY

ft(u)

* TIME * UPTAKE

ft(u)

If we c o m p a r e this file wirh t h e S t e l l a e q u a t i o n s above, we see t h a t it c o n t a i n s inform a t i o n about all t h e parameters that we had t h e r e . In t h e e q u a t i o n s : alpha - 1 beta - 1 k - 0.1 mu = 0 1 we find the same values in t h e R _ G 1 . c o n f file. W h a t we have lost are t h e initial c o n d i t i o n s . T h a r

is because

in S t e l l a we

defined t h e initial c o n d i n o n s in t h e state variables b o x e s , rather t h a n as parameters. S M E does nor like t h a t . Let us q u i c k l y go back to S t e l l a and fix it by d e h n i n g initial c o n d i t i o n s in terms o f some auxiliary parameters: INIT Grass = G j n i t INIT Rabbits -

Rjmt

Gjnit = 2 Rjnit = 1 N o t e t h e tiny difference between this set o f equations and what we had above. W e will now have to do a n o t h e r > S M E import and > S M E build. Keep in mind that whenever we alter the equations, we need to do a re-import and a rebuild. W e do nor need to reimport and rebuild if we only modify t h e parameters in the config file. However, if any o f the parameters are redefined as spatial, a rebuild is needed. W e will get back to this later. S o a n o t h e r S M E import modifies t h e R _ G I . M M L . c o n f i g file -

but w h e n we

run S M E build t h e R _ G l . c o n f file will n o t be c h a n g e d . T h i s is a level o f protection to m a k e sure t h a t t h e config file with all t h e valuable spatial i n f o r m a t i o n is n o t i n a d v e r t e n t l y o v e r w r i t t e n , by r e - i m p o r t i n g and r e r u n n i n g t h e S t e l l a e q u a t i o n s t h a t

Simple Model, Complex Behavior

183

d o nor c o n t a i n this daca. T h i s might he a little confusing; however, it is importanc co p r o t e c t t h e spatial version ot che config file. T h e output from t h e last rebuild c a n he found in R _ G 1 . S I . c o n f . o u t , and if this is really what you w a n t t o d o , you c a n d e l e t e your R _ G l . c o n f file and r e n a m e t h e R _ G I . S I . c o n f . o u t into R _ G l . c o n f . T h i s is what we will d o n o w co get t h e following as che config file for t h e m o d e l . # global D S d . 0 , 4 8 )

n<1) s(4332) ngliO} op(0) OTi 1.0,0.0,20.0) d( 0) U T M ( 0 , 0 . 0 , 0 0)

UTM(1,1.0,1 0) $ R_Gl_module * ALPHA

pm(1)

* BETA

pm(1)

* GRASS

s(1) sC(C)

* GRAZING

ft(u)

* G_GROWTH

ft'u)

* GJNIT

pm(2)

*K

pm(0.100000)

* MU

pm(0.100000)

* RABBITS

s(1) sC(C)

* RJNIT

pm(1)

* R_MORTALITY

ft(u)

* TIME * UPTAKE

ft(u)

N o t e that t h e initial c o n d i t i o n s are n o w properly defined in chis file. W e are ready to run t h e model in S M E . However, first let us take a n o t h e r look ac t h e config file. W e have already guessed thar p m ( ) is a parameter in S t e l l a . W h a t e v e r value the parameter had in Stella, it was automatically transferred into t h e config file. Also, t h e state variables ( G R A S S and R A B B I T S in this c a s e ) are described by two c o m m a n d s , s ( ) and s C ( C ) . W h a t are they? T h e best available d o c u m e n t a t i o n for S M E is o n the web at htcp://www.uvm.edu/giee/SME3/ftp/Docs/UsersGuide.html. Most o f the c o m m a n d s are described chere, chough n o t in rhe most foolproof way. For t h e state variables, we learn that s ( l ) m e a n s that we will be using t h e first-order precision numeric method. W e mighc also learn that t h e rwo c o m m a n d s that were generated by t h e S M E build c o m mand are actually n o t quite consistent with the latest d o c u m e n t a t i o n : the s C { C ) c o m mand could be erased and instead the s c o m m a n d should be s ( C l C ) . However, S M E will scill run with t h e s C ( C ) c o m m a n d . " C " means that che variable should be clamped chat is, ic will n o t be allowed to b e c o m e negative. It is n o t unusual to find these kinds o f glitches in open-source code; after all, these guys are nor paid to write che fancy tutorials and d o c u m e n t s to make cheir software useful! W e have to either bear with them (after all, t h e software is free) or, even better, help them. W e c a n always c o n t r i b u t e our bug reports and pieces o f d o c u m e n t a t i o n chat we put cogecher while exploring t h e program. N e x c we need to configure t h e oucput. S o far ic is undefined; we do n o t k n o w what che program will output and where will it go. L e t us use che P ( 0 , 0 ) c o m m a n d to see h o w the state variables c h a n g e . T h e lines for G R A S S and R A B B I T S will now be: * GRASS

P(0,0l S(C1 C)

* RABBITS

P(0,0)S(C1C)

N o t e thac we have also g o t rid of the outdated s C ( C ) c o m m a n d , j u s t o n e more t h i n g before we run t h e model. T a k e a look ac t h e first line in t h e config file, t h e o n e that starts with #global. T h i s is a sec o f general configuration c o m m a n d s thar are

mmmm^-"

—

Systems Science and Modeling for Ecological Economics placed there by default by the translator. T h e two important o n e s t h a t we may want to c h a n g e right away are t h e O T { ) a n d t h e d ( ) c o m m a n d s . C h e c k o u t t h e S M E docum e n t a t i o n to learn more about t h e m . T h e d ( ) c o m m a n d sets up t h e debug level - t h a t is, the a m o n n t if information t h a t will be provided i n t o t h e c o m m a n d line interface. W h e n we h a v e d ( 0 ) , that is t h e m i n i m a l a m o u n t . It we want co see what e q u a t i o n s are solved in w h i c h order and what actually h a p p e n s during our model run, we probably n e e d to bump up t h e debug level, m a k i n g it d( I } or d( 2 ) . T h e O T c o m m a n d defines t h e t i m e - s t e p , t h e start a n d the e n d ot the s i m u l a t i o n Kighi n o w we h a v e O T ( 1 . 0 , 0 . 0 , 2 0 . 0 ) , w h i c h means tfutt we will run [ h e model with a tune-step o f I , statting trom day 0 and finishing o n day 2 0 . T h i s will n o t allow us t o »o beyond 2 0 . If we wish to h a v e a longer s i m u l a t i o n t i m e , we n e e d to c h a n g e it t o , say, O T { l . C , 0 . 0 , 1 0 C . 0 ) . N o w we c a n m a k e up to 1 0 0 steps. Finally we c a n run t h e model, using > S M E run S e e what h a p p e n s In t h e c o m m a n d line i n t e r f a c e we get |AV-Compuier:SME/Projects/R_GI v o i n o v % S M E run Spatial M o d e l i n g E n v i r o n m e n t , C o p y n g h l IC) 1995 (TXU-707-542), Tom M a x w e l l * "

S M E c o m e s w i t h ABSOLUTELY MO W A R R A N T Y

""

This is free s o f t w a r e , and y o u are w e l c o m e to i s d i s t n b u t e it

* " * under the t e r m s of t h e G N U General Public License Current proiect dnecsory is / D o c u m e n t s / S M E / P r o j e c t s / Current proiect is R_G Current m o d e l is R_G1 Current scenario is xxx Running S M E m o d e l R_G1 in serio 1 m o d e , c m d : /Documents/SME/Projects//R_G/Dnver/R^Gl •ppath / D o c u m e n t s / S M E / P r o j e c l s / -p R_G - m R_G1 -ci / D o c u m e n t s / S M E / P r o j e c t s / / R _ G / C o n f i g / R _ G l . c o n f -pause 0 -seen xxx info. Setting Project Nanne t o R_G mfo. Allocating m o d u l e R _ G 1 _ m o d u l e . ignorable 0 info Reading Config Files info: Opening config frie: /Documerit5/SME/Projects/R_G/Config/R_G1 confinfo R e a c i n g config file w a r n i n g : 1h!s p r o g r a m uses gets)}, w h i c h is unsafe. SME> Here, t h e driver scops a n d waits for us t o tell it what to do n e x t . It looks like gibberish, but m a y actually c o n t a i n some i m p o r t a n t i n f o r m a t i o n - especially if we run inco errors. T o run che model for 5 days, we use SME>r 5 Ii we h a v e t b e debug level set at d( 1). we will probably getinfo: Setup Events info: C r e a t e E v e n t L i s t s info ProcessTempora D e p e n d e n c i e s info: P.'ocessSpat a l D e p e n d e n c i e s info: C r e a t e E v e n i L i s t s

Simple Model, Complex Behavior

185

info: FilNnitishzBtionList mfo: Split & Sort Lists info. Setup Variables i n t o . Setting U p Frames & S c h e d u l e s info: Allocating M e m o r y into Posting Events mlo

Opened

xm!

File

,'D o c u m e n t s/S M E /Pro j e c t S/fl _ G/ M o d e i s/ R _G 1 /x x>: / R _ G l _

module, xmi. !njo

Executing

Event

R_Gl_module.StateVarlnit

ai t i m e

Executing

Event

R_G1_moduie:FinalUptfats_S_

0.000000 in(o

at

time 5.000000 TCL> 5 SME> T h e m o d e l n o w stops again, a n d a n o t h e r r c o m m a n d is required to c o n t i n u e . Let us run it till day 1 0 0 : S M E > r 100 Now ir stops and waits again. To quit, we do SME > X It is i m p o r t a n t ro ensure thac E n t e r is pressed after e a c h of t h e s e c o m m a n d s . T h i s is it. N o w where a t e t h e r e s u l t s ' G o t o P r o j e c t s , f R _ G L / D r i v e r G u c p u t . H e r e , we might n o t i c e t h a t t w o m o r e tiles have h e e n g e n e r a t e d : GRASS.PTSP_p_0 RABBlTS.PTS.P_p_. 1 T h e s e hies c a n n o t he seen until we h a v e quit t h e model r u n ; they appear o n l y after t h e X c o m m a n d has h e e n issued N o « ; that we h a v e e x i t e d SMF., t h e files should he there. T h e s e are simple timeseries, with output for G R A S S a n d R A B B I T S respectively. T h e first c o l u m n is t h e t i m e , t h e s e c o n d c o l u m n is t h e value of t h e state vari ahle. O n e way to look at these results is ro simply copy a n d paste t h e hies i n t o E x c e l or a n o t h e r spreadsheet program

W e c a n draw' t h e graph a n d see t h a t , after a c o u p l e

of oscillations, t h e G R A S S population c r a s h e s followed hy t h e slow dying off of t h e RABBITS

T h i s is n o t e x a c t l y what we w'ould e x p e c t from a standard p r e d a t o r - p r e y

m o d e l . W h e r e a r e those n i c e population n u m b e r s , going up and down i n d e f i n i t e l y ' O f course, we were r u n n i n g t h e model with t h e first-order Euler m e t h o d . T h a t is a pretty rough a p p r o x i m a t i o n . Let us switch t o a more a c c u r a t e n u m e r i c m e t h o d W e o p e n che config file a n d c h a n g e t o t h e fourth-order m e t h o d : * GRASS

P{0,01 s K M C )

* RABBITS

P{0,0) SIC4C)

N o t e t h a t previously we had s ( C l C ) , now we h a v e s ( C 4 C ) . T h i s does ir. It we rerun t h e m o d e l ( S M E run, t h e n r 1 0 0 } , exit ( X ) , go to t h e D r i v e t O u t p n directory and paste t h e output hies i n t o E x e c I, t h e n we will get what we. were e x p e c t i n g - n i c e lasting oscillations o l both variables. However, where are the spatial d y n a m i c s ' W e could get all this in S t e l l a without che trouble o f setting up t h e model in S M E . B u t how c a n we e x p e c t a n y t h i n g spatial if

186

Systems Scienrf? and Modeling fo' Ecologicst Economics

we h a v e n o r d e n n e d a n y t h i n g spatial i n o u r model? So far, we have s i m p l y r e p l i c a t e d the Scella m o d e l . N o w let us go spatial. First of a l l we w i l l need some maps t o describe rhe areas lor grass a n d rabbits. Suppose we choose t h e area s h o w n i n Figure 5.36. T h e s e days, t h e simplest way t o generate these maps is t o use A r c l n f o o r A r c G I S , t h e m o n o p o l i s t o n t h e G I S m a r k e t . H o w e v e r , if we r u n G R A S S , a n o p e n - s o u r c e G I S (do n o t confuse w i t h o n e of che variables i n this m o d e l ) t h a t w i l l also w o r k . A n y w a y , w h a t we need co d o is generace a s i m p l e ascii hie t h a t w i l l first o f a l l describe che study area i n o u r m o d e l . T h i s w i l l have I s inside t h e study area a n d Os e v e r y w h e r e else. Ic may l o o k like this i n o n e of t h e f o r m a t s thar S M E rakes, i.e. t h e M a p l I f o r m a t :

FILETYPE=INTERCHANGE ROVVS=62 COLUMNS=67 C E L L S I ZE=200 000000 FORMA T=DEC INFO=°.hunl wsh" DATA=0OOOQOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOQOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOO

00000000000000000000000000000000000 00000000000000000000000000000000000 0000000000000000000000000000000000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOO 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

0000000000000000000000000000000) OOOOOOCOCOOOOOCOOOOOOOOOOOOOOOOl

000000000000000000000000000011 1 1 oooooooooooooocoooooooooooo 1111 1111 00000000000000000000000000 1111 OOOOOOOOCOOOOOO00000000000 1111 0000000000000000000000000 1111 000000000000000000000000 I 1111 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 M M 000000000000000000000000 I M M 00000000000000000000000' 1 OOOCOOCOOOOOOOOOOOOCOCOI

I

OOOOOOOOOOOOOOOOOOOOOD1 11 OOOOCOOOOOOOOOOOOOODOl 00 0 00 00 0 0 00 0 C C 00 0 00 0

000000000000000000 ooooooooooooooo oooooooooooo oooooooooooo ooooooooo 00000000 0000000 0000 0 00 0 0 00000 ooooooo 00 oooo1 0 0 0 0 0 0 1

0000001

00 0 000 1

0000001 00 0 0001 0 000001 0000C0 1 0 0 0 0 0 0 1 00 00 00 1 00 00 0 0 1

I 11

1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111

1111 1111 1111 1111 1111 00 1111 00 00 0 ooooooo 1111 0000000001 I 1 1 00000000000111

0000001 0 0 00001 000000 1 000 0 001 0 000001 0 000001 ooooooo 0 0 0 0 0 0 0 0 0 0 0 0 111 ooooooo 0000000000000 1 I 1 ooooooo 00000000000000111 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

<

11 11 M M M M

1111 1111 M M I I M I 1 M 1 1 M

1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 111M i l

1111 1111 1111 1111 1111 1111 1111 1111 1111 M M

000000000000000000000000000000 00000000000000 1 0 0 0 0 0 0 0 0 0 0 0 0 0 000000000001 0000000000 0 00 M 1 1 1110000000 11111 11110 0 00 00 11111 1110000000 11111 1 100000000 11111 1 1 0 1 1 00000000 11111 0000 1100000000 11111 00000000000000 111 i i ooooooooooooooo 1 1 1 I oooooooooooooooo 1 1 1 00000000000000000

1 1 tooooooooooooooooo 1 1 100000000000000000 I 1

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

I 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1111 oooooooooooooooo 11111 ooooooooooooooo 111111 ooooooooooooOo II 11 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 I 1 IC 00 0 00 00 00 00

11)11111100 00000000 1111 oooooooo 1111 1 0 0 0 0 00 1111 1 0 0 0 0 00 1111 1 0 0 00 1111 1 I) 0 0 0 1111 1 D O 00 1111 11000 1111 11000 1111 11000 1111 110 00 MM 1 1 000 1111 10 0 0 0 1111 1 1 1 00C0 1111 00 0 00 1111 10 00000 MM I 00000 0 11M 0C001 I 100000 1111 oooooooooooo 000001 I 1 I 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

oooooooooooooooooooooooo ooooooooooooooooooooooooo oooooooocoooooooooooooooo ooooooooooooooooooooooooo ooooooooooooooooooooooooo oooooooooooooooooooooooooc 00000000000000000000000000 00000000000000000000000000 0000000000000000000000000000 ooooooooooooooooooooooooooooo

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Systems Scienrf? and Modeling fo' Ecologicst Economics c h o s e n . N o w we h a v e initialized t h e model as being spatial, but h a v e n o t identified any spatial variables: all o f t h e m a r e still treated as single numbers. T o d o that we c a n use a n o t h e r S M E c o m m a n d

- o i ( ) . It is called " o v e r r i d e i n i t i a l i z a t i o n , " a n d h a s t w o

parameters. Ii t h e first p a r a m e t e r is positive, then t h e variable is assumed t o be c o n s t a n t . It t h e s e c o n d p a r a m e t e r is positive, t h e n l h e variable is assumed t o be spatially distributed. S o if we configure, say, C R A S S as "GRASS

s!C4C) oi(0.1)

we should get what we want Next

• a spatially distributed variahle.

lec us deal with t h e g i a p h i c output

T h i s is h a n d l e d by t h e s o - c a l l e d

Viewserver, w h i c h we n o w n e e d t o start up. Let us add yet a n o t h e r c o m m a n d t o t h e previous line: •GRASS The

DDO s{C4C) oi(0,1) D D ( ) c o m m a n d establishes a c o n n e c t i o n with t h e Viewserver - a very

important piece o f software used t o display t h e results of spatial s i m u l a t i o n . T h e Viewserver should be started using t h e c o m m a n d startup_viewserver. It is b e t t e r t o d o it from a separate t e r m i n a l window, s i n c e t h e V i e w s e r v e r g e n e r a t e s a long c o m m a n d line o u t p u t t h a t will clog t h e t e r m i n a l that is b e i n g used t o run S M E Let us also g e n e r a t e spatial output for t h e o t h e r m o d e l variable, R A B B I T S : • RABBITS

DDO s(C4C>

N o t e t h a t in this case we d o n o t e v e n n e e d t o d e c l a r e t h e variable as spatial. In t h e m o d e l it is d e p e n d e n t upon a n already spatial variable ( G R A S S ) , so ir will l i e c o m e spatial automatically. O n c e we h a v e started t h e Viewserver and d o n e a n o t h e r S M E run, we c a n see t h a t t h e Viewserver receives output from t h e running model a n d a n e w data set is added to t h e list o n t h e left panel o f t h e Viewserver. If we h i g h l i g h t o n e of t h e data sets and t h e n c h o o s e a 2 D a n i m a t i o n viewer a n d click t h e " C r e a t e " b u t t o n , we will get an image o f t h e m a p that is now d y n a m i c a l l y c h a n g i n g as t h e variables c h a n g e their values across t h e whole area. You c a n watch h o w t h e C r a s s a n d R a b b i t s a l t e r n a t e t h e i r biomasses, c h a n g i n g Ironi minimal ( b l u e ) t o m a x i m a l ( r e d ) numbers (Figure 5 . 3 7 ) .

4

A

WiW* +.M+.M B

. 3X 7f l BF d ii gl LuirUe i 5 KK

Simple spatial dynamics when all cells are the same.

A. Grass (max. 2.043, mm. 0.32791, B Rabbits (max. 1 285. mm 0 7642).

r i

Simple Model, Complex Behavior

189

Let us now make the model spatially heterogeneous. SupiKise we have a spatially heterogeneous initial condition tor the G R A S S biomass. and that grass is not uniformly distributed but has different biomass in different locations. W e will initialize the G R A S S variable with
diA./Documents/SME/Projects/R_G/Data/Maps/Biomass.arc, /Documents/SME/Projects/R_G/Dala/Maps/Area.arc)

Again, we have to provide the full path to the map lile that we want to use. There is actually a better way to do it using the Environment file. This hie should reside in the Data directory, and it contains all the paths that we may wish to use in the configuration files. For this model, we will put: the following two lines into the Projects, / R_G/ Data/Environment tile: MAPS = /Documents/SME/Projects/R_G/Dala/Maps RMAP = /Oocumems/SME/Proiects/R_G/Oata/Maps/Area.asc T h e first line defines t h e Maps directory, which we seem to be constantly refer ring to. T h e other line is rhe full name of the reference map, or the scudy area map. which is used to crop all t h e other maps in the project. Now some of the lines in the configuration hie c a n be much shorter: $ R_G1 . m o d u l e giA.$(RMAP}.oefault.S!RMAP!) AL(O.O) • GJNIT

Figure 5.38

dlA,SjMAPS}/Biomass.arc.$|RMAP})

Another map used to define spatially heterogeneous initial conditions for Gross

190

Systems Scienrf? and Modeling fo' Ecologicst Economics M o r e o v e r , t h e biomass m a p that w c used t o initialize t h e model has values b e t w e e n 0 and 6 1 . T h e initial c o n d i t i o n thai we used before was 2 . It would b e n i c e if we could s c a l e t h e m a p ro s o m e values thar would be closer ro t h o s e we had originally, a n d we c a n use t h e S O c o m m a n d t o d o t h a t . T h e syntax of this c o m m a n d is S ( a , b ) , w h i c h m e a n s that if x is t h e input value t h e n r h e result of this c o m m a n d is y — a * x -I b. S o finally it we use t h e c o m m a n d • G . I N I T d(A.${MAPS}/B>omass.arc,$fRMAP)) S( O l e + 00.1.0) this m e a n s rhar we will input t h e map from t h e Biomass.arc hie, t h e n e a c h value will be multiplied by 0 . 0 1 a n d added t o I. T h a t will be t h e result used in t h e s i m u l a t i o n s . A l s o n o t e that we n o longer n e e d t h e o i ( 0 , I ) c o m m a n d , s i n c e we n o w h a v e t h e initial c o n d i t i o n chat Initialized che variable as a spatial o n e , w h i c h ensures that all l h e rest of t h e variables c o n n e c t e d t o rhe s p a n a l o n e will also b e spatial. T h i s is a little more interesting: n o w there are s o m e spanal variations, and t h e r e are s o m e differences in h o w various cells e v o l v e (Figure 5 . 3 9 ) . However, t h e r e is still n o u u e i a c i H i n b e t w e e n cells, a n d t h e real spatial c u n t e x t is n o t present. W e simply h a v e a w h o l e b u n c h of models running in sync, but they do n o t i n t e r a c t with e a c h other. M a k i n g c e l l s " t a l k " t o e a c h o t h e r is a little more c o m p l e x t h a n a n y t h i n g we h a v e d o n e s o far. W h e r e a s until n o w we h a v e simply used s o m e predefined c o m m a n d s , and rhe model we built m S t e l l a , from now o n :f we are to d e f i n e s o m e meaningful spatial i n t e r a c t i o n we will need t o do some p r o g r a m m i n g . T h e r e are some modules that we c a n use in rhe Library o f Hydro-Ecological Modules ( L H E M -

http://giee.uvm e d u / L H E M ) ; however, there arc n o t t o o m a n y

things we c a n d o with those pre-designed modules. If we really want t o lie able t o build c o m p l e x spatial models, we will probably need t o be capable o f some level u f C + + programiiung. S M E supports so-called L s e r C o d e and offers full access t o its classes and methods, which c a n significantly help us in designing our o w n c o d e for spatial dynamics. S u p p o s e for t h e R a b b i t s

Grass model we wish t o allow rabbits to m o v e

b e t w e e n cells in search ot b e t t e r grazing c o n d i t i o n s

W e will assume that w h e n e v e r

rabbits find that there is more grass in t h e n e i g h b o r i n g cell, a c e r t a i n p r o p o r t i o n of

A

B F iBg u r ea 5i K . 3i M 9 • E I E

Spatial dynamics with no migration.

A. Grass (max. 1.490, min. 0 5), B. Rabbits tmax. 1.142. mm. 0.861.

Simple Model, Complex Behavior

191

rabbits from the torrent cell will move to the cell with more grass. Let us write the code that will describe this behavior of the predator: I

/ • " <

i n c l u d e " Rabbit.h"

void MoveRabbits( CVanable& Rabbits. CVanab'eSi Grass.. CVanable& Rate i // moves rabbits toward more grass, if there are less rabbits there ii arguments come from MML.config fi e, first arg is always variable being configured. 1 Grid_Direction il; float fi, R_moved = 0.. DistributedGnd& grid • Rabbits.Gndi): grid.SetPointOrdermgiO), // sets grid ordering to default ordering (row-coll (ordering #0) Rabbns.LinkEdgesO; Grass LinkEdgesi): static C Variable" R Flux = NULL, if(R_ Flux = = NULL ) R._F ux - Grass.GetSimilarVanablei"R_Flux"}; :// intermediate increment to Rabbits R.Flux- > Set(0.0}; for! Pix p — grid.rirstO, p; grid next(p) ) f

cons* OrderedRtjnt& pt = gnd.GetPoin:(p); II sets current Poifij if! !gricf.onGricffptl 1 continue; // (onGrid = = False) • > Ghost Po'nt float g_max - Grass(pT>; Pix p_max = p: // for each point calculate where is the max Grass in the vicinity fori il = first GDI); moreGDull; merGD(il|) { // e n u m Grid_Direction (NF = 2. EE, SE SS. SW. W W . NW, NNI Pix rp = gud.NeighborPixl p, .'I 1, !i relative t o pt, takes e n u m Gnd_D rection as arg ftt rp ] ( const OrderedPoint& rpt =•• grid.GeiPomtfrpI; if ( Grass(rpt) > g_rria>! j I

3_max - Grassirpi). p_max - rp,

) ) I

208 Systems Scienrf? and Modeling fo' Ecologicst Economics

const O r d e r e d P o m t & p t _ m a x = grid GetPoint(p_max); // sets c u r e n t P o m t // if t h e r e is a cell in t h e vicinity w h e r e t h e r e is m o r e Grass, t h e n a // p o r t i o n of Rabbits m o v e s t o that cell if ( g _ m a x > G r a s s ( p t ) ) fr = ( R a b b i t s ( p t ) > Rabbits(pt_max) ) ? ( R a b b i t s ( p t ) - R a b b i t s ( p t _ m a x ) ) * Rate(pt) : 0; (*R_Flux)(pt_max) + - fr; (*R_Flux)(pt) — - fr: R _ m o v e d + = fr; }// e n d area loop Rabbits. A d d D a t a ( * R _ F l u x ) , printf ( " \ n m f o : Rabbits m o v e d = %f," R_moved);

}

S o here we have only R a b b i t s m o v i n g horizontally from o n e cell t o a n o t h e r in search o f a b e t t e r life. H o w d o we tell S M E that t h e r e is s o m e t h i n g n e w t h a t t h e m o d e l wants t o take into a c c o u n t ? First, we g o all t h e way b a c k to t h e M M L . c o n f i g file t h a t we c a n find in t h e C o n f i g directory. In this hie we add a c o m m a n d for R a b b i t s : * RABBITS

UF( Rabbit,MoveRabbits,GRASS,RATE)

H e r e , R a b b i t is t h e n a m e o f t h e file t h a t c o n t a i n s t h e a b o v e C + + c o d e . A c t u a l l y its n a m e is R a b b i t . c c , and it resides m t h e U s e r C o d e directory. M o v e R a b b i t s is t h e n a m e o f t h e f u n c t i o n in this hie t h a t we use. G R A S S and R A T E are two variables that are passed to this f u n c t i o n . W h i l e G R A S S

h a s always been there, R A T E is

new. T h e way we get it i n t o t h e config hie is by modifying t h e S t e l l a m o d e l and adding a n o t h e r variable. O n c e again, we h a v e to e x p o r t t h e e q u a t i o n s a n d t h e n d o t h e " S M E i m p o r t " c o m m a n d . A l t e r n a t i v e l y , we c a n modify t h e e q u a t i o n hie that we c r e ated earlier from S t e l l a e q u a t i o n s . W e simply n e e d to add o n e line: rate = 0.5 and t h e n we c a n also d o this by h a n d in t h e R _ G 1 . M M L . c o n f i g file. N o t e , however, that this is s o m e w h a t risky, s i n c e it is very easy to forget about s o m e o f these small m o d i f i c a t i o n s o f che e q u a t i o n hie, and there is n o way we c a n import these modifications from t h e e q u a t i o n s to t h e S t e l l a model. A s a result, o n c e we have finally decided that we wish ro modify t h e S t e l l a model for s o m e o t h e r reason later o n , most likely we will forget a b o u t these modifications. W h e n

taking t h e e q u a t i o n s from

S t e l l a and c r e a t i n g a new e q u a t i o n file, we will lose all these previous c h a n g e s . T h e m o d e l will suddenly perform q u i t e c o n t r a r y t o e x p e c t a t i o n s , a n d it will take a while to figure o u t why a n d to redo all t h e little updates. S o while every n o w and t h e n it seems very simple to modify just t h e e q u a t i o n file, actually it is m u c h better if all t h e modifications are d o n e directly to t h e S t e l l a m o d e l . A s we remember, w h e n e v e r t h e e q u a t i o n s or t h e M M L . c o n f i g hie is c h a n g e d we need to do t h e S M E import c o m m a n d . T h e n we c a n d o t h e S M E build c o m m a n d , and update t h e config file to add t h e R A T E parameter to it as well. R e m e m b e r - e i t h e r

Simple Model, Complex Behavior

193

it h a s to he d o n e by hand, or t h e hie c a n he r e n a m e d t o use t h e R _ G L S I . c o n f . o u t instead. A s a result, we get: * RATE

pm(0.5)

A r e we ready t o r u n ' A l m o s t , but there is still o n e g l i t c h t o fix. T h e variables that we have b e e n passing t o t h e newly designed f u n c t i o n t o m o v e R a b b i t s are all assumed to be spatial. M o v e R a b b i t s f CVariable& Habbus. CVanable& Grass. CVan3bie& Hate ) However, t h e R A T E p a r a m e t e r as we defined it a b o v e is a scalar. T h e r e is an easy fix. Just add t h e override c o m m a n d * RATE

p m ( 0 5) oi(O.I)

and you will he back in t h e game. A l t e r n a t i v e l y , you could also define this p a r a m e t e r as a map: * RATE

d ( A . $ ( R M A P ) . S { R M A P } | S|.5e + 00.0.0)

H e r e we used t h e study area m a p to initialize this parameter, w h i c h , with this s c a l i n g factor, is identical t o what we did a b o v e

However, this could be any m a p ,

w h i c h would probably be t h e o n l y r e a s o n a b l e way t o define this p a r a m e t e r if we w a n t e d it to be spatially h e t e r o g e n e o u s A l t e r n a t i v e l y , if we d o n o t want this p a r a m e t e r t o be spatial, we must n o t refer to it as if it were spatial in t h e c o d e . R e p l a c e R a r e ( p t ) for R a t e V a l u e ( ) . R a t e . V a l u e O is a scalar, it will n o t n e e d t o b e initialized by a m a p o r a spatial variable. It will take pm(0.5). Finally, we are ready t o h i t t h e " S M E r u n " c o m m a n d a n d w a t c h s o m e t h i n g m o v ing across che landscape - rabbits h o p i n g from o n e place t o a n o t h e r , grass dying a n d regrowmg back w h e n t h e predators leave, and s o o n (Figure 5 . 4 0 ) . C e r t a i n l y , this was n o t as easy as putting t o g e t h e r a m o d e l in S t e l l a , o r e v e n S i m i l e . However, for s o m e b o d y c o m f o r t a b l e with C + + it would n o t be a big deal and actually may r u m out to be simpler t h a n learning t h e n e w formalism required for Simile

O n c e we are in r h e programming language m o d e , we h a v e all t h e power we

A

B

• Fat«i»K l '0l i g u r eJ 5- .t 4

Spatial

dynamics with migration towards the cells with higher density of Grass.

Clusters of high density are formed when Rabbits from several cells jump into a cell with higher Grass abundance. A Grass (max 1 490. min 0.5), B. Rabbits (max. 1.142, mm 0 861

194

Systems Scienrf? and Modeling fo' Ecologicst Economics n e e d to create a n y c o m p l e x model S o in a way, S M E m a y be treated as a n i c e interface b e t w e e n S c c l l a and C + + power m o d e l i n g .

5.5

Conclusions A very simple model c a n p r o d u c e an amazingly diverse c o l l e c t i o n o f b e h a v i o r patterns. T h e fact that t h e p r e d a t o r - p r e y m o d e l c o n t a i n s n o n - l i n e a r i t y makes it a very e x c i t i n g system t o e x p l o r e . A f t e r many g e n e r a t i o n s o f m a t h e m a t i c i a n s a n d modelers studying t h e system, it still every n o w a n d t h e n produces s o m e i n t e r e s t i n g results, especially if we add s o m e detail in e i t h e r t h e structural or t b e spatial i n t e r p r e t a t i o n . T h e r e are probably hundreds if n o t thousands of papers about t h e d y n a m i c s in such o r similar t w o - s p e c i e s systems. W h a t is always most intriguing about m o d e l s is w h e n we find some e m e r g e n t properties t h a t were not at all e x p e c t e d when we first looked at t h e system. F o r e x a m p l e , t h e fact that pure species i n t e r a c t i o n s may produce persistent o s c i l l a t i o n s in p o p u l a t i o n n u m b e r s could b e hardly e x p e c t e d . W i t h e v e r y t h i n g c o n s t a n t in t h e system, with n o e x t e r n a l forcings, n o c l i m a t i c or e n v i r o n m e n t a l c o n d i t i o n s i n v o l v e d , we still get variability tn species populations. S y s t e m s w i t h linear f u n c t i o n a l response a r e usually more p r e d i c t a b l e . Ic is when we find f e e d b a c k s t h a t h a v e a n o n - l i n e a r e f f e c t in t h e system chat we should e x p e c t surprises. T h e s e systems need especially careful analysis. T h e y are also hardest t o analyze analytically. Providing for spatial h e t e r o g e n e i t y o n l y adds t o t h e list o f surprises. W h y would t h e spatial distribution m a k e t h e p r e d a t o r - p r e y o s c i l l a t i o n c o n v e r g e . ' W h y does it stabilize che system:' H o w g e n e r a l c a n these c o n c l u s i o n s lie? D o e s this m e a n

chat

more diversity in t h e system also means more stability? I low far c a n we go in this sort of generalization? T h e s e are all e x c i t i n g questions t h a t beg further research.

Further reading I lolling, C.S. (1959) Some characteristics of simple types o f p red at ion and parasitism. The Canadian Enromoirjjp.ti, 91: 385-389 - A classic work on trophic tnwracuvms w

GII* A

qudit ELL;desc r fpEion oj functional

when trophic functions

are considered

resportces

and toed,

.'N

especially

poptuuaon eco.'o^F. for

example

oj what iinaiNdcoi iiiidies l u h riri ;n research

Ttvo-jpecies communities, including VoJierra model art

always

cited

modeling.

Svireihev Yu, M and Lugofec, D.O. (19S ) SwMuy of 3'Ohs^cal Communities. T h o is an excellent

One of the .first

AITTCCJ.SI

M i r Publishers -

uj population

;i>nainiC5.

twll presented, tu u^H as the theory of

trophic chains. The hunk akacmiert the contents oj the clciuic paper by Kolmogorov, A . N . which was published in Italian i n 1936: Sulla Teoria di Volterra della Lotta per I'Esisrrcnia. G i u m Imntuw Ital. Attwm, 7, 84-80. Unfortunately the bonk is also quite hard to gel Some of the ideas have been further develu|ied jn Logofet D . O i 1993). Mam'cej and Graphs Stability PnMerns in Mathematical Ec^ogy, C R C Tress H-i read more dewiis ahum the wotwn in (he Ye/iotustone

rhe article by Virginia Morel I (2C07).

Aspens Return to Yellowstone, W i t h Help From Some Wolves. Science, Vol, 317(5837): 4 3 8 439. The amoving wry about the effect of bine crnh on riune /wmoacm is reported by Cheryl Dybas on an ,\'5F web sire at http://www.nsf fj:)v/tfcl/lpa/news/C2/[ip0209l6.hcir> .Si'niie can be fcn
Simple Model, Complex Behavior The Spatial

Modeling

Environment,

S M E , is an open

sourceForge.nec/projects/smodenv. Some example

sou ice project

195

cm SourceForge

See hcrp.//

fyrujects and latest developments

related

to ihe

SME cein be found at h t t p : / / V v v w . u v m . e d u / g i e e / I D l i A S / Some ideas about

me role of spatial

interactions

M a y n a i ' d - S m i t h , J. (197S). Models in Ecology were studied for so-called species.

There

metapopulanoivi,

is even special software

in adding

stability

u'liicli are collections

packages

to the system

can be found

C a m b r i d g e U n i v e r s i t y Press. Latei on these developed

cf interacting

populations

to study such populations.

oj the same

R A M A S is one

of those (.see hrtp^/wvvu'.rainas.com/inpmodcU.htin). To learn more about metapojmlauons example,

Hansla, I., Gaggiotci, O . eds. ( 2 0 0 4 ) . Eco/o&y. generics, and evolution

nons. blseviei A c a d e m i c Press.

in

effects

of

see foi metapopula-

6. W a t e r 6.1

Modeling as a hydrology primer

6.2

Unit model

6.3

Spatial model

6.4

Conclusions

SUMMARY T h e r e are critical natural resources that ate essential for human survival, and water is certainly one of them. T h e dynamics of water, its quantity and quality mirror what is happening at the watershed, and can serve as an indicator of overall e n v i r o n m e n t a l quality. NX/e fiist consider various parts o f the hydrologic cycle, and some o f the different processes that move water and that define its quality and quantity in diffeient storages- W e then put these processes together into a unit model lhat can describe dynamics of water in a small, confined and spatially homogeneous plot or cell. A variety of temporal, spatial and structural scales and resolutions may be considered, as dictated by the goal of the modeling effort. W e then presenc several ways in which water can be described over spatially heterogeneous area. T h e lumped modeling approach uses relatively large spatial compartments or hydrologic units, which are then c o n nected over a stream network. In the grid-cell approach, local dynamics are replicated across an array of grid cells i hat are driven by raster maps for variables and parameters It time is not important, it is better to focus on spatial aspects using a C I S approach.

Keywords Excludable and rival resources, scoping model, rainfall, snow/ice, surface water, groundwater, unsaturated zone, infiltration, precipitation, Julian day, evaporation. N a t i o n a l C l i m a t i c Data C e n t e r , photoactive radiation, bi-flow, porosity, transpiration. percolation, field capacity, soil moisture, hydraulic, conductivity, soil types, M e L i l e u c a , Delay function, T R 55, retention, curve number, surface roughness, horizontal water tiansport, vertical water transport, lumped models, hydrologic units, HSr-'b; S W A T , grid-based models, S M E , C I S - b a s e d models, .scorn".water, ram barrel, retention pond, rain garden, U D A R , A r c G I S , watershed management *

* *

Water, energy and land are the three most crucial limiting resources on this planet. T h i s makes it especially important t o understand how the systems related to these resources operate, the most elficient ways to control the depletion of these resources, and how the resources can be restored if damaged. In this chapter, we start with water. W a t e r is essential for life o n this planet. T h e water c o n t e n t of a human body is about 6 0 percent. Humans can survive for more than ) weeks without food, but for only

197

198

Systems Scienrf? and Modeling fo' Ecologicst Economics 3 days without water. There are some reports ol longer survival times, up to as many as 1 5 days; however, irreversible damage to the organism is most likely to occi.it earlier than thac, and in any case it will be thirst rather than hunger that will kill first. Water is also required tor other organisms and plants to persist. It is an important transport mechanism that delivers nutrients to the plants. A t rhe same tune, ir provides a mechanism tor pollution reduction through dilution W h i l e most ecologists wil tell you that "pollution dilution is not a solution," until recently it was probably the main - it not the only - way to remove toxins and waste from our environment. O r rather to make them less toxic, since dilution ceitainly does not remove them. In 2000, Fortune

magazine predicted thac water "will he to the 21st century

what oil was to t h e 2 0 t h . " Note that as long as we rely upon purely renewable waiter (as well as energy), ir is noivtival and non-excludable. T h a t is, solar eneigy and mmfall are available, more or less uniformly, over vast territories. W h o e v e r is there has access to thac waccr and energy W e c a n n o t prevent our neighbor from having equal access to sunshine or rainfall, or collecting it in some way W e cannot exclude someone from using it, and since there is n o rivalry ir makes no sense to attempt n> do so. Certainly there may be geographical differences. W e know that there is very much more water in the Pacific North West than in the Sahara, but these are regional distinctions. Locally, everybody in the Pacific North West still has equal access to rainfall and sunshine, just as everybody in the Sahara has equal access t o the rainfall and sunshine there. However, as soon as we need t o dip into reserves, into fossil water or energy, or even into the temporary reserves (lakes, reservoirs, or forest and crop biomass), immediately the resources become excludable and rival (Daly and Farley, 2 0 0 4 ) . W e can put a fence around a reservoir, privatize a forest, or outlaw pumping water from underground - like Israel did in Palestine. Tins changes the whole political landscape, and requires dillerent types of management. As resources become scarcer and we dip into stocks, we are creating potential for conflict situations (water and energy wars) Let us consider some simple models related to the water cycle, and figure out how they can he used to increase our understanding of what is happening with water

6.1

Modeling as a hydrology primer As in other models, we should first decide on the spatial and temporal scales that are to be used in our hvdrologic model A t varying temporal scales processes look fairlydifferent Consider a major rainfall event when, say, during a thunderstorm there is a downpour that brings 1 0 c m of rain in I hour, then the storm moves away and there is no more rain over the next 23 hours. If we assume a 1-minute rime-step in our model, we will need t o take into account che accumulation o f water on the surface, irs gradual infiltration into the soil, and the removal of water hv overland flow. If we look more deeply into the unsaturated layer, we c a n see how the front of moisture produced by the infiltrating water will be moving downwards through the layer of soil, eventually reaching the saturated layer. After the ram stops, m a while all the surface water will be removed, either by overland flows or by infiltration. A new equilibrium will be reached in rhe unsaturated layer, with some of rhe water accumulating on top of the saturated layer and effectively causing us level to rise somewhat, and che rest o f che water staying in the unsaturated layer, increasing the moisture c o n t e n t of soil,

Water

199

N o w suppose rhat rhe model time-seep is I day. T h e picture will be rotally dilferenr. In l day we will see no surface water ar alt, e x c e p t in rivers or stream*

In

o t h e r parrs o f rhe landscape, rhe warer will already have either got into t h e soil or run downhill to a nearby srieam ot pond. T h e unsaturated layer will nut show any water-front propagation; ir will have already equilibrated at the new state of moisture c o n t e n t and groundwater level. T h e processes look quite different in che model. A n d we probably already needed ro know something about rhe hydrologic processes in our system ro figure all this out Similarly, the spatial resolution is important. If all rhe variables are averages over a certain area, then within this area wc do n o t distinguish any variability, and the amounts of surface water, snow/ice, unsaturated and saturated water ;ire considered lo be rhe same If we are looking at a l - n r cell this does not cause any problem, and ic is easy to imagine how to measure and track these variables. However, il we are c o n sidering a much larger area - say I km" - then within a single cell we may find hills, depressions, rivers and ravines. T h e geology and soils may be also quite different, and need to be averaged across che landscape W e may be able to crack many more processes, but t h e model cost will increase accordingly as we will need far more data and greater computer power to deal with these spatially detailed models. For the first iteration of our modeling process, let us assume that the area of interest is a small watershed with quite uniform geo-morphologicnl conditions, with more, or less homogeneous soils, and let us suppose that we wish to figure out the amount of water that drains off this watershed into t h e river downstream. W i t h this goal in mi nil, we can probably consider the syscem using a daily time-Step - at least as a first iteration. A simplified conceptual model o f hydrologic processes for this system is presented in Figure 6 I T h i s diagram is only the tip of the iceberg, with a lor of fairly complex processes that may be further described in much more detail

At

this point, it is important to decide o n the most important featmes of t h e system that need be considered. W e chose rhe following four variables for this general model: 1. S U R F A C E W A T E R - water on t h e surface o f the land {in most cases it is in rivers, creek-, ponds and depressions). 2 . S N O W / I C E - at freezing temperatures surface water becomes ice. which then melts as temperature rises above 0 ° C . J . U N S A T U R A T E D W A T E R - the amount of water in the unsaturated laver o f ground. Imagine the ground as a sponge; when we pour water o n t o it the sponge will hold a certain amount before it starts dripping. All rhe time water can still be poured o n t o and held by the sponge, it is in the unsaturated c o n d i t i o n , 4 . S A T U R A T E D W A T E R - the amount of water tn the saturated ground O n c e the sponge can no longer hold additional water, ir becomes saturated. A s with surface water, if we add water to the saturated zone, it' level increases. T h e s e variables are c o n n e c t e d by a variety of processes that we also need to understand in order Co build a meaningful model. W h e n working on c o m p l e x models, it helps considerably if we split lhe whole system into c o m p o n e n t s , or modules, anil develop some simplified models for these modules. It is very likely chat some modifications

will be needed when pulling all the modules together again; however, as

previously discussed, it is so much easier to deal with a simplified model than to get lost in the jungle of a spaghetti diagram o f a complex model with numerous processes and interactions, and n o clear understanding nf what affects what

200

Systems Scienrf? and Modeling fo' Ecologicst Economics

\ /

Transpiration

JI Evaporation

Snow Overland llow

Infiltration Unsaturated waler Percolation saturated T exchanges Saturated waler

• a t « J i W i B

Groundwater flow

Conceptual model ol unit hydrology

Note that this diagram describes certain processes as if they were spatially distributed with a horizontal dimension present Irunoff "moves" water from rainfall to a pond, saturated water also movesl. In fact, when we run the model we assume that all these variables are uniformly distributed over the whole area and are represented by "point" quantities or concentrations.

Modeling is truly an iterative orocess As stated msny times before, w e want to know the spatial and temooral scales before w e start building the model. Bui h o w do w e figure them out if w e have only a vague idea about the system? What are l h e processes involved? At w h a t t'mes are they important, and do w e want t o include t h e m at all? Or perhaps there are s o m e other important processes that w e a ' e simply unaware of Indeed, there is no prescribed sequence of events Perhaps you want t o start w i t h a socalled 'scooing m o d e " - a model that would put together whatevei you already know about the system in a ratner qualitative format, omitting al1 the details ihat a'e not dear, outlining the system in general and lhe processes that we think are important. This you can start discussing w i t h colleagues and w i t n potential future users of the model. These users are the ones w h o formulated the initial goal of the study, so they are most likely t o know something about the system Start talking to t h e m or. even better, engage them in a participatory

mod-

eling process - something w e will be discussing in a lot more detail in Chapter 9. In any case, do not think that there is anything final in your a e o s ons a b o j t t h e scales and processes There will always be a reason end a chance to come back and make improvements. Thai is the beauty of computer models: they exist in virtual reality, to build t h e m you do not have t o have something cut. ploughed, extracted or destroyed, and you can easily modify or relocus them if necessary

W a t e r on t h e surface T h e surfocc tuaier variable is used to model water on the surface o f the land. If we are looking at an area with n o steep gradients and fairly high potential rainfall (for example, the Florida Everglades 01 other wetlands), then surface water can accumulate

Water

201

in significant amounts before it is absorbed by the soil. In this case it is necessary to consider the process that connects the accumulated surface water and the underlying unsaturated layer. This process is known as infiltration.

In most terrestrial areas with

steeper slopes, most of the surface water will drain oft into rivers, creeks, ponds and depressions in which n will accumulate over a layer of saturated water. Therefore, there will be no infiltration. Instead, there will be an exchange process between the surface waler and the saturated layer. It is hard to isolate a unit of surface water without c o n n e c t i n g it with the surrounding neighborhood. Much uf the surface-water transport is due to horizontal fluxes, and therefore a box-model approach will be only approximate when modeling surface-water dynamics. However, with appropriate spatial and temporal scaling we can think of an aggregated unit model to represent surface water in a homogeneous unit ccll, assuming that wc are modeling the total amount o f water over a large enough area and one that can somehow be isolated from the other territories. T h i s can be a small watershed, or an agricultural held, tor which we can monitor the inflows and outflows. A simple conceptual model can lie described as in Figure 6.2 T h e r e arc two major processes involved: precipitation Precipitation

and

infiltration

is probably the process that is intuitively most obvious. W e deal

with precipitation in our everyday lives when wc decide whether we might need an umbrella on going out for the day. T h e amount of precipitation is what we are concerned with when building a hydrologic model. It is also important 10 know m what form (liquid or solid - rain or snow) the precipitation will arrive. Precipitation is recorded, by most o f the meteorological stations, in millimeters or inches per day A sample data sheet for precipitation registered at Baltimore Washington Airport, M D in 1 9 9 6 is shown in Figure 6 . 3 . In Figure 6.3 0 . 0 T stands for titices, which means that the precipitation was recorded at levels below measurement accuracy. In many cases it is possible to find meteorological data for a specific area at the National Climatic Data C e n t e r ( N C D C : http://www.ncdc.noaa.gov/)

For example, on entering this site and choos-

ing Maryland, then the station at Baltimore Washington Airport, the relevant data can be found. A graphic can also be generated for a cable such as that reproduced here. T h e data can be downloaded in numeric format to use in a model. Temperature is important for us to decide whether the precipitation is rain or snow. T h e Snow/tee model below describes this process. Infiltration

is the process by which water from the surface is taken into the

ground by means o f gravitational and capillary forces. T h e rate of infiltration defines how much water will be left on the surface to contribute to the rapid runoff, and how

Precipitation

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Evaporation

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To/From snow/ice

Surface water

I

Infiltration

upflow

202

Systems Scienrf? and Modeling fo' Ecologicst Economics DAILY Station BALTIMORE W AHPT Pariwnetw ftcp PO Code MD Latitude N39:11:00 Sbi ID 465 l o n u i t u d e W076:40:00 C o u n t y ANNE ARUNDEL Elevationtfn.) 45.1

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13

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Motice the treacherous inches/day used as a unit in this data set.

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A bare-bones Stella model for local surface hydrology, and output from this model.

m u c h will g o i n t o t h e ground and t h e n travel slowly t h r o u g h t h e porous media. W e will c o n s i d e r infiltration in m o r e detail below, w h e n discussing t h e unsaturated water storage. A S t e l l a m o d e l t h a t corresponds t o this c o n c e p t u a l model o f surface hydrology is p r e s e n t e d in Figure 6 A . W e h a v e o n l y o n e s t o c k a n d t w o flows, a n d n o

Water

203

feedbacks. In this case we assume rhar the surface warer is delivered by rain and then gradually infiltrates into the ground. T h e rainfall is fast, whereas infiltration is slow. However, rainfall occurs only sometimes, whereas infiltration is continuous. T h e equations are:

SurfaceJA/alerltl = Surface_Water(t - dtl + (Ramfail - Infiltration) * dt INIT Surface .Water •= 0.01 DOCUMENT: The surface water is assumed to be a function of t w o processes. Rapid rainfall provides surface water, which then gradually infiltrates into the ground. Rainfall = Precipitation "0 0254 DOCUMENT: Converting rainfall m inches/day to m/day Infiltration = 0 01 DOCUMENT. Infiltration rate (m/day!

In reality tins tale depends upon soil charactenstics

habitat type, slope, panem of rainfall. DayJul = mod(time-1.365) + 1 DOCUMENT Julian day, 1 thru 355 This is a counter that resets the day to zeio after 365 iterations Needed to use tbe same graph function for several years of model runs Piecipitation = GRAPH (DayJul) (1.00, C 02), (2.00, 0.34), (3.00, 0.00). (4 00, 0 07), (5 00, 0 00). (6.00. 0 18). (700. 0 46) (354, 0 00), (355, 0.001, (356, 0.15). (357 0.00). (358. 0 00). (359. 0 001. (360, 0 02), (361 0 00). (362. 0.00). I363. 0 00). i364, 0 001, (365, 0.001 DOCUMENT: RainfalMrom Beltsville M D 1969 (m/dl

Note a few interesting features here, which may be helpful in other models First, notice the units. W e have put together the model in meters and days, as would normally be the case in science. However, the data came from a U S meteorological station where they still use inches lor measurements. Therefore, we need the converter Rainfall = P r e c i p i t a t i o n * 0 . 0 2 5 4 where we use the conversion factor 1 inch = 0 . 0 2 5 4 ni. It is extremely important to make sure that all units are consistent throughout the model. W h i l e Stella offers some background functionality to help track the units, it is really in your best interest to

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Systems Scienrf? and Modeling fo' Ecologicst Economics finding ii station that is located close to the Site being modeled. It is most problematic to obtain data on solar radiation (also known as photoactive radiation - P A R ) For some reason it is not o n e of the standard observations, and direct measurements are rare. Therefore, in our model we will estimate solar radiation based on the latitude o f a site and the generally available information about precipitation. All these factors arc put together to estimate evaporation in a simple model in Figure 6.6. T h e corresponding equations arc as follows:

A = 720 52-6 6 8 * L a l D e g A i r j e m p _ d e g C = ((Air_temp_clegF-32)*5/9 - Air_temp_minC)/2 Air_lemp_nnnC - (Ait_temp_minF-32l*5/9 B - 105 94"(LatDeg-1748) / v 0 27 C = 175-3 6 " L a t D e g cloudy = if Precipitation > 0 then maxt0.10-1 155"(vap_press/tPrecipitation'25 4 * 3 0 ) ) A 0 5) else 0 CLfactor = 0 15 DayJul = moditime-DT.365} + 1 Evap_M = EvapBeU "0.0254 Hyd_evap_calc = *Hyd.evao_rc*SolRadGr/585*pan_CW*pan_CT"pan_CH LatDeg = 39 0 pan_CH = 1 035 + 0.240*(Humidity/60) A 2-0 2 7 5 , ( H u m i d i t y / 6 0 ) A 3 pan_CT = 0.463 + 0.425" (Air_temp_degC/20j + 0 112'!Air_temp_degC/20)' A -2 pan_CW = 0 672 + 0 4 0 6 * I Wind/6 7) + 0 078"(Wmd/6 7 ) * 2 SolRad = A + B'COS(T) + C*SINi.T)*2 SolRadGr - maxfO.SolRad'il-CIJactor'ci'oudy)) T = 2 / 3 6 5 ' PI * (DayJul-173) vap_press = Humidity*6 1078*EXP(17269 < Air_temp_degC/lAir_temp_degC + 2373)) Wind

Wmd_speed* 1 852/24

•Hyd_evap_rc = 0.0028 Air_temp_degF = GRAPH (DayJul) !1 00. 44 0). (2 00. 42 0). (3.00. 51.0), (4.00, 42.0), (5.00. 3 8 0i. (6.00, 43.0). (700. 4 4 0). A i r j e m p _ m i n F = GRAPH (DayJul) (1.00, 19.0), (2 00, 21.0), (3 00. 22.0), (4.00, 26.0), (5.00, 19.0), (6 00, 21 0), (700, 32 0). EvapBeit - GRAPH (DayJul) (0 00, 0.00), (1.00, 0.00), <2.00, 0 00), (3 00, 0.00), (4 00. 0 00). (5.00. 0.00), (6 00, 0.00), . Humidity = GRAPH (DayJul) (1 00. 670). (2 00. 71.0), (3.00, 69.0). (4.00. 50.0). (5.00. 65 0), (6.00. 88 0). (7.00. 90 0).. Precipitation = GRAPH (DayJul) (1 00. 0.00). (2.00, 0 00), (3.00, 0.00), (4.00. 0 00), (5.00, 0 00), (6.00. 0.05), {7.00. 0.41!,... W m d . s p e e d = GRAPH (DayJul) (1.00. 129), 12.00. 113). (3 00. 1481, (4 00, 160). (5 00, 102), (6 00, 66.0). (700. 179),..

T h e climatic data are entered as graphs to represent the time series downloaded from the N C D C website. Note that in this model we do not have any state variables; we only reproduce some empirical relationships that correlate evaporation with known data. W e do not really need to use Stella; all this could be done in a spreadsheet program such as Excel or Open Office. However, in this case Stella is useful to describe the cause-effect links that ate important to estimate evaporation. T h e model is based o n an empirical relationship by Christiansen (see S a x t o n and McGuinness,

Water

Eva

DayJul

207

PBelt.

A • emp T i n \ ' v

i \ \

rf X

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V

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Air lemp degC

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\ Piec'pila'io^

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i

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a

cloudy

L > < ) < , ) s ! IRad

6.6

vap press

pan CH

jLalDeg

Figure

EvaoM

VJ Air lemp mmC

A

Stella model for evaporation With no state variables this could easily be a spreadsheet

in Excel

£]

00

1; hyd evap calc

2. Evap M

365 00

Figure

6.7

Goodness of fit for evaporation.

Comparison of model results with available data for evaporation for Beltsville. MO. 1991

1 9 8 2 ) . T h e solar radiation is e s t i m a t e d by a simplified version ol an algorithm developed by N i k o l o v and Zellcr ( 1 9 9 2 ) . W c c a n c o m p a r e the results of this analysis with e x i s t i n g m e a s u r e m e n t s oi e v a p o r a t i o n t o see how well the m o d e l works (Figure 6 . 7 ) . T h e r e is a lot of variability in evaporation caused by the differences in climatic data. In the model we have managed to obtain a good estimate of the general trend, but have failed t o reproduce all the c h a n g e s in evaporation T h e data for wind speed, precipitation

I HHP IIM III

208

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S y s t e m s Science and Modeling for Ecological Economics 2 Wind speed

1 Precipitation

3: Humidity

1 2 00 2 450 00 3 100 00

h 250 88 3: 65.00

1 00 50.00 30.00 0

91.25

182.50

273.75

365.00

Time Figure

6.8

Variability ol climatic data

Data measured at Beltsville, MD, meteorological station in 1991. There is hardly any seasonal pattern in the data for rainfall, humidity and wind.

1: 800 00

1: SoiRadGr

1: 450.00

1 100. 0 00

91.25

182.50

273.75

365 00

Time Figure

6.9

Estimated solar radiation lor Beltsville, MD, 1991 (Latitude 39°).

and humidity show significant variability (Figure 6 . 8 ) T h e model of solar radiation also shows significant variability caused by the cloudiness effect (Figure 6 . 9 ) . T h e basic bellshaped trend for radiation that is defined by t h e latitude o f the site is s m o o t h . A d d e d t o tt is the s t o c h a s t i c pattern of climate that generates t h e cloudiness in our model. A l s o n o t e t h a t this model c a n be formulated as a pre-processor t h a t is run to g e n e r a t e t h e missing t i m e series to run t h e full m o d e l . T h e r e are n o f e e d b a c k s that would p o i n t i n t o this m o d u l e from a n y w h e r e else. T h e o n l y purpose is to generate t h e missing t i m e series for P A R based o n t h e e x i s t i n g c l i m a t i c time series and the latitude/longitude o f t h e site we are m o d e l i n g . W e may w a n t to run this m o d e l only

209

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2. Chock out tna sensitrvnv 0' trie r«00«« '.0 cnono<»i m I M l«- f - B ' O W ' O M T»%e effect o l c t o u a n e a s What n i p p e r a ,t tna alt»i;1 .•»! c o u t H -K n' i»wi7wl y n f rr1 I h " imraatnrir tmliavi-K in t i n i n x l n l I t u ! w n observe n this c a s e ' 3. W h « is m o t t , m o o r t * r t ic* ' i t e o« M D O f f t i C •vuiWi c o n t f n o n s / W h a t c r a r ^ e s r o * r a < e can c o - n o * o i « t o r * versa' 4. Include tt>C F A C E S

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210

Systems Scienrf? and Modeling fo' Ecologicst Economics part o f that model. T h e accompanying Stella model is shown in Figure 6.11

W e will

want to supplement rhe equations above with the following:

Snowlce(t) = Snowlcelt - dt) + (Freeze + Snowfall) * dt INIT Snowlce - 0 DOCUMENT The amount of snow and ice c n ihe surface (ml Freeze = if Temperature < 0 then Surface_Water/DT else -Melt D O C U M E N T Freezing/melting of water/snow

Formulated as a biflow W h e n temperature is

above 0 D C snow (if available) is melting ai a constant rate. O t h e r w i s e water is freezing All available water is assumed to freeze immediately. Snowfall = if Temperature< - 0 then Precipitation*0.0254 else 0 DOCUMENT: Snow accumulation from precipitation; use 0.0254 t o transfer inches into TI. Melt = 0.01 DOCUMENT: How much snow can melt per day (m/d) Temperature = 25 * SIN (Day Jul* PI/180/2) A 2-5 + RANDOMf-3,3) DOCUMENT: Temperature (°C) is modeled by a combination of the SIN function and the R A N D O M function. The amplitude of the SIN is ncreased t o 25. Power 2 is used t o make it always positive The DayJul'PI/180/2 conversion is used to switch t o radians and stretch the SIN period over the w h o l e year - 5 is the lowest temperature generated All temperatures are modified by a random value b e t w e e n - 3 and 3

Notice here that we aie using a so-called bi-flow to describe the conversion of water into ice and back - the "Freeze" tlow. Stella allows only positive flows. Whenever a tlow becomes negative, it is clumped to zero. Sometimes this is a useful feature, but it can cause a lot of confusion if it is forgotten. If ir is clear that the

P'ecipiiaiion

Q Figure 6.11

A Stella model with snow/ice formation added

One important process, called sublimation, is missing from this model. This is not important HI warm climates, when snow does not stay on the ground for long periods of time.

flow

is

supposed

to

he

negative

sometimes,

it

is

important

ro

ensure

that

it is described as a hi-l1ow by c l i c k i n g o n t b e radio b u t t o n at t h e top o f t h e flow dialogue box A n o t h e r feature t o n o t e h e r e is i n s t a n t a n e o u s c o n v e r s i o n o f all available surface water i n t o s n o w or ice w h e n e v e r the temperatures tall below zero. R e m e m b e r why we divide S u r f a c e _ W a t e r / D T ? A l s o n o t e t h e effort m a d e t o provide proper docum e n t a t i o n d i r e c t l y in the body of the model

T h i s c a n save a great deal o f trouble

later o n , when we return to y o u r model after a period o f rune and are trying t o figure out o n c e again what an e q u a t i o n was for and why a particular p a r a m e t e r looks so weird. Also

notice

that

tempera-

ture is described as a formula in a similar way t o that

described

There « no u^ch dung

in C h a p t e r Z. W h i l e the formula

doTMneiUaHjm..

ii somewhat different, t h e result

c4m**etUUuj

is q u i t e the same: cycles of warm

ivfuUyoii

3nd cold t e m p e r a t u r e over a 3 6 5 dav

period

noise

with

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W h i c h o f t h e two f o r m u l a s is bett e r ? ! t is really hard t o say. T h e model results are s h o w n in Figure 6 I 2. W e c a n see t h a t surface water is delivered by rain and t h e n gradually infiltrates i n t o t h e ground. It will freeze i n t o snow/ice w h e n the t e m p e r a t u r e is below 0 ° C , and under freezing c o n d i t i o n s precipi t a t i o n also arrives as snow. W e o b s e r v e a rapid a c c u m u l a t i o n of s n o w during t h e early, c o l d m o n t h s of the year. Later on snow/ice disappears, and t h e d y n a m i c s are similar t o t h o s e g e n e r a t e d by t h e surface water d y n a m i c s model Towards the end o f rhe year there are again freezing temperatures, and thus s o m e snow/ice is produced

1: Surface Water

2 Snowlce

3. Precipitation

0.06 2 0 15 3 4 00 1

1: 2: 3

RF iWg u^ r^e J 6J .U1 2J

Output from the snow/ice model

Snow/ice is present only during the first few cold months and then quickly disappears. More snow appears at the end ol the year when the temperature drops below 2ero

212

" " sy i t S m l Science en-d

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Exercise 6.3 1.

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i/

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Water

213

T h e Factors that influence infiltration may be grouped into three categories (Figure 6 . 1 4 ) : 1. T h o s e related to climatic conditions. T h e amount of water infiltrated depends upon the duration and intensity of rainfall. A 24-hour dri:rle can be entirely a c c o m m o dated by the soil, whereas che same amount of water received during a 20-minuce downpour will most probably end up to the surface-water runoff. Temperature also matters. W h e n the ground is frozen, the intensity of infiltration is reduced. 2 . T h o s e related to surface characteristics. Landuse and land cover translate into die imperviousness o f che surface. A parking lot will leave little water Co infiltrate, whereas a forest may capture the e n t i r e amount ol water arriving. O n the o t h e r hand, forests can intercept the incoming rainfall with leaves and trees in such a way thac a certain portion of the incoming water never reaches the ground. T h i s moisture is only exposed co evaporation. Slope also matters. In a flat area there is more time for water to enter the ground, while on a lull it starts traveling downwards along the surface as soon as it hits the ground. 3 . T h o s e related to soil characteristics. S a n d is an excellent medium for infiltration. O n rhe contrary, clay can block almost all infiltration. Moreover, it the soil is already saturated with water ( t h e soil moisture c o n t e n t is h i g h ) there will be little space left in the pores for additional water to infiltrate. A typical infiltration e v e n t evolves in both space and time (Figure 6 . 1 5 ) . As the rainfall starts, some water begins to seep into the ground, gradually increasing the soil water concent (curves 1 - 3 ) ac che cop o f che soil layer. A s more water comes with the rain it keeps entering the soil pores. T h e gravitation removes some wacer from the top layers and makes it tiavel further deeper into the ground. If this vertical movement is fast enough to tree up space on the top for the additional incoming water, then all che rain is absorbed. If che soil characteriscics do not allow water to travel fast enough through the soil, then the pores on the top are all tilled (curve 4 - 6 ) and the additional water will be left on the surface to cravel with overland flows. This is when ponding may occur. T h e wave ot saturated water propagates downwards through the soil O n c e che rain stops, the pores at the top start to dry out and get ready co accommodate a new rainfall event Loss of water from the unsaturated layer occurs by transpiration (upwards) and percolation (downwards). Transpiration

is a process that removes water from the soil

and transfers it as water vapor into the atmosphere - just a; in evaporation. T h e major difference is that in transpiration plants are responsible for water cransporc. T h e y suck moiscure from che soil with their roots, move it up into the canopy and

214

Systems Scienrf? and Modeling fo' Ecologicst Economics

Figure 6 . 1 5

Propagation of a water front

through the unsaturated layer during a rainfall event. Note that this simple model does not describe the spatial dynamics in the vertical. See how the amount of water in unsaturated storage (ul changes as a function of depth (hi. Cu^ve I, start of a rainfall event; curves 2-4, increase of unsaturated moisture until saturation is reached; curves 5-6. propagation of the saturation front downwards, curve 7, end of rainfall event dryoutfrom top.

•

Water

Soil panicles

Figure 6.16

Pores

How water travels through porous media. When all the pores are filled with water,

soil is saturated.

t h e n release it i n t o t h e a i r through t h e i r leaves. A s a result, t h e r e is m o r e w a t e r a v a i l a b l e for transpiration t h a n t h e r e is for e v a p o r a t i o n - w h i c h o n l y picks up moisture from t h e surface and t h e very few c e n t i m e t e r s towards t h e surface o i t h e soil. T r a n s p i r a t i o n c a n access water as d e e p as the roots e x t e n d . S o transpirarion is a f u n c t i o n u f t h e plant biomass w h i c h c a n c h a n g e uver rite s i m u l a t i o n period. Percolation

is r h e process by w h i c h water from t h e unsaturated storage e n t e r s t h e

saturated layer by m e a n s o f gravitational a n d capillary forces. S o i l c o n s i s t s o f m a t e r i a l particles with air in b e t w e e n (Figure 6 . 1 6 ) , and these voids or pores c a n p o t e n t i a l l y he filled by water. W h e n all t h e pores a r e filled t h e soil is referred t o as saturated, ar.d v e r i . c a l m o v e m e n t o f water is very m u c h slowed down. W h i l e t h e pores are n o t

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from 1 m/d in sandy soils to 1 rara/d in clav T h e percolation rate is also affected by the soil moisture c o n t e n t : p = f(U)hc. higher the rate of percolation

T h e more moisture there is in the soil, the

If U < F/P, the moisture is at field capacity and perco-

lation is C. It tends to 1 when U approaches I. As the water percolates downwards, it adds to the amount of water already present in the unsaturated storage. It takes only P - U water to till in the unsaturated storage so that it becomes saturated. T h e Stella model for water in unsaturated layer is presented in Figure 6.18. T h e corresponding equations are as follows:

Unsat_Depthi.t) = Unsat_Depth(t - dt) + (UD_plus - UD_mmus) ' dt INIT Unsat_Depth = 1 2 UD_plus = Transpiratio-i/Poros'ty DOCUMENT

Unsaturated depth is increased by the effect of transpiration, which removes

water from the saturated layer and can make it unsaturated (m/day) NB. Noie h o w porosity comes inio play. Why do w e do that? UD_minus = it = Unsat_Depth"Porosity) then LJnsat_Depth/DT else Percolation/ Porosity DOCUMENT Unsaturated depth is decreased due to the percolation (m/d! of water from the unsaturated zone tc the saturated, which raises the water table If the amount of unsaturated water exceeds the potential unsaturated capacity (Unsat_Water> -

Unsat_Depth*Porosily),

this means that no unsaturated layer can remain. all soil becomes saturated, u n s a t u r a t e depth becomes zero. Unsat_Waterlt) = Unsat_Watei(t - dt) + (Infiltration - Percolation - Transpiration) * dt INIT Unsat_Water = 0 11 DOCUMENT: Amount of water m the unsaturated layer measured as height of water column if "squeezed" from the soi! (m). Infiltration =

m,ni.lnfili_rate.Precipitation*0.0254,Porosity*Unsat_Depth-Lnsat.Wateri

D O C U M E N T The amount of water infiltrated is the m i n i m u m of infiltration rate, the amount of precipitation available .0 0254 converts mcnes to m). and the unsaturated capacity (m/d! The

Water

217

unsaturated capacity is the potential capacity lihe volume of pores in the soil) minus Unsat_ Water ithe space already occupiecii Perco'ation = if Unsat_Depih = 0 then Unsat_Water/DT else if Unsat_Wate-< - Field_cap'Unsat_Depth then 0 else Perc_rate DOCUMENT Percolation flow (m/d). The amount of water removed by gravity from the unsaturated 'ayei. This process can remove only water m excess of field capacity. Transpiration - NPP*Transp_rate DOCUMENT: The transpiralion flow (m/day! DayJul = modltime-1,365) + 1 DOCUMENT: Julian day, 1 thru 365 This is a counter that resets the day to zero after 365 iterations Needed to use the same graph function for several years of model runs Field_cap = 0.13 DOCUMENT: The amount of moisture in soil that is in equilibrium w i t h gravitational forces (dimiess) lnfilt_rate = 0.5 DOCUMENT Rate of in'iltration - the amount of water (ml that can be moved nto the unsaturated layer from the surface Perc_rate = 0.01 DOCUMENT Rate o? water removal by gravitation (m/day) Deoends upon soil characteristics Porosity = 0.35 DOCUMENT Proportion of pores m the soil They can be potentially filled w i t h water (dimlessi Transp_rate = 0 005 DOCUMENT The amount of water that planis can remove from soil by the sucking action of their roots (m of water/kg b i o m a s s " m 2 / d l NPP = GRAPH (DayJul) (0 00. 0.00). 133.2. 0.00), (66 4, 0.00), (99 5. 0 04). (133, 0 4). (166. 0.925), 1199, 0.975), (232. 0.995). (265. 0.985). (299, 0 855). (332, 0.105). (365. 0 00) DOCUMENT: An estimate of plant g r o w t h over the year (kg/m 2 ) Precipitation = GRAPH (DayJul) (1.00, 0.02). (2.00. 0.34). (3.00. 0.00). 14.00. 0.07). (5.00. 0.00), (6.00, 0 18), (700, 0.46), (354, 0.00), (355, 0.00), (356, 0.00), (357 0.00). (358. 0 00), (359, 0 00), (360, 0 02). (361. 0 00), (362. 0.00). (363, 0.00), (364, 0.00), (365, 0.00) DOCUMENT: Rainfall from Beltsville. M D 1969. (in/d)

In this model wc reproduce the dynamics that may be observed in a wetland that gets flooded during the wet season and dries out during the dry period. T h e vegetation that is removing significant amounts o f water by transpiration controls the state of the wetland T h e resulting dynamics of unsaturated water and unsaturated depth are shown in Figure 6 19. W h e n the transpiration rate is 0.C05 rn/kg m'/d, the plants can remove almost all the water and keep the area dry for most of the year. W h e n the transpiration rate declines to 0.C03, there is a succession of wet and dry periods. Certain species are known to be more effective in sucking the water out of the soil (e.g. Melaleuca

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Figure 6.22

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The layer of clay underlies the unsaturated layer. Clay has very low permeability, and w a t e r travels horizontally on top until it r e a c h e s the shore of Chesapeake Bay. Note the dry unsaturated layers on top of the w e t saturated l a y e ^ below.

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Water

221

saturated and unsaturated zones m o v e s and che two srorages b e c o m e closely related. W h a t was previously the saturated zone may turn to be unsaturated, and vice

versa.

T h e model e q u a t i o n s are as follows:

Sat_Water(t) - Sat_Water(t - dt) + (Percolation& - Recharge - Transp_Sst) * dt INIT S a t _ W a t e r = 4 D O C U M E N T A m o u n t of w a t e r in t h e saturated layer, m e a s u i e d in m f r o m s o m e base d a t u m Percclation& = if U n s a t _ D e p t h > 0 t h e n Of U n s a t _ W a t e r < ^ F i e l d _ c a p * U n s a t _ D e p t h t h e n 0 else Perc_rate) + M Unsat_De!ta > 0 t h e n U n s a t _ W a t e r * U n s a t _ D e l t a / U n s a t _ D e p t h / D T else U n s a t _ D e l t a * P o r o s i t y / D T ) else U n s a t _ W a t e r / D T D O C U M E N T : Percolation f l o w (m/d) + the c o m p e n s a t i o n for the change in the w a t e r table height First t e r m is percolation, the a m o u n t of w a t e r r e m o v e d by gravity f r o m the unsaturated layer This process can r e m o v e only w a t e r in excess of field capacity. S e c o n d t e r m tells h o w m u c h w a t e r w a s a d d e d to (or l e m c v e d f r o m -

hence the biflow) the unsaturated zone w h e n w a t e r table

w e n t d o w n (up). Recharge = S e e p a g e * $ a t _ W a t e r D O C U M E N T : Loss of saturated w a t e r t o d e e p e r aquifers (m/d) Transp_Sat = Transp_Unsat D O C U M E N T : A s s u m i n g that transpiration f r o m t h e saturated layer o c c u r s at a rate equal t o that f r o m t h e unsaturated layer Unsat_Water(t) - U n s a t _ W a t e r ( t - dt) + (Infiltration - P e r c o l a t i o n ^ - Transp_Unsat) * dt INIT U n s a t _ W a t e r = 3 D O C U M E N T A m o u n t of w a t e r in the unsaturated layer m e a s u r e d as height of w a t e r c o l u m n if " s q u e e z e d " f r o m the soil (m). Infiltration = m i n | l n f i l t . j a t e . P r e c i p i t a t i o n * 0

0254,Porosity*Unsat_Depth-Unsat_Water/DT)

D O C U M E N T The a m o u n t of w a t e r infiltrated is the m i n i m u m of infiltration rate, t h e a m o u n t of precipitation available (0.0254 c o n v e r t s inches to m), and the unsaturated capacity (m/d) The unsaturated capacity is the potential capacity (the v o l u m e of pores in t h e soil)

minus

U n s a t _ W a t e r (the space already occupied) Percolation& - if U n s a t _ D e p t h > 0 t h e n (if U n s a t _ W a t e r < - F i e l d _ c a p * U n s a t _ D e p t h t h e n 0 else Perc_rate) + (if Unsat_Delta > 0 t h e n

Unsat_Water*Unsat_Delta/Unsat_Depth/DT

else U n s a t _ D e l t a * P o r o s i t y / D T ) else U n s a t _ W a t e r / D T D O C U M E N T . Percolation f l o w (m/d) + the c o m p e n s a t i o n for the c h a n g e in the w a t e r table height. First t e r m is percolation, the a m o u n t of w a t e r r e m o v e d by gravity f r o m the unsaturated layer. This process can r e m o v e only w a t e r in excess of field c a p a c i t y S e c o n d t e r m tells h o w m u c h w a t e r w a s added to (or r e m o v e d f r o m - h e n c e the biflow) the unsaturated zone w h e n w a t e r table w e n t d o w n (up). Transp_Unsat = NPP*Transp_rate D O C U M E N T The transpiration f l o w (m/d) DayJul = m o d ( t i m e - 1 , 3 6 5 ) + 1 D O C U M E N T Julian day, 1 thru 3 6 5 This is a c o u n t e r that r e s e t s the day t o zero after 3 6 5 iterations. N e e d e d to use t h e s a m e g r a p h f u n c t i o n for several years of m o d e l runs.

222

Systems Scienrf? and Modeling fo' Ecologicst Economics

Elevation — 3 0 D O C U M E N T Elevation of surface f r o m base d a t u m (m) Field_cap = 0 13 D O C U M E N T : The a m o u n t of m o i s t u r e in soil t h a i is in e q u i l i b r i u m w i t h gravitational forces (dimless) lnfilt_rate = 0 . 0 5 D O C U M E N T : Rate of i n f i l t r a t i o n — t h e a m o u n t of w a t e r (m/d) that can b e m o v e d into t h e unsaturated layer f r o m t h e surface Perc_rate - 0 . 0 0 5 D O C U M E N T Rate of w a t e r removal by gravitation (m/d). D e p e n d s upon soil characteristics Porosity = 0 3 5 D O C U M E N T . Proportion of pores in t h e soil. They can b e potentially filled w i t h w a t e r (dimless) Seepage = 0 0001 D O C U M E N T : Rate of loss of s a t u r a i e d w a t e r to d e e p aquifers (l/d) Transp_rate = 0 . 0 0 5 D O C U M E N T ; The a m o u n t of w a t e r that plants can r e m o v e f r o m soil by t h e sucking action of tneir roots (m of w a t e r / k g b i o m a s s * m 2 / d a y ) Unsat_Delta -

DELAY(Unsat_Depth,DT)-Unsat_Depth

D O C U M E N T : I n c r e m e n t in w a t e r table height (m) over one DT. U n s a t _ D e p t h = Elevation-Sat_Water/Porosity D O C U M E N T . D e p t h of u n s a t u r a t e d zone (m), d e f i n e d as Elevation - a m o u n t of saturated w a t e r * porosity N o t e that sat w a t e r is t h e w a t e r " s q u e e z e d " out of the ground, by m u l t i p l y i n g it by porosity w e g e t t h e actual height of saturated layer NPP - G R A P H (DayJul) (0.00, 0.00), (33.2, 0.00), (66 4, 0.00), (99.5, 0.04), (133, 0 4), (166, 0.925), (199. 0.975), (232, 0.995), (265, 0 985), (299, 0.855), (332, 0.105), (365, 0.00) DOCUMiENT: A n e s t i m a t e of plant g r o w t h over t h e year (kg/m 2 ) Precipitation - G R A P H (DayJul) (1.00, 0 02), (2 00, 0 34), (3 00, 0.00), (4 00, 0 07), (5 0 0 . 0.00), (6.00, 0 18). (700, 0 46), (8 00, 0.22), (9 00, 0 08), (10 0, 0 00), (11 0, 0 00). (12 0. 0.38), (13 0, 0.1), (14 0, 0.00], (354, 0.00), (355, 0.00), (356, 0 00), (357, 0.00), (358, 0.00), (359, 0.00), (360. 0 02), (361, 0.00), (362, 0 001, (363, 0 00), (364, 0 00), (365, 0 00) D O C U M E N T Rainfall f r o m Beltsville, M D 1969 {in/d}

W e c o n s i d e r rhe a m o u n t o f water in t h e saturated zone, as if ir. were squeezed out o f t h e ground. T h e actual h e i g h t o f t h e saturated layer will rhen be S a t _ W a t e r / Porosity, where porosity is t h e p r o p o r t i o n o f pores in t h e ground. T h e d e p t h o f t h e unsaturated layer U n s a t _ D e p t h is n o w c a l c u l a t e d as t h e d i f f e r e n c e b e t w e e n t h e e l e v a t i o n and t h e h e i g h t o f the saturated layer. N o t i c e t h e use o f t h e D E L A Y f u n c t i o n in this m o d e l . T h e U n s a r _ D e l t a is c a l culated as t h e difference b e t w e e n t h e unsaturated d e p t h before and t h e depth now. If U n s a t _ L ) e l t a is positive, it m e a n s t h a t there was a deeper unsaturated layer before t h a n t h e r e is now. T h i s c a n o n l y be t h e case if t h e water table is rising, so we need to m o v e s o m e water t h a t previously was in che unsaturated storage into t h e saturated

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Exercise 6.5 1.

d o t h © d . n a m c s o c o m o ® « e d to d * m o d H * *

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224

Systems Scienrf? and Modeling fo' Ecologicst Economics

a d o p t e d s o far. It m a k e s s e n s e t o k e e p a r e c o r d o f t h o s e , s i n c e m a n y t i m e s a m o d e l e r c a n get carried away with t h e process a n d forget a b o u t s o m e o f t h e simplifications t h a t were m a d e a t o n e o f t h e earlier stages. It also adds credibility t o t h e model if you c a n a l w a y s e x p l a i n a l l t h e a s s u m p t i o n s co t h e m o d e l users. T h e m a j o r p r o c e s s e s a n d a s s u m p t i o n s we m a d e t o c r e a t e a m o d e l a r e as f o l l o w s : •

P r e c i p i t a t i o n c o m e s w i t h r a i n f a l l a n d s n o w f a l l . If t h e t e m p e r a t u r e is b e l o w 0 ° C ( 3 2 ° F ) , t h e p r e c i p i t a t i o n 15 c h a n n e l e d i n t o t h e s n o w / i c e v a r i a b l e . O t h e r w i s e p a r t o f it i n f i l t r a t e s i n t o t h e u n s a t u r a t e d w a t e r a n d t h e rest g o e s i n t o t h e s u r f a c e w a t e r .

•

W e assume immediately

that

rainfall

into

the

infiltrates

unsaturated

l a y e r a n d o n l y a c c u m u l a t e s a s surf a c e w a t e r if t h e u n s a t u r a t e d becomes

saturated

layer

or if t h e daily

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-ustre

built.

i n f i l t r a t i o n r a t e is e x c e e d e d . •

S u r f a c e w a t e r m a y b e p r e s e n t in r i v e r s , c r e e k s , s t r e a m s o r p o n d s . S u r f a c e w a t e r is r e m o v e d b y o v e r l a n d flows a n d by e v a p o r a t i o n .

•

S u r f a c e water flow rates are a f u n c t i o n o f d y n a m i c a l l y varying plant biomass, d e i v sky, a n d m o r p h o l o g y m a d d i t i o n t o surface a n d water e l e v a t i o n . H o w e v e r , at this p o i n t we i g n o r e d e t a i l s o f s u r f a c e w a t e r flow.

•

W a t e r f r o m t h e u n s a t u r a t e d layer is f o r c e d by g r a v i t y t o p e r c o l a t e d o w n t h e s a t u r a t e d layer. A s it a c c u m u l a t e s ,

cowards

t h e level o f t h e saturated water goes up

w h i l e t h e a m o u n t of w a t e r i n t h e u n s a t u r a t e d l a y e r d e c r e a s e s . •

T r a n s p i r a t i o n is t h e p r o c e s s o f w a t e r r e m o v a l f r o m s o i l by t h e s u c k i n g a c t i o n o f roots. T r a n s p i r a t i o n fluxes d e p e n d o n p l a n t g r o w t h , v e g e t a t i o n type a n d r e l a t i v e humidity.

•

S a t u r a t e d g r o u n d w a t e r c a n r e a c h t h e s u r f a c e a n d f e e d i n t o t h e flow o f surface, w a t e r . T h i s p r o c e s s is w h a t feeds t h e s t r e a m s a n d r i v e r s b e t w e e n

t h e rainfall e v e n t s

-

t h e s o - c a l l e d baseflow. After looking a t individual processes a n d variables, we c a n put together t h e w h o l e m o d e l f o r t h e h y d r o l o g i c c y c l e , a s s u m i n g t h a t we c a n s i n g l e o u t a n a r e a t h a t is m o r e o r less i n d e p e n d e n t o f t h e a d j a c e n t r e g i o n s . W e a s s u m e t h a t we a r e l o o k i n g a t a n a r e a o f less t h a n 1 k m 2 , l o c a t e d i n r e l a t i v e l y flat t e r r a i n t h a c is n o t c o o m u c h a f f e c t e d by h o r i z o n t a l fluxes o f g r o u n d w a t e r . T h e r e is a c e r t a i n g r a d i e n t o f e l e v a t i o n t h a t is suffic i e n t t o r e m o v e all t h e excess surface water thac did n o t g e t a c h a n c e t o infiltrate into t h e g r o u n d o v e r o n e t i m e - s t e p . T h e g r o u n d w a t e r t a b l e is r a t h e r s t a b l e a n d t e n d s t o b e a t e q u i l i b r i u m a t t h e i n i t i a l c o n d i t i o n s . T h e c l i m a t i c d a t a t h a t w e h a v e a r e at a d a i l y t i m e - s t e p , a n d t h e r e f o r e t h e r e is n o r e a s o n t o a s s u m e a finer t i m e - s t e p in t h e m o d e l . T h u s , we c a n a g r e e t h a t o u r t i m e - s t e p is 1 d a y a n d o u r s p a t i a l r e s o l u t i o n is J k m 2 . T h e m o d e l diagram in Figure 6 . 2 5 is q u i t e c o m p l e x , b u t y o u will c e r t a i n l y recognize

some

o f the modules and

submodels previously considered. T h e Globals sector (Figure 6 . 2 6 ) c o n t a i n s c l i m a t i c d a t a t h a t are input

Keep your

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as g r a p h s a n d t h e e m p i r i c a l m o d e l for solar radiation. H e r e , we also define t h e e l e v a t i o n o f t h e area c o n s i d e r e d . T h i s m i g h t n o t b e v e r y i m p o r t a n t f o r t h e u n i t m o d e l , b u t it will b e c o m e c r u c i a l i f w e d e c i d e t o c o m b i n e t h e unit models into a spatial simulation.

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X )^

225

SGiobals f \ X •Datum to MSI ( v s •

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Forcing (unctions for the hydrologic model collected in a separate submodel called Globals.

Input/Output I : SAT WATER 23.50

Initial C o n d i t i o n s

; __ ' • li It "V 'n

2: SNOW ICE 3: SURFACE WATFR4: UNSAT WATER

0.02

pr

0.01 2.S0

Rale C o e f f i c i e n t s & Constants

•C iVv' iml* •C * =p •C PrtC M4I-C4P

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L.J.

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274.00

3SS.OO

Input/output section for the hydrologic model.

Note that it is easier to manipulate parameters if they are collected in ore place using the "ghost" feature in Stella.

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i:

u ' . t p ' d ove- seve'*'

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3 . Can y o u fmo o oarameier

several years that destaoe'ees -.he sys'.c*-

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Systems Scienrf? and Modeling fo' Ecologicst Economics H e r e we need to use some c a u t i o n and r e m e m b e r o n e o f t h e c o m m a n d m e n t s above: mind the units. A s for any product c o m i n g from a U S Federal A g e n c y , T R - 5 5 is designed in imperial units -

in this case, inches. U S Federal A g e n c i e s do n o t

acknowledge that t h e rest of t h e world has adopted metric standards, w h i c h causes a lot o f confusion and errors. S o take care w h e n e v e r dealing with a product that c o m e s from there! In t h e case o f t h e e q u a t i o n s above, t h e units did n o t m a t t e r until we arrived at t h e relationship between S (measured in units o f l e n g t h ) and C N (a dimensionless empirical curve n u m b e r ) . T h e curve numbers C N are designed to produce S in inches. S o in order to stay w i t h i n t h e universally accepted m e t r i c c o n v e n t i o n s , a conversion is needed:

c/ , 2540 S cm ^ CN

2}.4

A l l t h e c o m p l e x i t i e s o f t h e hvdrologic c y c l e t h a t we have e x p l o r e d

become

e m b e d d e d in this magical e m p i r i c a l parameter. If there is n o r e t e n t i o n c a p a c i t y o f t h e watershed, C N -

1 0 0 , S = 0 a n d Q = P, all rainfall b e c o m e s runoff. T h e larger

t h e r e t e n t i o n capacity, t h e smaller t h e c u r v e number, t h e less runoff is seen. C u r v e n u m b e r s are produced from e m p i r i c a l studies for various land covers and soil types. A sample o f curve n u m b e r s is presented in T a b l e 6 . 1 .

A sample of runoff curve numbers for urban areas. Similar tables exist for a gricuttural and 1 other types of land uses. See the full TR-55 publication Cover d e s c r i p t i o n

H y d r o l o g i c soil g r o u p A

B

C

D

Poor condition (grass cover < 5 0 % )

68

79

86

89

Fair condition (grass cover 5 0 - 7 5 % )

49

69

79

84

Good condition (grass cover > 7 5 % )

39

61

74

80

98

98

98

98

98

98

98

98

83

89

92

93

Average percent

Cover t y p e a n d h y d r o l o g i c c o n d i t i o n

i m p e r v i o u s area Open space (lawns, parks, golf courses,

Impervious

cemeteries,

etc.)

areas.

i Paved parking lots, roofs, driveways, etc (excluding right-of-way) ! Streets and roads: Paved; curbs and storm sewers (excluding right-of-way) Paved; open ditches (including right-of-way)

•[Continued)

Water

B Q ^ ^ X X I IContinued) I — Cover description

229

Hydrologic soil group A

B

C

D

Gravel linclLding right-of-way)

76

85

89

91

Dirt (Including right-of-way)

72

82

87

89

Cover type and hydrologic condition

Average percent impervious area

Urban districts Commercial and business

85

89

92

94

95

Industrial

72

81

88

91

93

1/8 acre or less nown houses!

65

77

85

90

92

1 /4 acre

38

61

75

83

87

1/3 acre

30

57

72

81

66

-

Residential districts by average lor size

Infiltration lates of soils vary widely, and are affected by subsurface permeability as well as surface intake rates. Soils are classified into four Hydrologic Soil Groups ( H S G ) - A, B, C and D - according to tlien minimum infiltration rate, which is obtained for bare soil after prolonged wetting. Roughly, the H S G soil textures are as follows: A - sand, loamy sand, or sandy loam, B - silt loam or loam; C - sandy clay loam; and D - clay loam, siltv clay loam, sandy clay, silty clay, or clay. Comparing the two models, the process-based Stella mode! and the empirical T R 55, tbe simplicity of the latter can be appreciated. Note, however, how little the empirical model tells us about the actual processes - about how various forcing functions (temperature, wind, etc.) affect the system. While it is certainly a useful tool for some particular applications, especially where quick estimates are tequned, it is unlikely to advance our understanding of how the system works. O n the other hand, it is quite easy ro become buried in all the complexities of the process-based approach, especially if we considei all the parameters we will need to figure out to make it run, and all the data for forcing functions that we will need lo find. In some cases, a bicycle is all you need to get there, in other cases, a Boeing-777 would be a better choice. Note, however, that in most situations when a bicycle is a good solution, a Boeing would be a ridiculous or even impossible option. T h e same applies with different kinds of models. Also note that both models have quite limited application, since they assume a very small watershed and no horizontal movement of water If we want to cover larger watersheds, we need to explore how water gets routed and what spatial algorithms are needed to make the models work.

6.3

Spatial model In reality, hydrologic processes are very much spatial and their desctiption within the framework of a spatially uniform unit model is quite limited. Water, both on the

230

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Water

231

defined by horizontal hydraulic c o n d u c t i v i t y . T h i s rate .s very m u c h d e p e n d e n t o n the soil type, and c a n vary hy several orders of m a g n i t u d e . A s with surface water transport, g r o u n d w a t e r flow is c e r t a i n l y spatial hy Spatial gradients and t h e spatial c h a r a c t e r i s t i c s o f soil

It is driven

In fact, o f t h e foui m a j o r

variables in t h e unit model considered a b o v e , only two (unsaturated water and snow/ i c e ) are a t t a c h e d t o a c e r t a i n area a n d c a n b e modeled locally

For t h e o t h e r two

m a j o r actors (surface water and saturated w a t e r ) , wc need s o m e r e p r e s e n t a t i o n o f spatial dynamics. A s we saw i n C h a p t e r 5 . S t e l l a is certainly n o t a proper tool t o build spatial m o d e l s t h a t may b e c o m e very c o m p l e x and are likely t o require d i r e c t links to maps and G e o g r a p h i c I n f o r m a t i o n S y s t e m s ( G I S ) . T h e r e are two basic a p p r o a c h e s used for m o d e l i n g spatial hydrology (Figure 6 . 2 9 ) : 1. Lumped o r n e t w o r k - b a s e d hydrologic units. Here, t h e space is represented as a n u m b e r o f hydrologically h o m o g e n e o u s areas that are linked t o g e t h e r by a linear n e t w o r k , representing rhe flow of water in streams. 2 . G r i d - b a s e d units. H e r e , the space is represented as a uniform o r n o n - s t r u c t u r e d grid of square, triangular or o t h e r cells. E a c h o : t h e two a p p r o a c h e s has its advantages and disadvantages.

Lumped models When

using network based segments, t h e n u m b e r of individual hydrologic

units

that arc c o n s i d e r e d spatially may be q u i t e small. T h e w h o l e area is subdivided i n t o regions, based o n c e r t a i n hydrologic c r i t e r i a . T h e s e may be subwatersheds of c e r t a i n sue, hill slopes, areas wirh similar soil and h a b i t a t properties, e t c . In most cases it is up t o t h e researcher t o identify t h e ranges within w h i c h factors are aggregated,

Figure 6.29 Lumped network approach and the grid approach. Each subwatershed or hydrologic unit is presented as a combination of cells,

232

Systems Scienrf? and Modeling fo' Ecologicst Economics and therefore decide o n t h e number o f spatial units that are t o be considered in t h e model T h i s decision is made based o n : • T h e goals o f the model - how m u c h spatial detail do wc need about the system, and what are the maior processes wc want to analyze and understand within t h e framework o f t h e model? • T h e available computer resources - how much memory (here is to handle the spatial arrays, and how fast is the C P U to run t h e full model? • T h e available data - how much do wc know about t h e study area, and what is the spatial resolution o f the data? O n c e the spatial units have been chosen, they are assumed to be h o m o g e n e ous, and the geometry of the area is hxed T h i s is also t h e major disadvantage of t h e lumped or t h e unstructured grid approach. If for some reason we need to reconsider the geometry o f the watershed and switch to other hydrologic units, it may require a considerable effort to develop a new grid or routing scheme. O n c e the routing network is defined, the procedure is more or less t h e same. C e r t a i n empirical or process-based equations are derived to define the amounts of water and constituents that each hydrologic unit may generate. T h e s e quantities are then fed into a network model that represents the transport along t h e river and its tributaries. T h e network model links together the individual models for the spatial units. One

of the classic

examples of this approach

is the H S P F

(Hydrological

Simulation Program Fortran), which is available for download from a variety o f sites (http://water.U5gs.gov/software/hspf h t m l )

I he model was developed in the early 1960s

as t h e Stanford Watershed Model. In the 1970s, water-quality processes were added. H S P F c a n cover extended periods of tune with time-steps ranging from 1 minute to I day. It has been used to model various spatial areas, from small sub-catchments o f several hundred square meters to the 1 6 6 . 5 3 4 - k n r Chesapeake Bay watershed. T h e model simulates the hydrologic and associated water-quality processes on pervious and impervious land surfaces, and in streams and well-mixed impoundments. Ir uses standard meteorological records to calculate stream flow hydrographs and pollutographs. T h e list o f processes that are covered by various versions of H S P F is long and impressive: interception, soil moisture, surface runoff, interflow, base flow, snowpack depth and water content, snowmelt, evapotranspiration, groundwater recharge, dissolved oxygen, biochemical oxygen demand ( B O D ) , temperature, pesticides, conservatives, fecal coliforms, sediment detachment and transport, sediment routing by particle size, channel routing, reservoir routing, constituent routing, pH, ammonia, nitrite-nitrate, organic nitrogen, orthophosphate, organic phosphorus, phvtoplankton and zooplankton. Probably one o f the best-elaborated versions of H S P F became part o f the B A S I N S suite developed at the U S Environmental Protection Agency ( E P A ) (http://www. epa.gov/OST/RASlNS/). A major improvement is the user-friendly interface, which allows users to build a project for a watershed that they are interested in. A t this site there may even he data sets that are needed for a model for almost any watershed in the U S A , T h e latest version of B A S I N S also includes the S W A T model - Soil and Water Assessment Tool (http://www.brc.tamus.edu/swat/index.html) - another wellknown spatial hydrology model that is also based on t h e same lumped subwatcrshed paradigm. Both models are generally able to simulate stream flow, sediment, and nutrients loading. According to some reports, H S P F simulates hydrology and water-quality c o m p o n e n t s more accurately than S W A T , however, H S P F is less user-friendly than

Water

233

S W A T , o w i n g t o t h e r e b e i n g e v e n m o r e p a r a m e t e r s t o c o n t r o l . H S P F is a n e x t r e m e l y d a t a - i n t e n s i v e and over-parameterized information. S W A T

m o d e l , a n d r e q u i r e s a large a m o u n t

of

site

is s o m e w h a t s i m p l e r ; it e s t i m a t e s t h e s u r f a c e r u n o i i I r o m d a i l y

r a i n f a l l u s i n g t h e c u i v e n u m b e r m e t h o d w e d i s c u s s e d a b o v e , a n d s e d i m e n t yield is c a l c u l a t e d with t h e Modified U n i v e r s a l Soil Loss E q u a t i o n

(MUSLE)

Y e t a n o t h e r t w o m o d e l s t h a t are w o r t h m e n t i o n i n g a r c 1. T O P M O D E L

- a classical m o d e l t h a t has been used tor a variety ot rivers a n d

watersheds (see 2,

http'//www.es.lancs.ac.ut:/hklg/freeware/hfdg_lr
R H E S S y s - t h e R e g i o n a l H y d r o - E c o l o g i c a l S i m u l a t i o n S y s t e m , w h i c h is a C I S based, hydro-ecological

modeling framework designed to simulate c a r b o n ,

water

a n d n u t r i e n t fluxes. R H E S S y s c o m b i n e s a set ot physically-based process m o d e l s a n d a m e t h o d o l o g y lor p a r t i t i o n i n g a n d p a r a m e t e r i z i n g t h e l a n d s c a p e ( s e e http:// geography.sdsu.edu/Rcsearch/Piojects/RHESSYS/) D e s c r i b i n g a n y o f t h e s e m o d e l s in a n y d e c e n t a m o u n t o f d e t a i l c a n t a k e a s m u c h s p a c e as t h i s w h o l e h o o k

H o w e v e r , t h e basic: c o n c e p t

i l l u s t r a t e d by t h e s a m e T R - 5 5

is q u i t e s i m p l e a n d c a n

be

m o d e l c o n s i d e r e d a b o v e . A s we h a v e s e e n , w e c a n

c a l c u l a t e t h e a m o u n t ot r u n o f f f r o m a c e r t a i n d r a i n a g e a r e a for e a c h r a i n f a l l e v e n t . B y d e f i n i t i o n , t h i s r u n o f f d o e s n o t stay in p l a c e - it r u n s . N o w we n e e d t o look at t h e h o r i z o n t a l d i m e n s i o n a n d figure o u t t h e f a c t o r s t h a t c a n i m p a c t t h i s run, s i n c e o n c e it s t a r t s r u n n i n g it s t a r t s co a c c u m u l a t e w a t e r f r o m v a r i o u s a r e a s , a n d t h a t is w h a t c o n f i g u r e s t h e flow h y d r o g r a p h , o r t h e p a t t e r n ol f l o w in a s t r e a m o r river. T R - 5 5 h a s b e e n d e v e l o p e d t o e s t i m a t e t h e p e a k flow t h a t an a r e a c a n g e n e r a t e in r e s p o n s e t o v a r i o u s r a i n f a l l e v e n t s . It t a k e s t h e r u n o f f , c a l c u l a t e d a b o v e , as t h e p o t e n t i a l a m o u n t of w a t e r t h a t t h e a r e a c a n p i o d u c e , a n d t h e n t a k e s i n t o a c c o u n t v a r i o u s s p a t i a l c h a r a c t e r i s t i c s tit t h e w a t e r s h e d ( s u c h

as s l o p e , c h a n n e l i z a t i o n , s u r f a c e

characteristics,

e t c . ) a n d t h e t e m p o r a l c h a r a c t e r i s t i c s of r a i n f a l l ( d u r a t i o n ) t o e s t i m a t e t h e m a x i m a l flow t h a t s h o u l d be e x p e c t e d f r o m t h i s a r e a O n e c r u c i a l i n d i c a t o r is che t i m e o f c o n c e n t r a t i o n ( T f ) , w h i c h is t h e t i m e tor r u n o f f t o t r a v e l f r o m t h e h y d r a u l i c a l l y m o s t d i s t a n t p o i n t of t h e w a t e r s h e d to a p o i n t o f i n t e r e s t w i t h i n t h e w a t e r s h e d . T c is c o m p u t e d by s u m m i n g all t h e t r a v e l t i m e s t o r c o n s e c u t i v e c o m p o n e n t s of t h e d r a i n a g e c o n v e y a n c e s y s t e m . T r a v e l t i m e ( T , ) is t h e t i m e it t a k e s w a t e r t o t r a v e l f r o m o n e l o c a t i o n t o a n o t h e r i n a w a t e r s h e d . T r a v e l t i m e is a f f e c t e d by s e v e r a l f a c t o r s , s u c h as s u r l a c e r o u g h n e s s , c h a n n e l s h a p e , a n d s l o p e of surface. F o r e x a m p l e , u n d e v e l o p e d v e g e t a t e d a r e a s w ill h a v e a h i g h d e g r e e of r o u g h n e s s a n d very s l o w a n d s h a l l o w o v e r l a n d tlow. A s f l a w is d e l i v e r e d t o streets, g u t t e r s a n d s t o r m sewers, r u n o f f d o w n s t r e a m b e c o m e s far m o r e rapid. U r b a n i z a t i o n will g e n e r a l l y s i g n i f i c a n t l y d e c r e a s e t h e t r a v e l t i m e t h r o u g h a w a t e r s h e d . T h e s l o p e will t e n d t o i n c r e a s e w h e n c h a n n e l s a r e s t r a i g h t e n e d , a n d d e c r e a s e w h e n o v e r l a n d flow is directed through storm sewers, street gutteis and diversions T h e t i m e o f c o n c e n t r a t i o n ( r c ) is l h e s u m of T , v a l u e s tor t h e m v a r i o u s c o n s e c u t i v e flow s e g m e n t s :

T

- T ,

l +

T

{ 1

+ -

+

Tm.

T r a v e l t i m e ( i n h o u r s ) is t h e r a t i o of flow l e n g t h t o flow v e l o c i t y . W a t e r

moves

t h r o u g h a w a t e r s h e d as s h e e t flow, s h a l l o w c o n c e n t r a t e d flow, o p e n c h a n n e l flow, oi s o m e c o m b i n a t i o n o f t h e s e . S h e e t flow is t h e How o v e r p l a n e s u r f a c e s , a n d u s u a l l y o c c u r s in t h e h e a d w a t e r o f s c r e a m s . W i t h s h e e t flow, t h e f r i c t i o n v a l u e

(Manning's

?i)

is a n e f f e c t i v e r o u g h n e s s c o e f f i c i e n t t h a t i n c l u d e s t h e e f f e c t o f r a i n d r o p i m p a c t , d r a g

234

Systems Scienrf? and Modeling fo' Ecologicst Economics o v e r rhe plane surface; o b s t a c l e s such as litter, crop ridges a n d locks; a n d erosion a n d transportation of sediment. M a n n i n g ' s k i n e m a t i c s o l u t i o n , w h i c h works for travel t i m e o v e r 1 0 0 m or less, is 0.007(nL)° p0 5

<

8

w h e r e n = M a n n i n g ' s roughness c o e f f i c i e n t ( T a b l e 6 . 2 ) , L = flow length ( f t ) , P = 2-year, 2 4 - h o u r rainfall ( i n ) , s = slope o f hydraulic grade line (land slope, ft/ft). N o t e again t h e c o n f u s i o n with units here. A f t e r a m a x i m u m o f 1 0 0 m, s h e e t flow usually b e c o m e s shallow c o n c e n t r a t e d flow, It is driven by slope, so for c o n c e n t r a t i o n time we h a v e

1

3600V

where: L = flow l e n g t h ( m ) , V = average velocity (m/s) and 3 6 0 0 -

conversion

f a c t o r from s e c o n d s to hours. For slopes less than 0 . 0 0 5 and unpaved c o n d i t i o n s ,

Roughness coefficients (Manning's n) for sheet flow

Surface d e s c r i p t i o n

n

Smooth surfaces (concrete, asphalt,

0 011

gravel, or bare soil) 1 Fallow (no residue) Cultivated

0 05

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lic grade line (watercourse slope, m/m). For steeper slopes t h e e q u a t i o n s ate similar, but t h e c o e f f i c i e n t s will be different. W h e n flow b e c o m e s c h a n n e l i z e d the e q u a t i o n is different:

=

I -49r^35i;j n

So w h y d o e s 1 49 appear in f r o n t of t h e M a n n i n g ' s e q u a t i o n ? W h a t a strange w a y t o w r i t e an e q u a t i o n W h y not i n c l u d e t h e 1 4 9 in t h e empirical c o e f f i c i e n t n. w h i c h is also t h e r e ' W h a t ' s so special a b o u t 1.49? Well, your g u e s s is probably correct Of c o u r s e , it is the unit conversion. The real M a n n i n g ' s e q u a t i o n is n w h e r e r is m e a s u r e d in m e t e r s and s. t h e slope, is m e a s u r e d in m / m W h i l e n is an empirical c o e f f i c i e n t a n d is usually p r e s e n t e d as d i r n e n s i o n l e s s . in tact it has u n i t s it w e w a n t to have V in m/s, w e n e e d to have n in s / W 3 - v e r y w e i r d u m t s indeed. But n o w it is clear t h a t if w e w i s h to u s e l h e s a m e empirical values for n, but get t h e result in ft/s. w e ' l l n e e d s o m e t w e a k i n g Indeed. sfm,a

= sA'3.281/3 f t ' ' 0 ) = s / f l . 4 9 f 1 i r j l A n d t h e r e is our 1.49!

The b o t t o m une is. if y o u really n e e d t o use i m p e r i a l units, brace yourself 1or a lot of fun.

H e r e , r is t h e hydraulic radius ( f t ) and is equal to it//),,, a is t h e cross s e c t i o n a l flow area ( f t " ) ,

is the wetted p e r i m e t e r ( f t ) , s is che slope o f t h e hydraulic grade

line ( c h a n n e l slope, ft/ft), and n is t h e M a n n i n g ' s roughness c o e f f i c i e n t for o p e n c h a n n e l flow. T h i s is also k n o w n as t h e M a n n i n g ' s e q u a t i o n Finally, t h e peak discharge ( t t V s ) e q u a t i o n is:
Fj, is just an adjuscmenr factor if che pond and

s w a m p areas are spread t h r o u g h o u t che watershed a n d are not c o n s i d e r e d in che T r c o m p u t a t i o n . T h e unic peak discharge qu is what requires most efforc co work out. Ic takes inco a c c o u n t T c , che 2 4 - h o u r rainfall ( i n ) , and o n c e again t h e curve number, C N . S t e p p i n g through a series ot rabies and graphics, T R - 5 5 finally gees che answer. T h e r e is a S t e l l a i m p l e m e n t a t i o n o f T R - 5 5 developed by E v a n Fitzgerald thac c a n be downloaded from che book website or from t h e " R e d e s i g n i n g t h e A m e r i c a n Neighborhood"

project

website

(http://www.uvm.edu/~ran/ran/iesearchers/ran55.

php). In this simplified version, t h e standard r a i n f a l l - r u n o f f

relationships

a n d equa-

tions used in T R - 5 5 models h a v e b e e n written into t h e S t e l l a model to produce nearidentical results to t h e N R C S models. T h e s e relationships include che c u r v e n u m b e r approach as well as che rainfall curve used for che northeast T h e time c o n c e n t r a t i o n v a u a h l e was excluded in this version, s i n c e the model did not appeal to he sensitive ro it. A c o m p a r a t i v e analysis b e t w e e n T R - 5 5 and S t e l l a model results was performed for t h e t i m e of c o n c e n t r a t i o n variable at che fixed scale of 10 acres, and ir was d e t e r m i n e d

236

Systems Scienrf? and Modeling fo' Ecologicst Economics

that the effect o f not including this variable in the Stella model was negligible for peak flow rate calibration. T h e model also piovides a good example o f the use o f the modeling interface thar comes with Stella, fn rhis case, the goal was to explore various alternative management practices for stormwater in a small Vermont neighborhood. T h e r e are all sorts o f switches and sliders, knobs and graphics that allow rhe user to define easily the various scenarios and m a n a g e m e n t solutions to compare results in search o f a he tter understanding o f the system and an optimal design o f management practices. It is also interesting to n o t e that we have solved a spatial problem by a fairly local Stella model, although we have actually simplified it to the greatest e x t e n t possible. In reality, what makes a system really spatially distributed are the variations in data and processes. S o far, we are still assuming chat all the landscape characteristics (soil and landuse, expressed m the curve number, slope, rainfall pattern, e t c . ) are spatially uniform. W e have provided for some spatial proxies by describing how water gets routed and lemoved from t h e unit area, but that is nor really spatial. W h a t the models like those listed above (HSPF, S W A T , R H E S S y s ) and others do is replicate a version o f local T R - 5 5 or our Stella U n i t Hydrology model for a series o f nodes. T h e y then use similar delivery algorithms like the Manning's equation over the network o f c h a n n e l s that c o n n e c t s rhose nodes. T h i s takes care o f the delivery m e c h a n i s m over a large and spatially heterogeneous watershed.

Grid-based models In grid-based models, t h e homogeneous spatial units are defined

mechanistically,

by representing the study area as a grid o f cells. T h e major decision in this case is the size and form o f the cell. T h e size defines t h e spatial grain - the resolution o f the model, ideally, the smaller the cells, the finer the resolution and the more detail regarding the landscape c a n be accounted for. However, the reverse side is again the model complexity and the tune needed to run the model. T h e decision about the size and configuration o f cells is usually based o n pretty similar principles to those above: • T h e goals o f the study - what is the spatial resolution needed to meet those goals? • T h e available computer lesources - how much memory is there to handle the spatial arrays, and how fast is the C P U to run the full model? • T h e available data - how much d o we know about the study area, and what is the spatial resolution of the data? T h e r e is yet o n e more consideration that may be important. Grid-based models generate huge arrays o f outpuc information. T h e y may he quite useless unless there are good data processing and visualization tools that can help to interpret this output. Imagine a model of, say, 10 variables running over a grid of, say, 5 , 0 0 0 cells. A n d suppose we are running this model for 1 year at a daily time-step. T h i s is probably an average complexity for spatial hydrology models. As an output we will be generating time series of maps, o n e for each state variable, every day. S o potentially we will be obtaining some 3 , 6 5 0 maps for state variables in each o f the 5 , 0 0 0 cells, plus as many more as we may want for intermediate variables. W h a t do we do with all this information? Keep in mind thai methods o f spatial statistics and analysis are quite rudimentary. W e also need to remember that it is hardly possible to expect to have anything close to that in terms of experimental data to compare our results and calibrate our model. S o chances are that much of the spatial grain that we will be producing will be

Water

237

left u n u s e d , a n d m o s t likely w e will b e g e n e r a t i n g s o m e i n d i c e s a n d s p a t i a l l y a v e r a g e d i n d i c a t o r s ro a c t u a l l y use i n o u r study. N e v e r t h e l e s s , it is g o o d t o h a v e the of

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esses. M o r e o v e r , s p a t i a l o u t p u t l o o k s so n i c e m p r e s e n t a t i o n s a n d reports - p e o p l e like t o see c o l o r f u l m a p s o r a n i m a t i o n . J u s t m a k e s u r e s u c h o u t p u t is n o t b e i n g m i s u s e d o r m i s i n t e r p r e t e d ! In C h a p t e r 5 we v i s i t e d w i t h t h e S p a t i a l M o d e l i n g E n v i r o n m e n t

- SME -

and

s h o w e d h o w it c a n b e used t o e x t e n d l o c a l S t e l l a m o d e l s o v e r a s p a t i a l d o m a i n . H e r e , w e will t a k e a q u i c k l o o k at a r e a l - l i f e a p p l i c a t i o n o f t h i s a p p r o a c h t o w a t e r s h e d m o d e l i n g . T h e P a t u x e n t L a n d s c a p e M o d e l ( P L M - h t t p : / / g i e e . u v m . e d u / P L M ) is a gridb a s e d s p a t i a l l a n d s c a p e m o d e l t h a t was b u i l t u p o n t h e S M E p a r a d i g m

The

model

uses a n e c o s y s t e m l e v e l " u n i t " m o d e l b u i l t in S t e l l a t h a r is r e p l i c a t e d in e a c h ol t h e unit c e l l s r e p r e s e n t i n g t h e l a n d s c a p e { F i g u r e 6 3 0 ) . For e a c h different h a b i t a t

type

t h e m o d e l is d r i v e n hy a d i f f e r e n t s e t o f p a r a m e t e r v a l u e s ( e . g . p e r c o l a t i o n r a t e , infilt r a t i o n r a t e , e t c . a r e d i f f e r e n t for a f o r e s t v s a n a g r i c u l t u r a l field vs a r e s i d e n t i a l

lot)

( F i g u r e 6 . 3 ! ) . A c t u a l l y , ir is n o t o n l y o n e m o d e l in S t e l l a b u t a w h o l e s e r i e s ot t h e m . S M E s u p p o r t s m o d u l a r i t y in s u c h a w a y t h a i y o u c a n t a k e s e v e r a l S t e l l a m o d e l s , e a c h r e p r e s e n t i n g a c e r t a i n s u b s y s t e m , a n d run t h e m in c o n c e r t , e x c h a n g i n g

information

between the different modules. A s a c o m p a n i o n tool to S M E , the Library of Hydro-Ecological M o d u l e s ( L H E M

-

h t t p : / / g i e e . u v m . e d u / L H E M ) h a s b e e n d e v e l o p e d t o r e p r e s e n t m o s t of t h e p r o c e s s e s i m p o r t a n t for w a t e r s h e d d y n a m i c s a n d m a n a g e m e n t remarkable with tins approach

(Figure 6 . 3 2 )

What

is t h a t it l e n d s u l t i m a t e t r a n s p a r e n c y t o t h e

is m o s t model

U n l i k e t h e watershed m o d e l s described a b o v e , w h e r e t h e c o d e may not be easily a v a i l a b l e o r i n d e e d a v a i l a b l e at a l l ( a s in s o m e p r o p r i e t a r y m o d e l s ) , a n d all t h e i n f o r m a t i o n a b o u t t h e m o d e l i n t e s t i n e s h a s t o be e i t h e r figured o u t f r o m r h e d o c u m e n t a t i o n p r o v i d e d o r g u e s s e d u s i n g c o m m o n s e n s e , h e r e w e h a v e t h e a c t u a l m o d e l at o u r f i n g e r t i p s . W e c a n e x p l o r e e a c h m o d u l e , r u n it as a s e p a r a t e S t e l l a a p p l i c a t i o n , u n d e r s t a n d t h e d e p e n d e n c i e s a n d a s s u m p t i o n s , o r e v e n m a k e c h a n g e s if w e h a v e b e t cer i d e a s r e g a r d i n g h o w t o p r e s e n t c e r t a i n p r o c e s s e s . T h e l o c a l h y d r o l o g y m o d e l in L H E M is s i m i l a r t o r h e u n i t m o d e l in Figure 6 2 5 . In a d d i t i o n t o t h a t , t h e r e a r e m o d u l e s for n u t r i e n t c y c l i n g , d e a d o r g a n i c m a t e r i a l , p l a n t g r o w t h , e t c . F u r t h e r , t h e r e are a l s o s p a t i a l a l g o r i t h m s t h a t c a n b e used t o m o v e w a t e r a n d c o n s t i t u e n t s b e t w e e n c e l l s . T h e r e is a c h o i c e o f a l g o r i t h m s o f s p a t i a l l l u x i n g t h a t link t h e c e l l s t o g e t h e r ( F i g u r e 6 . 3 3 ) . In e f f e c t , t h e y a r e s o m e w h a t s i m i l a r t o t h e p r o c e dures discussed a b o v e when wc were m o v i n g water o v e r the network b e t w e e n nodes. H e r e t o o w e n e e d t o d e c i d e h o w far a n d h o w fast t h e w a t e r will t r a v e l , e x c e p t , a s i n t h e c a s e o f P L M , t h e n e t w o r k is d e g e n e r a t e d t o a s i m p l e c a s e o f c e l l - t o - c e l l p i p i n g . T h e m e t h o d s used in L H E M a r e g r e a t l y s i m p l i f i e d in o r d e r t o h a n d l e large a r e a s and c o m p l e x ecological models

T h e y may he considered as a n empirical a p p r o a c h to

surface-water routing. T h e y a i e verv m u c h based o n e m p i r i c a l a s s u m p t i o n s and c o m m o n s e n s e . In a l a n d s c a p e - m o d e l i n g f r a m e w o r k , h y d r o l o g y is o n l y a p a r t o f a m u c h m o r e c o m p l e x a n d s o p h i s t i c a t e d m o d e l s t r u c t u r e . T h e r e f o r e w e h a v e r o try t o k e e p

238

Systems Scienrf? and Modeling fo' Ecologicst Economics

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A standard retention pond such as is built in most new developments to comply with the

stormwater regulation It requires huge investment, and needs to tie maintained prooerly.

far m o r e surface runoff during storm events. Rivers and streams b e c o m e raging torrents, causing erosion and flooding over vast areas. A t the same time, there is less flow in b e t w e e n t he storms. T h e so-called baseflow dries out, since all t h e water has been already drained and there is not m u c h stored in the ground and wetlands to feed t h e streams. T h e high flows result in highly incised landscapes, with streams digging deeper i n t o t h e ground, taking o u t lots o f s e d i m e n t and d u m p i n g it i n t o t h e rivers and lakes. The

water quality also dramatically deteriorates. T h e

sediments

themselves

are

a n u i s a n c e tor adult fish, a n d c a n destroy spawning grounds. T h e y also carry large a m o u n t s ol n u t r i e n t s . N u t r i e n t s also c o m e Irom fertilizers used to improve residential lawns T h e lawns are also treated w i t h c h e m i c a l s - herbicides and pesticides - w h i c h all end up in estuaries and lakes. The

bottom

line

is

that

residential n e i g h b o r h o o d s

have

a s t r o n g impact o n

stormwater

q u a n t i t y and quality, and need to start taking c a r e of t h e i r runoff. S o far, most o f t h e solutions h a v e b e e n q u i t e centralized. In one,

the

water

is captured

in

large r e t e n t i o n ponds, where it is h e l d for a while, losing sedim e n t s a n d partially

infiltrating

i n t o t h e soils (Figure 6 . 3 4 ) . T h i s s o l u t i o n is q u i t e e x p e n s i v e b o t h

The river n e t w o r k is d e v e l o p e d by t h e landscape. However, it is n o t |ust the geology and height that matter; land cover is also a factor If w e have forests, t h e y can absorb m o s t of t h e rainfall, so t h e r e is not m u c h left for runoff It forests are replaced by i m p e r v i o u s or less pervious surfaces, then there is m o r e surface runoff and obviously m o r e s t r e a m s a n d rivers are required to c o n d u c t all that water. Besides, the m o r e water

is channeled t h r o u g h

these

s t r e a m s , the w i d e r and d e e p e r t h e y b e c o m e

it

is interesting t c realize that perhaps m o s t of the existing rive.' n e t w o r k , especially the

smaller

s t r e a m s and rivers, have been d e v e l o p e d as a result of our land-cover changing activities.

242

Figure 6.35

A ram barrel

This is a simple device to capture water collected from rooftops. It intercepts only the first few centimeters of a rainfall event Idepending upon the area ol the roof and the size of the barrel) However, it may be quite useful in improving water quality, since it is usually the first flash of runoff that contains most of the constituents, and the more of it we can retain, the belter. Checkout http.//www likbez.com/AV/barrel/for h o w t c make your own rain barrel

t o install a n d t o m a i n t a i n . T h e s e s u p e r - p o n d s c a n b e b u i l t during t h e c o n s t r u c t i o n p h a s e , w h e n t h e r e are c l e a r regulations a n d c o n t r o l s w i t h w h i c h t h e d e v e l o p e r s n e e d t o c o m p l y . H o w e v e r , they are p r o h i b i t i v e l y e x p e n s i v e t o install later o n , w h e n t h e n e i g h b o r h o o d is already in p l a c e and t h e h o m e o w n e r s a r e e x p e c t e d to a b s o r b all rhe a d d i t i o n a l costs of redesign. An

alternative

solution

that

is m o r e d i s t r i b u t e d a n d d o e s not

require

huge

u p f r o n t i n v e s t m e n t starts right at t h e door. F o r e x a m p l e , h o m e o w n e r s c a n install soc a l l e d rain barrels (Figure 6 . 3 5 ) , w h i c h a r e s i m p l e c o n t a i n e r s t h a r c a p t u r e t h e d r a i n a g e off t h e h o u s e roots

H o w e v e r , t h e s e c a n i n t e r c e p t o n l y low- a n d mid-size s t o r m

e v e n t s , and t h e y c a n be d a m a g e d in w i n t e r , w h e n t e m p e r a t u r e s are b e l o w freezing. A n o t h e r s o l u t i o n for larger v o l u m e s o f rainfall are t h e s o - c a l l e d rain g a r d e n s T h e s e are artificial and n a t u r a l d e p r e s s i o n s w h i c h are p l a n t e d w i t h v e g e t a t i o n t h a t r e m o v e s water t h r o u g h t r a n s p i r a t i o n . T h e c o n c e p t is n o t f a m i l i a r for m o s t h o m e o w n e r s . and it is s o m e t i m e s hard t o persuade t h e m to c o n s i d e r this as an o p t i o n

A simple

s p a t i a l m o d e l c a n h e l p in d o i n g t h a t . For e x a m p l e , it m i g h t be n i c e to s h o w w h a t t h e flows o f ' s u r f a c e water look like, w h e r e t h e y g o . h o w w a t e r is a c c u m u l a t e d , a n d w h e r e t h e rain g a r d e n s are most likely t o work best.

This c a n be a c c o m p l i s h e d with (.51$

m o d e l i n g , p r o v i d e d t h a t we h a v e significantly h i g h - r e s o l u t i o n e l e v a t i o n d a t a . T h e latest L I D A R ( L i g h t D e t e c t i o n a n d R a n g i n g ) p o i n t d a t a offer e x a c t l y t h a t o p p o i t u n i t y . F o r e x a m p l e , for t h e w h o l e C h i t t e n d e n C o u n t y in V e r m o n t , t h e r e are h i g h - r e s o l u t i o n data sets. T h e y are c o l l e c t e d w i t h a i r c r a f t - m o u n t e d lasers c a p a b l e of r e c o r d i n g e l e v a t i o n m e a s u r e m e n t s ar a rate o f 2 , 0 0 0 t o . 5 , 0 0 0 pulses per s e c o n d , a n d h a v e a v e r t i c a l p r e c i s i o n of 15 c e n t i m e t e r s . T h i s i n f o r m a t i o n c a n be pulled i n t o a

Water

Figure 6.36

243

The drainage network as generated by ArcGIS.

The reef line is the main stem of the stream, the yellow lines are engineered drainage pipes l h e blue lines are the surface flows We can see individual houses, and how surface flow is channeled from each property

G e o g r a p h i c I n f o r m a t i o n S y s t e m ( G I S ) , such as A r c G I S , w h i c h has s o m e q u i t e elabo r a t e h y d r o l o g i c m o d e l i n g t o o l s e m b e d d e d in it. First, we c a n build a Digital E l e v a t i o n M o d e l ( D E M ) using t h e I n v e r s e D i s t a n c e W e i g h t e d ( I D W ) i n t e r p o l a t i o n tool Irom rhe A r c G i s 9 . 2 T o o l B o x . A n o t h e r A r c G I S t o o l c a n be used t o c a l c u l a t e t h e stream n e t w o r k and s u b w a t e r s h e d d e l i n e a t i o n o n t h e basis o f t h e s e D E M s (Figure 6 . 3 6 ) . Results o f analysis s h o w t h a t t h e m o d e l e d water d r a i n a g e n e t w o r k follows t h e s t o r m w a t e r p i p e l i n e s and s t r e e t curves - e v e n depressions a l o n g t h e property lines- If we f u r t h e r d e c r e a s e t h e t h r e s h o l d , we will g e n e r a t e a m i c r o - d r a i n a g e n e t w o r k that gives us e v e n m o r e detail a b o u t t h e r o u t i n g of surface water ( F i g u r e 6 . 3 7 ) . T h i s k i n d ol i n l o r m a t i o n helps us to visualize t h e fate o f s t o r m w a t e r o n i n d i v i d u a l properties a n d tn t h e n e i g h b o r h o o d , a n d c a n also serve as a c o m m u n i c a t i o n t o o l to h e l p several n e i g h b o r s t o agree o n w h e r e it will be ino.st efficient a n d c o s t - e f f e c t i v e t o l o e a t e t h e t r e a t m e n t area

In m o s t cases, a bigger s h a r e d rain garden

will be m u c h c h e a p e r t h a n several smaller g a r d e n s on different properties. It's said t h a t " a picture's w o r t h a hundred words." I n d e e d , w h e n they look at these images t h e h o m e o w n e r s c a n actually recognise t h e i r houses a n d properties a n d see h o w t h e water flows o v e r their land. It is also c l e a r w h e r e t h e rain gardens can be l o c a t e d in order to be most e f f i c i e n t . T h e s e niixJels are powerful tools for d e l i b e r a t i o n a n d decis i o n - m a k i n g . In fact, in t h e " R e d e s i g n i n g t h e A m e r i c a n N e i g h b o r h o o d " p r o j e c t - an a t t e m p t t o find s t o r m w a t e r m a n a g e m e n t solutions at t h e scale ol small t o w n s , c i t i e s and d e v e l o p m e n t s in V e r m o n t - such visuals d e v e l o p e d by H e l e n a V'ladich w o r k e d very well in d i r e c t i n g t h e a t t e n t i o n of h o m e o w n e r s in t w o small n e i g h b o r h o o d s toward t h e distributed a l t e r n a t i v e e n g i n e e r i n g solutions. A f t e r seeing h o w t h e

rain-garden

m e t h o d could be i m p l e m e n t e d , and c o m p a r i n g costs with t h e super-pond o p t i o n , t h e citizens agreed t h a t t h e s m a l l - s c a l e distributed a p p r o a c h would be more promising a n d decided t o pursue t h a t t e c h n i q u e . M o d e l s d o n o t n e e d t o h e d y n a m i c if w e a r c mostly i n t e r e s t e d in t h e spatial c o n t e x t . T h e G I S f r a m e w o r k o f f e r s n u m e r o u s t o o l s for spatial m o d e l i n g t h a t are q u i t e simple t o i m p l e m e n t , a n d s h o u l d c e r t a i n l y be c o n s i d e r e d w h e n t h e t e m p o r a l d o m a i n is n o t i m p o r t a n t o r is n o t s u p p o r t e d by a n y d a t a or i n f o r m a t i o n .

244

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7.

Adding Socio-Economics

7.1

Demographics

7.2

Dynamics on the market

7.3

Corporate rule

7.4

Sustainability

7.5

The end of cheap oil

7.6

The World

SUMMARY W h a t m o d e l s c a n we build t o d e s c r i b e social and e c o n o m i c systems? E c o n o m i c s h a s d e v e l o p e d its o w n m o d e l s , a n d h a s b e c o m e o n e o f che most m a t h e m a t i z e d b r a n c h e s o f s c i e n c e . H o w e v e r , most, o f t h o s e m o d e l s d o n o t t a k e i n t o a c c o u n t t h e n a t u r a l side, t h e e c o l o g y . C a n we apply s o m e o f t h e m o d e l s and m e t h o d s t h a t work g o o d in t h e n a t u r a l world co d e s c r i b e e c o n o m i c processes? W o u l d t h e s e m o d e l s t h e n work for e c o l o g i c a l e c o n o m i c s ? In m a n y cases, che a n s w e r is yes. W e c a n use p o p u l a t i o n m o d els t o d e s c r i b e t h e d y n a m i c s o f h u m a n p o p u l a t i o n s . W e c a n try t o m i m i c s o m e of che w e l l - k n o w n p r o p e r t i e s o f t h e m a r k e t e c o n o m y , s u c h as t h e d y n a m i c s o f supply, d e m a n d and p r i c e . H o w e v e r , we i m m e d i a t e l y realize t h a t t h e t r a n s i t i o n regimes are q u i t e difficult co r e p r o d u c e . W h e r e a s c l a s s i c e c o n o m i c s o p e r a t e s in t h e m a r g i n , we stare c o n s i d e r i n g s o m e s u b s t a n t i a l c h a n g e s in che s y s t e m . T h i s turns ouc co be s o m e w h a t hard to m o d e l . S o m e s i m p l e q u a l i t a t i v e m o d e l s c a n h e l p us t o u n d e r s t a n d p r o c esses e m b e d d e d in our s o c i o - e c o n o m i c a n d p o l i t i c a l systems. For e x a m p l e , we c a n e x p l o r e h o w l o b b y i n g works to p r o m o t e big c o r p o r a t i o n s , and h o w t h i s c a n

allow

s u c h c o r p o r a t i o n s t o " r u l e t h e w o r l d . " W e c a n e v e n c o m b i n e s o m e of che processes f r o m che s o c i o - e c o n o m i c field wich natural c a p i t a l and try co c o n s i d e r s c e n a r i o s o f s u s t a i n a b l e d e v e l o p m e n t . A n a l y z i n g t h e s e i n t e g r a t e d e c o l o g i c a l a n d e c o n o m i c syst e m s , we find a n e w m e a n i n g in t h e m o d e l t i m e - s t e p . It c a n be related to t h e effic i e n c y of che d e c i s i o n - m a k i n g process, s i n c e t h i s is t h e t i m e over w h i c h t h e system r e a c t s to c h a n g e , t h e t i m e o v e r w h i c h p r o c e s s e s are u p d a t e d in che m o d e l . If t h e rates of p r o c e s s e s in t h e system grow, it is e s s e n t i a l that t h e t i m e - s t e p d e c r e a s e s -

other-

wise che g r o w i n g system is likely t o c r a s h . S i m i l a r l y , s i m p l e analysis of che p e a k oil p h e n o m e n o n g i v e s us s o m e insight i n t o che possible fucure of che e n d o f c h e a p o i l . It s e e m s likely t h a c in che g l o b a l s c a l e , w h e r e we d o noc h a v e easily a v a i l a b l e substit u t e s , t h e t r a j e c t o r y o f oil e x t r a c t i o n may e x t e n d s o m e w h a t f u r t h e r t h a n t h e peak at o n e - h a l t o f e x t r a c t e d r e s o u r c e . H o w e v e r , che f o l l o w i n g c r a s h will be sceeper a n d

249

250

Systems Scienrf? and Modeling fo' Ecologicst Economics harsher. T h i s could he avoided if sufficient investment: were piped inro a l t e r n a t i v e energy resources early e n o u g h , while fossil resources a r e still a b u n d a n t . Finally, we will look at a different class of models; those t h a t are used to study r h e d y n a m i c s in rhe global scale. T h e s e models c o n t a i n m u c h i n f o r m a t i o n about different processes, and should he Treated as k n o w l e d g e bases o f a kind

S o m e scenarios o f futures and

a p p l i c a t i o n s to ecosystem services are also described here.

Keywords P o p u l a t i o n dynamics, natality, mortality, migration, C a n a d a , M a l t h u s , age c o h o r t s , population pyramid, population s e n e s c e n c e , S o c i a l Security crisis, supply and d e m a n d , price, corporations, c o m p e t i t i o n , subsidies, carrying capacity, Terra C y c l e ,

Miracle-

G r o , lobbying, liquid c o a l , sustainable d e v e l o p m e n t , i n v e s t m e n t , production, chaos, fossil fuel, n o n - r e n e w a b l e , biofuel, c h e a p oil, H u b b e r t curve, C r i t i c a l N a t u r a l C a p i t a l , Energy R e t u r n o n Energy Invested, a l t e r n a t i v e energy, c o n s e r v a t i o n , global dynamics, ecosystem services, scenario, futures.

7.1 W e have already considered several p o p u l a t i o n models earlier in this book. M o d e l i n g a h u m a n population may be quite similar to m o d e l i n g a population o f woozles, as long as we h a v e t h e same i n f o r m a t i o n a b o u t t h e factors t h a t affect t h e p o p u l a t i o n d y n a m i c s . In most cases, w h a t we n e e d t o c o n s i d e r are primarily t h e growth due t o births ( n a t a l i t y ) , die d e c l i n e due to d e a t h s ( m o i t a l t t y ) , and c h a n g e due to in- a n d out-migration. C o n s i d e r , for e x a m p l e t h e data that a r e available at t h e S t a t i s t i c s C a n a d a w e b page (see T a b l e 7 . 1 ) . T h i s table presents t h e d y n a m i c s o f t h e p o p u l a t i o n of C a n a d a o v e r t h e past cenrury ( i n t h o u s a n d s ) . Based o n those data, we h a v e e s t i m a t e d and added ro t h e table t h e per capita

natality a n d mortality rates.

A simple S t e l l a model c a n be put together based o n this data. L e t us assume first t h a t there is n o m i g r a t i o n , a n d formulate t h e model o f e x p o n e n t i a l growth with varying birth and d e a t h coefficients:

dt w h e r e .\ is t h e population size, b ( t ) is t h e birth rate and m ( i ) is rhe d e a t h rate. U s i n g the " T o G r a p h " o p t i o n in S t e l l a , it should be easy to insert t h e data regarding t h e t i m e - d e p e n d e n t birth and d e a t h factors into t h e S t e l l a model a n d run it. ( A c t u a l l y , it is n o t as easy as it should b e . B e c a u s e o f a bug in some versions o f S t e l l a , it is impossible to copy and paste t h e numbers from t h e E x c e l file c o l u m n i n t o t h e G r a p h description tn S t e l l a . For s o m e reason this o p e r a t i o n supports only rhree digits, and all t h e n u m b e r s t h a t are larger t h a n t h a t will be split i n t o rwo lines. Ir is i m p o r t a n t to be aware of this, s i n c e ir may o c c u r o n a line that is n o t visible in t h e o p e n e d window a n d therefore all t h e graph data may be shifted and treated incorrectly. It seems to be much easier to do ir in M a d o n n a - so maybe that is h o w we will do tt n e x t time.) W e c a n e i t h e r put t o g e t h e r t h e m o d e l ourselves, or d o w n l o a d it from t h e b o o k website.

Add ing Soc io-Econom ics

251

Figure 7.1 gives a c o m p a r i s o n of a m o d e l run with t h e data for t h e total populat i o n numbers. T h e m o d e l s e e m s to perform quite n i c e l y for the first 11 decades, but t h e n it c o n s i s t e n t l y u n d e r e s t i m a t e s the population g r o w t h . If wc look at the difference b e t w e e n in- and o u t - m i g r a t i o n in t h e table, we c a n s e c that it has a p r o n o u n c e d

Table 7.1 Period

Dynamics of the population of Canada over the last century

Census

Total

at e n d

growth

Births

Deaths

Immigration

Emigration

Births/ year

Deaths/ ind./ year

ind./

1851-1861

3,230

793

1.281

670

352

170

0.04

0 021

1861-18/1

3,689

459

1,370

760

260

410

0.037

0 021

1871-1881

4.325

636

1,480

790

350

404

0.034

0.018

1881-1891

4,833

508

1,524

370

680

826

0.032

0 018

1891-19C1

5,371

538

1,548

880

250

380

0.029

0.C16

1901-1911

7,207

1.836

1,925

900

1.550

740

0.027

0 012

1911-1921

8,788

1,581

2,340

1,070

1.400

1.089

0.027

0 012

1921-1931

10.377

1,589

2,415

1,055

1,200

970

0.023

001

1931-1941

11,507

1,130

2,294

1,072

149

241

0.02

0 009

1941-1951

13,648

2,141

3.186

1,214

548

379

0.023

0.009

1951-1961

18.238

4.590

4.468

1,320

1,543

463

0.024

0 0G7

1961-1971

21..568

3,330

4.105

1,497

1,429

707

0 019

0.007

1971-1981

24,820

3,253

3,575

1,667

1,824

636

0.014

0 007

1981-1991

28,031

3.210

3,805

1,831

1,876

491

0.014

0.007

l Population

F i g u r e 7.1

2: DATA

Modeling population dynamics with no migration.

252

Systems Scienrf? and Modeling fo' Ecologicst Economics

1600 1400 1200 1000

800 600 400 200 0 -200

Net migration or difference b e t w e e n immigration and emigration rates in Canada There is a substantial increase of immigration in the s e c o n d half of the t w e n t i e t h century, w h i c h explains w h y it is hard to m a t c h the data w i t h o u t taking migration into a c c o u n t .

2 DATA

1 • Populalon

li

1

.

;

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—

Figure 7.3

M o d e l i n g population w i t h migration included.

Actually there is a n error in this model Can you figure out w h a t it i s 9

g r o w t h t r e n d o v e r t h e y e a r s ( F i g u r e 7 . 2 ) . It b e c o m e s e s p e c i a l l y large o v e r t h e past five d e c a d e s , w h i c h q u i t e c l e a r l y m a t c h e s t h e p e r i o d w h e n o u r m o d e l s t a r t e d t o fail. Ir s e e m s t o m a k e p e r f e c t s e n s e t o b r i n g t h e m i g r a t o r y p r o c e s s e s i n t o t h e p i c t u r e a n d i n c l u d e t h e m in t h e m o d e l . T h e s i m p l e s t w a y is j u s t t o add t h e i n c o m i n g p o p u l a t i o n a n d s u b t r a c t t h e n u m b e r of p e o p l e l e a v i n g : dx

dc

(b(c)

— m(r))x + in(t) — Out(t)

w h e r e l n ( c ) is t h e n u m b e r o f i m m i g r a n t s a n d O u t ( r )

is t h e n u m b e r o f e m i g r a n t s .

H o w e v e r , if w e r u n t h e m o d e l n o w , t h e r e s u l t s t u r n o u t t o b e e v e n less s a t i s f y i n g . First w e u n d e r e s t i m a t e d

t h e p o p u l a t i o n size, a n d t h e n

we o v e r e s t i m a t e d

it q u i t e

considerably (Figure 7 3 ) . W e may begin t o speculate that perhaps migrants are affecting natality a n d mortality i n a d i f f e r e n t w a y t h a n t h e a b o r i g i n e s . T h i s m a y b e e i t h e r b e c a u s e o f a s p e c i f i c a g e s t r u c t u r e o f t h e m i g r a n t p o p u l a t i o n ( p e r h a p s t h e y a r e a r r i v i n g l a t e r in t h e i r r e p r o d u c t i v e life a n d t h e r e f o r e g i v i n g b i r t h ro f e w e r c h i l d r e n , o r m a y b e e x a c t l y t h e o p p o s i t e

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S o m e -very basic e x p a t u r c n t s w i t h t h e population m o d e immediately s i o w u s w h a t expon e n t s grcNvtt- o , end r«vis t«vwj to toko thai traiectory As long as t h e birth rate is I v g h e ' than the death rate, the population gtov.-. • and grows quite du»c*ly Th-s is what m a d e Vlalthus worry as u.n v lis 1796 w h e n h n wrote "Tho powor of population is so supe-or to t h e power of the earth to produce subsistonco 'or man. that prematiire death must in s o m e shape or other visit the human l a o r The world's cutront population g r o w i h -ate -. -ibout I 14 ocicont. repruserihng a doubling tiinrj of 01 ycara W o can expect r e wo-id's copulatior o f 6.5 billion to become 13 billion by 2 0 0 7 if current j i t w l h c o n t n j e s The w w k J V gr /.virate p e e i e d in the 1960s at 2 percent, w i t h a doubling time o l 35 years M o s t Europesr countries have l o w g r o w t h rates In tne Unitud Kingdom, the -a'e C 2 porcont in Germany it '3 O.C cercent. and in France It's 0 4 percent. G e r m a n s zero rata of y e w m m o u d e s a natur» r c ? o a $ e o f - 0 . 2 Peicont Without i r T u n ^ ' a t o - Germany's p o p j U - n ^ o u l d b s s' • i>r,g. as that of t h e Czech Repuolic Maoy As j r . are Afncen c o u n - w s have N g f p o p u i o ' i c ^ y ^ v . — rates Afghanistan has a current g r o w t h r a w . ' 4 8 o-- c e n t representing a dc ^ t ^ n g w r e of 14 5 years' W e can also see f r o m me model that it's r o t the g'.vwth c* death r, *» n » r -rasers. b u t lathei The -i g r o w t h m o r e or "ess under control m d e - e o p e a COWTKIIM. death rales h M decreesed
r

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tim-a b m fotcn havo s o r t e d to decrease as wall as o 'o- utt ot bcr.fc' edtxat-oo

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port of the tiliiorfv As t> result ine net growth has d a o w M d »>rt05t tc » r o . as in G e r m * - , Problem*. in ti. volot>og counvias b c ^ n w o n ocrta." • .>?.,••
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as vaccination, v t b n The death rate t r a n s u m to d e c i r « e v » V * the t w t h rate remains the san-iu And in Paul Ehr'-ch's t e r n s . a p o p ^ - s t o o b e m h e .ptodes Dooc mm m t o " t h i t we sno-ita stop h - j m a n t r u n a r o motlical aid to uncetdeveiopec African c o u n t i n g

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that v. ti

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- .iw.tv Otherwise copulation*; will grow oxponontally. wiping cut natu

iai resource*. cieMrtg civil unrest and refugee problems Eventually population growth will adjust b>r i M ini.>-t be a long and quite an ugly process involving r>^es. t*>od*h?d and relocation

A m thr- i v i i M r t s p L i i J L h m I i I V . ill l n r

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256

Systems Scienrf? and Modeling fo' Ecologicst Economics

H o w e v e r , as t i m e goes by the d i s t r i b u t i o n graph b e c o m e s q u i t e distorted, r e p r e s e n t i n g t h e arrival of t h e b a b y - b o o m e r s in the 1950s a n d clearly s h o w i n g t h e t r e n d t o w a r d s a p r e d o m i nantly older population.

2001

30 Million

J

i

L

|k L

J M A 4

i

_

J

i

IF

j

k

fe

2J5 2-y> iv.• '.*

I !<

110 75

!u

15

2!

F H — !u ?S m 13: i*. 17; i j i

C o n s i d e r tour age groups: c h i l d r e n (aged 0—15 years), a d u l t s l

(16—40 years),

adults2 V41 —65 years) and retired adults (over 6 6 years of age). T h e goal o f looking at t h e s e age groups is two-fold

First, we want t o separate t h e c h i l d b e a r i n g group

( a d u l t s l ) ; secondly, we wish t o distinguish b e t w e e n t h e working adults (adultsl

+

a d u l t s ! ) a n d t h e rest ( t h e n o n - w o r k i n g p o p u l a t i o n ) . In s o m e cases we may n e e d t o c o n s i d e r more age groups (also called c o h o r t s ) , but usually it makes sen>e to differe n t i a t e only b e t w e e n t h e o n e s that have different f u n c t i o n s A f t e r all, why m a k e t h e m o d e l more c o m p l e x ' T h e S t e l l a diagram for this model c a n be s e e n in Figure 7 . 5 . W e h a v e four s t a t e variables with transfer functions, t l , t2 a n d t 3 . E a c h transfer f u n c t i o n should

D

o mm3

r - f .y

m3 Aye" 11065

rn4

Age66above

i

X O^D 'OO t .

d4

CO 3

Cd4

8 Figure 7.5

An age-structured model, with four state variables representing lour age groups or cohorts.

Add ing Soc io-Econom ics

257

be designed in such a way that ir moves all che individuals in one age group to the n e x t over rhe time period that an individual stays in the age group. For example, if a baby is born and put in the A g e 0 - 1 5 group it will stay in this group for che n e x t 15 years and then be transferred to the next group, A g e 16—40. T h i s means rhat every year one-fifteenth of che number of individuals m this age group will be moved to rhe next group. Therefore, rl = (AgeO—15)/l 5. Similarly, r2 = ( A g e l 6 - 4 0 ) / 2 5 , and t3 = ( A g e 4 1 - 6 5 ) / 2 5 . Individuals stay longer tn these groups, and therefore only onetwenty-fifth of the group is transferred to the next group annually. T h e other processes are similar to those in rhe standard populacion model considered above, except chat the birch flow is proportional only to the number of individuals in the second childbearing age group, Age 1 6 - 4 0 , and nor the total populacion, as we saw before. If we want co use che birch rate from the daca sec thac we have, we need to add a scaling factor that will rescale the birth rate for the whole populacion co the bitch race controlled only by che Age 1 6 - 4 0 age class. Similarly, we need to figure out how the total deaths will be distributed among the different age groups. C o m m o n sense dictates that death rates in the younger age groups should be smaller than the population wide average. T h e death rate in the A g e 4 1 - 6 5 group should be close co the average, while che death race in che A g e 6 6 above group should be considerably higher than che average. Assumptions should also be made co distribute the overall migration data (— immigration — emigration) among the various age classes. Overall, we end up with che following sec of equations. Definitions of the four age groups are almosc identical: [transfer from t h e younger age groupl — [death] + [net m i g r a t i o n ] In addition, there is birth in the first age group. [AgeOtol5ft) - Age0to15(t - dt) ^ (b1 + m l - cM - t i ) * dt INIT Age0to15 = 600 INFLOWS: bl = cbl *bb*Age16to40 m l = mm1 OUTFLOWS: d1 = c d l * AgeOtol 5 t1 = AgeOtol 5/15 Age16to40(t) = Age16to40(t - dt) + (:1 + m 2 - d2 - t2) * dt INIT Age16to40 = 2100 INFLOWS: t1 - AgeOtol 5/15 m2 -

mm2

OUTFLOWS' d2 -

cd2*Age16to40

t2 - Age 16to4 0/25 Age41to65(t) - Age41 to65{t - dt) + (t2 + m 3 - d3 - t3) * dt INiT A g e 4 1 t o 6 5 = 300 INFLOWS: t 2 = Age16to40/25 m3 = mm3

272 Systems Scienrf? and Modeling fo' Ecologicst Economics OUTFLOWS d3 = c d 3 " A g e 4 U 6 6 5 13 = Age41to65/25 Age65aboveit! - Age66above!t - dt) +- U3 + m 4 - d
J

. -

m

-- • —-

Adding Socio-Economics

259

2: DATA

1: Total 30000 00

/ !

15000 00

000 1893 50

1861.00

Figure 7.6

1926 0C

1958 50

199100

Model results for total p o p u l a t i o n tn the age structured m o d e l after calibration of birth

and death-rate parameters.

2: Age 16IO40

1 AgeOtol 5 10000.00

3: Aye41 lo65

4 Age66ab0ve

5000 00-

0.00'

Figure 7.7

1861.00

1893 50

1926 00

1958.50

1991.00

Dynamics of differeni age c o h o r t s in the a g e - s t r u c t u r e d model.

Towards the end of the simulation, the numbers in the elderly c l a s s e s Stan to g r o w more rap'dly than in other c o h o n s A time bomb for a generational storm is set, unless some drastic measures are taken.

W h a t w e c a n d o n e x t wich r h i s m o d e l is e x p l o r e t h e d a u n t i n g p r o b l e m of p o p u l a t i o n s e n e s c e n c e t h a t is c u r r e n t l y l o o m i n g o v e r m o s t o t t h e s o c i e t i e s in d e v e l o p e d countries. W i t h

che g r o w t h o f a f f l u e n c e a n d e d u c a t i o n , p e o p l e a r e less inc l i n e d t o

h a v e c h i l d r e n . A s a result h i r r h r a t e s a r e d e c r e a s i n g , w h i l e t h e a d v a n c e s in m e d i c i n e a n d h e a l t h c a r e a r e d e c r e a s i n g t h e m o r t a l i t y rare. T h e s e c h a n g e s m a y n o t a f f e c t r h e t o t a l p o p u l a t i o n n u m b e r s ( a f t e r a l l , w e a r e d e c r e a s i n g b o t h t h e inflow a n d r h e o u t f l o w for t h e stcxrk o t t h e p o p u l a t i o n n u m b e r ) , b u t t h e y will h a v e a s u b s t a n t i a l e f f e c t o n t h e shape o f the a g e distribution

T h e n u m b e r o f old and retired people keeps growing,

causing a n increasing burden o n t h e welfare system. A t t h e s a m e time, che n u m b e r o f p e o p l e o f w o r k i n g age b e c o m e s r e l a t i v e l y s m a l l e r , s o t h e r e a r e f e w e r p e o p l e c o n t r i b u t i n g t o t h e s u p p o r t of t h e r e t i r e e s i n t h e s y s t e m . In o u r C a n a d i a n m o d e l w e a l r e a d y h a v e t h e d e c l i n e o f b i r r h a n d d e a t h

rates,

a n d w e c a n a l r e a d y s e e t h a t n u m b e r s in t h e e l d e s t a g e g r o u p , A g e 6 6 a b o v e , a r e s t e a d ily i n c r e a s i n g , m a k i n g t h i s a g e g r o u p d o m i n a n t

i n t h e p o p u l a t i o n . L e t us a d d r h e

s o c i a l s e c u r i t y s y s t e m i n t o o u r m o d e l . T h i s c a n e a s i l y b e p e r f o r m e d by i n t r o d u c i n g a n o t h e r s t o c k in t h e m o d e l t h a r will h a v e a n i n f l o w g e n e r a t e d b y p a y m e n t s f r o m r h e

260

Systems Scienrf? and Modeling fo' Ecologicst Economics A g e / 6 m 4 0 a n d Age4lto65 oi t h e Age66ahavc

groups, while t h e outflow will be in proportion t o t h e size

group (Figure 7 . 8 V

SS_tund(t) = SS_fundtt - d t l + (taxes - p e n s i o n s ) " dt INiT S S J u n d = 100 INFLOWS: taxes = p a y * ( A g e 1 6 t o 4 0 + A g e 4 1 i o S 5 } OUTFLOWS: pensions - p p ' A g e 6 6 e b o v e T h e " p a y " is t h e a m o u n t that individuals c c n t r i b u t e t o che S o c i a l S e c u r i t y fund while they are working, and " p p " is rhe size of t h e p e n s i o n that retired people r e c e i v e . W e c a n immediately see that if we k e e p " p a y " a n d " p p " c o n s t a n t , t h e " S S J u n d " will g o bankrupt some t i m e in t h e near future (Figure 7 . 9 ) . In this model w e h a v e assumed that t h e social security system has been in p l a c e s i n c e r h e b e g i n n i n g o f our data set in 1 8 6 1 . t h a t t h e p a y m e n t s t o a n d from t h e fund have b e e n c o n s t a n t o v e r these years, a n d chat t h e age o f r e t i r e m e n t has also r e m a i n e d c o n s t a n t . T h i s is c e r tainly not realistic, and for a b e l t e r model we should include all thebe historical data in our c o n s i d e r a t i o n . However, this is unlikely to c h a n g e t h e overall trend because, again, qualitatively it is q u i t e c l e a r that as t h e elder population group grows in size we will need more resources t o support it T h e model is an e x c e l l e n t tool ro quantify s o m e o f these q u a l i t a t i v e n o t i o n s .

^

taxes

SS fund

pension

( Age 16:o40

Age41 to65

Age63above

A simple submodel of a social security lund

1 S S lund

6000000 00,

3000000.00-

ooo-K 186100

BFfi fg fuTr e^ W 7 .P9B

1895 2 5

1935 5 0

1972 75

2010.00

Dynamics of the social security fund when population s senescing and there are less

people who work and more people who receive pensions.

1 m • • • ! ! Adding Socio-Economics

i

^r 261

The p r o b l e m t e e m i » b e a u n a s « « u ] l o r t h e USA A c a y d r g t o r o f i - k o * a n d B u r n s 0 0 0 5 1 b y m o - c e f f i i * y r « US c e n t e n a r « n p o p u l a w m c e r i e t a * * * 0 u > U >oo»y I n 1800.

e * c e e o 60C 0 0 0 Tnars t e n t i m e s i r t e w i w

e ^ e c t a x y « & n r \ w e a 47 y e a n * K J n e

or

•»««» o>

all Ame* tcaris IKo« x w n g e r haK o O e d v w s cn)y ? 2 3 y e w s On** 4 1 p e r c e n t o l t h e p o p u t o t o n wvw a g e o 6 6 o i m o r a "Soday. We a x 3 a c t * n c Y a t b r t h i s a t o . 1 ^

years - a g a m
t f e e x p e c t a n c y at 6 5 ts n o w l ? y e a n up t r o r r 12 years r> ' 9 0 0

H e e * p a c t t n c y at 6 6

1 * 1 " i o be » e e M e r i t i r \ g As a e e n w q u s n c a . t h e s e » « J 66 a n a cuer m a c e JO 1 2 * p e - c s r t C t h e p o c u f c t o n Cv Tonn - nearly 4&J3+

t h e 6 3 p e n * r « of popi*»t»xi

unde« 5

*• the

s a m e t i m e b n h r a t a s o l i / n m e t e d f r o m wefl t x » 2 1 c M c y e n ot-- e o w t e « * e l o n g - t e r m r*p«aeem e n t rata f o r t h e p c p U i t t o n ] and are r o w hovenng near t h e r e p w c e m e n i TO A s « result i * e « e are m o m p e o p l e r o w i n g n o w , w h i l e the numfcer o f tf-^dren c o m i n g of a p e and p m n g !Pe * * x t forca isn't nearly as ttrge T h e forces t h a i wo>Ja e x p a n d t n e younger ( a n d working? p o p u t o t o ' i p o y w g S o o n I Security end M e d i c a r e t a * e s are m reverse. demographic storm

tt>e

i* a k n d ol o

Rack in 1950. m a n u m b e r of w j x o r e per 5ociai Security b e r - e f c e r y w a s

f f i b; try > 0 0 0 . t h i s had c r o p p e d t o 3 e. i n i S c process, m o s t woricets f a t e d p a y r ^ m o r a m e m p l o y m e n t taxes than they pay in i n c o m e taxes, as e m p l o y m e n t tax r o s e t i v e ' o l d Tha A * g e s s u b j e c t t o Social Secuiitv tax r o s e as

rising f r o m $ 3 . o o n m 1950 t o $ 8 7 0 0 0 m 2 0 0 3

Q e t w u u n n o w a n d 2 0 3 0 w e ' l l have t h e tast bio surge

t h e r e t i r e m o n i of t i n b o o m

B y t h a n w e ' l l be d o s e t o having enly t w o cohered w o r k e r s pe* b e n e f i c i a r y

I n s t e o d of hev-

•ng 16 w o - l e r s chipping m to S j p p o r t each senior citizen, there w i l l onfc bo 2 W h o r a w e h i d 3 5 5 m * o n p e o p l e age 6 5 a r d older in 2 0 0 0 , w e II have 6U a m J l e n i n 2.130 D i x m g t h e s e 3 0 y e a n , tne d e p e n d e n c y ratio - lh# ratio of those ognd over 65 t o i h o s e aged 2 0 - t i a - w i l l rise f r o m 2> 1 D e t c c n t t o 35 5 p e r c e n t A o c o r o r g t o KotJikoff. t h e s i t u a n o n is so s c a r , '.hat he keeps reft?" ing t o o w e r |i
gptt-'^fl t l i l ' t " r'ffllhK

p r o j e c f t o n s w i t h o u r ve»y Simplified m o d e l Then « v j m b « i * -aiII n n e d l o b e d i a n o o d t o d e s c r i b e r e

U S Situetiori

but tno-e is g e e d w s o n to e » o e c t thai, n u e i i f e f n ^ y .

w i l l be g e t n r g

exactly fre s a m e scenario t h a t < o t l i k o f portrnvs

Exercise 7.3 H o w « « coss.oe k

o » e r « r t t->e S e e *

t i •_'•«: •

rate o f C0rar
b e t h e r e a r e < r i e n t a g e »O M « K » t t w K a x J U I I M *

•Kf h m .

U< J

• . j V u r g ' LX6>Oi> dYi-iying " e

J"«l d A i y r ^ " t ' m w - * ' U I K

•N'VGTU'H

I j >K/»tt

lit* nit mij.-Tjt i m m m

iu i > • H i - i i

Ji >i

I I

i t

it

1.1

.1.1.

i.i>

r.i

J:

J.. J

A " dt

«RN

j i j r W S i - v«u,iit SJJIT-

i h r r.fo.tr I R i n JTT umA

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1

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J,I.

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262

Systems Scienrf? and Modeling fo' Ecologicst Economics

Here,

X| = A g e 0 t o i 5 ,

x 2 = Agel6to4G,

X3 = Age4Ito65

a n d x4 =

Age66above\

b is t h e b i r t h r a t e , i, a r e t h e t r a n s f e r c o e f f i c i e n t s , a n d d, a r e t h e m o r t a l i t y r a t e s - j u s t as m t h e S t e l l a m o d e l a b o v e . W h e n r u n n i n g S t e l l a , w e d i d h a v e s o m e p i o b l e m s i d e n t i f y i n g t h e c o r r e c t v a l u e s for t h e c o e f f i c i e n t s . S o m e t i m e s t h e v a r i a b l e s w e r e g r o w i n g t o o fast, o r a l t e r n a t i v e l y t h e y w e r e d i m i n i s h i n g t o zero. A r e t h e r e a n y r e l a t i o n s h i p s t h a t w e s h o u l d k e e p in m i n d w h e n l o o k i n g for s u i t a b l e c o m b i n a t i o n s o f p a r a m e t e r s ? First, l e t us c h e c k for a n e q u i l i b r i u m . M a k i n g t h e l e f t - h a n d s i d e o f t h e e q u a t i o n s e q u a l t o zero, we g e t a s y s t e m o f a l g e b r a i c e q u a t i o n s :

0 = bx2

- t[X| -

djX,

0 - £|X, - t2x2

— d2x3

0 = t2xz -

t3x3

~ d3X3

0 -

d^Xq

t^Xj -

T h e first e q u a t i o n y i e l d s : X[ = bx 2 /(£] + d j ) . S u b s t i t u t i n g t h i s i n t o t h e s e c o n d e q u a t i o n , w e g e t (btlj(t[

+

— (t2 + d ? ) ) • x 2 = 0 . T h i s m e a n s t h a t w e g e t a n e q u i -

l i b r i u m o n l y if x 2 = 0 , w h i c h t h e n a u t o m a t i c a l l y m a k e s a l l r h e o t h e r v a r i a b l e s e q u a l t o zero. O r if (btll(tl

^

+ d\) — (ti + d2))

_

bx-, -

x» =

= 0 , in w h i c h c a s e X2 c a n b e a n y , a n d t2x2 —

,

and

x* ~

t3x3 ——

. ,. ^7.1)

N e i t h e r o f t h e s e s t a t e s is i n t e r e s t i n g , s i n c e t h e first is t r i v i a l , w h e n t h e r e is n o p o p u l a t i o n , a n d t h e s e c o n d is e x t r e m e l y u n l i k e l y b e c a u s e it r e q u i r e s t h a t t h e r e is a n e x a c t relationship between model parameters. Equality-type relationships are unreali s t i c f o r a n y r e a l - w o r l d s i t u a t i o n s , w h e r e t h e r e will a l w a y s b e s o m e u n c e r t a i n t y a b o u t m o d e l p a r a m e t e r s a n d ir is i m p o s s i b l e t o g u a r a n t e e t h e y w i l l b e e x a c t l y e q u a l t o s o m e combination biil(ii

+ dj) -

b e t w e e n o t h e r p a r a m e t e r s , as we r e q u i r e i n t h i s c a s e by a s k i n g

that

(h + dL) = 0 .

H o w e v e r , t h i s analysis is n o t w i t h o u t m e r i t . W h a t we c a n s e e is t h a t w h e n this c o n d i t i o n d o e s n o t h o l d a n d , say, bt\l(t\ + d\) > t 2 + d 2 , t h e n dx2ldt

> 0. This means that

in this c a s e x 2 will b e g r o w i n g . K e e p i n g in m i n d ( 7 . 1 ) , we c a n s e e t h a t all t h e o t h e r varia b l e s will also be g r o w i n g . If. o t h e r w i s e , bt\l{t\ + d\) < t2 + d2, all t h e m o d e l v a r i a b l e s will b e d e c l i n i n g . S o we h a v e found a s i m p l e c o n d i t i o n t h a t q u i c k l y tells us w h e n t h e p o p u l a t i o n b e c o m e s e x t i n c t ; a n d w h e n it survives. I n t e r e s t i n g l y , n o n e o f t h e p a r a m e t e r s from t h e third o r fourth e q u a t i o n s in t h e m o d e l a r e i n v o l v e d . N o t surprisingly, this m e a n s t h a t , for survival o f t h e p o p u l a t i o n , o n l y t h e first t w o a g e groups m a t t e r . T h e r e m a i n i n g t w o a r e a tail t h a t c a n b e c u t t o a n y l e n g t h . T h e p o p u l a t i o n still persists, as long as t h e c h i l d b e a r i n g g r o u p is in p l a c e . O n c e it g i v e s birth t o progeny, it c a n disappear. T h i s s i m p l e a n a l y s i s is q u i t e h e l p f u l w h e n l o o k i n g f o r t h e r i g h t c o m b i n a t i o n o f parameters to make the model a m , Instead o f t h e trial-and-error method, we c a n i d e n t i f y c e r t a i n p a r a m e t e r d o m a i n s w h e r e t h e m o d e l b e h a v e s as w e w o u l d w a n t it t o . If w e b r i n g in m i g r a t i o n , w e g e t a s l i g h t l y m o d i f i e d s y s t e m o f e q u a t i o n s : dx 1 ~T dt dxj dt

. =

-

bx>

, ~ lixi "

t , x , - t2x2

+

~ a2x2

+ m2

Add ing Soc io-Econom ics

263

dx, dc dx., _ — — - 13X3 at

a 4 v 4 + m4

H e r e , m, a r e che n e e m i g r a t i o n races i n c o che f o u r a g e g r o u p s . T h e y c a n b e p o s i c i v e o r n e g a t i v e . T h e o c h e r p a r a m e t e r s a r e a l w a y s p o s i t i v e . T h i s t i m e , we c a n s e e t h a t t h e r e exists an equilibrium

in che

m o d e l : s u b s t i t u t i n g x]

— (bxi

c o m e s f r o m che e q u i l i b r i u m in t h e first e q u a t i o n (dxjdt

-

+ rrij)/(t, + d , ) ,

which

0 ) , inco the s e c o n d e q u a -

tion at e q u i l i b r i u m , we i m m e d i a t e l y get a s o l u t i o n for x 2 :

x

miLi

=

+

(7.2)

(t2 + d 2 ) ( t , + d , ) -

2

6c,

T h i s c a n b e t h e n s u b s t i t u t e d b a c k i n t o che e q u a t i o n f o r _ ''

to p r o d u c e

+ m , ( t 2 -I- di)

bm2

(t 2 + d 2 ) ( t ,

+d

1

)-bc

(j

^

l

S u b s t i t u t i n g x 2 i n c o c h e t h u d e q u a t i o n , we c a n c a l c u l a t e _

t 2 X 2 + T7l}

T h e n , s i m i l a r l y , t h i s v a l u e for Xx c a n b e used t o c a l c u l a t e

x4

-

+

w h i c h follows from t h e fourth e q u a t i o n . O b v i o u s l y , t h e s e e q u i l i b r i a h a v e t o b e p o s i c i v e . If che m i g r a t i o n c o e f f i c i e n t s m j a n d m 2 a r e p o s i t i v e , it f o l l o w s f r o m ( 7 . 2 ) a n d ( 7 . 3 ) t h a t w h e n

b < " ^

)

(

' '

+

d

' >

(7.4)

w e h a v e all t h e e q u i l i b r i a i n t h e p o s i t i v e d o m a i n ; o t h e r w i s e we m o v e i n t o t h e n e g a t i v e d o m a i n . If t h i s c o n d i t i o n h o l d s , t h e o t h e r t w o e q u i l i b r i a for X3 a n d x 4 will a l s o be posicive. I f we n o w run t h e S t e l l a m o d e l u n d e r t h e s e c o n d i t i o n s , it a p p e a r s t h a t t h e e q u i l i b r i u m is s t a b l e : we c a n scavc m o d i f y i n g c h e i n i t i a l c o n d i t i o n s , a n d still will c o n v e r g e to che v a l u e s chac we h a v e i d e n t i f i e d a b o v e . H o w e v e r , if che e q u i l i b r i u m m o v e s i n t o che n e g a t i v e d o m a i n , e i c h e t w h e n ( 7 . 4 ) n o l o n g e r h o l d s o r w h e n m i g r a t i o n

becomes

n e g a t i v e , we gee e x p o n e n c i a l g r o w c h o r e x p o n e n c i a l d e c l i n e p a t c e r n s . T h o u g h

ana-

lycical a n a l y s i s c a n b e c o m e q u i c e c u m b e r s o m e , w i t h o u t it it m a y be h a r d to figure o u t chac che m o d e l c a n p r o d u c e a l l c h r e e t y p e s o f d y n a m i c s : e x p o n e n t i a l g r o w t h , e x p o n e n t i a l d e c l i n e , o r s t a b l e s t e a d y s t a t e . It all d e p e n d s u p o n t h e p a r a m e t e r s we c h o o s e . W e may o n c e again c o n c l u d e t h a t l o o k i n g at t h e e q u a t i o n s c a n be q u i t e helpful. Unfortunately,

t h e a l g e b r a b e c o m e s r a t h e r t i r e s o m e e v e n w h e n we h a v e o n l y

e q u a t i o n s a n d s o m e fairly s i m p l e i n t e r a c t i o n s . H o w e v e r , w h e n w e h a v e m a n y

four more

264

Systems Scienrf? and Modeling fo' Ecologicst Economics e q u a t i o n s a n d p a r a m e t e r s , it is still i m p o r t a n t to run a n d r e r u n t h e m o d e l for as m a n y c o m b i n a t i o n s o f p a r a m e t e r v a l u e s ;ind i n i t i a l c o n d i t i o n s a s we c a n afford. T h i s is t h e o n l y way t o a t t a i n c o n f i d e n c e a n d u n d e r s t a n d i n g o f t h e results we are p r o d u c i n g . B e f o r e w e c o n t i n u e w i t h s o m e l i n k e d m o d e l s o f d e m o g r a p h y a n d e c o n o m i c s , let us c o n s i d e r a few e x a m p l e s o f simple e c o n o m i c a n d s o c i o - e c o n o m i c m o d e l s .

7.2

Dynamics on the market Let us see if d y n a m i c m o d e l i n g is a n a p p r o p r i a t e tool t o m o d e l s o m e e c o n o m i c syst e m s . C o n s i d e r t h e basic d e m a n d - s u p p l y - p r i c e t h e o r y t h a t is discussed in m o s t c l a s s i c a l b o o k s o n m i c r o e c o n o m i c s , a n d a t a variety of w e b pages ( e . g . http://hadm.sph sc.edu/COURSES/ECON/SD/SD.html

or http://vcollege.lansing.cc.mi.us/econ201/

unit. 0 3 / l s s 0 3 l . h t m ) . In e s s e n c e , we are l o o k i n g at a system o f two state v a r i a b l e s , o n e r e p r e s e n t i n g t h e q u a n t i t y o f a g i v e n c o m m o d i t y ( G ) a n d t h e o t h e r o n e r e p r e s e n t i n g its p r i c e ( P ) . In a m a r k e t e c o n o m y , t h e t w o a r e supposed t o b e d e t e r m i n e d b y t h e r e l a t i o n s h i p b e t w e e n supply a n d d e m a n d . L e t us l o o k a t Figure 7 . 1 0 t o see h o w we derive t h e r e l a t i o n s h i p between price and the a m o u n t of c o m m o d i t y o n the market. Suppose that t h e price o f t h e c o m m o d i t y is s e t at

I n Figure 7 . 1 C A , we will graph t h e r e l a t i o n s h i p b e t w e e n

p r i c e a n d t h e a m o u n t of t h e c o m m o d i t y o n t h e m a r k e t ; in Figure 7.1 OB, we will s h o w che c h a n g e in p r i c e o v e r time. L e t us first draw t h e graph of S u p p l y T h e L a w o f S u p p l y states that t h e h i g h e r t h e p r i c e tor a c o m m o d i t y , t h e m o i e products will b e offered by t h e p r o d u c e r o n t h e m a r k e t . S o S should b e a n i n c r e a s i n g f u n c t i o n o f P . ( N o t e t h a t , m a t h e m a t i c a l l y , t h i s is s o m e w h a t d u b i o u s , s i n c e we h a v e just replaced t h e i n d e p e n d e n t v a r i a b l e in t h e graph. N e v e r t h e l e s s , this is t h e way e c o n o m i s t s d o ic.) On

t h e graph, we s e e thac che quancicy g| c o r r e s p o n d s co che price p j . T h i s

p r o j e c t s t h e first price p o i n t

in Figure 7 . 1 0 B H o w e v e r ,

t h e r e is also che L a w o f

D e m a n d t h a t scaces chat che p r i c e ot a c o m m o d i t y is inversely relaced co t h e a m o u n t d e m a n d e d p e r t i m e period. I n our case, t h e D e m a n d c u r v e s t i p u l a t e s thar ar a q u a n t i t y i>i t h e c o m m o d i t y c a n be sold o n l y at a price as low as

( t h e s e c o n d p o i n t in Figure

7 . 1 0 B ) . W i t h such h i g h supply t h e r e is simply n o t e n o u g h d e m a n d t o k e e p t h e price up, so c o m p e t i t i o n a m o n g producers i n c r e a s e s a n d they h a v e t o push che price d o w n co sell all t h e

stock thac was produced. H o w e v e r , a t p r i c e p2 t h e S u p p l y c u r v e tells

p pi Po

p4 Pi

g3 A • Figure 7.10

9,

G B

Converging of supply towards demand and equilibrating of price after several cycles

A. The demand (Di and supply IS) curves. B Dynamics of price as defined by the demand and supply.

Add ing Soc io-Econom ics

265

Its t h a t p r o d u c e r s will n n l y he w i l l i n g t o p r o d u c t : a n d s h i p ro t h e m a r k e t g; c o m m o d i ties. P r o d u c i n g t h e s e g o o d s at s u c h a p r i c e is n o t t h a t p r o h t a b l e a n d l u c r a t i v e for t h e p r o d u c e r , s o o n l y a few will r e m a i n , a n d t h e y will p r o d u c e m u c h less. O n c e a g a i n , p r u i e c t i n g t h e a m o u n t g j t o c h e D e m a n d c u r v e w e r e a l i z e chat n o w t h e d e m a n d for t h e c o m m o d i t y

is s o h i g h ( t h e r e is a n e x c e s s d e m a n d o n t h e m a r -

k e t ) c h a t it c a n sell at a p r i c e as h i g h as p v F o r s u c h a p r i c e t h e p r o d u c e r is o n c e a g a i n e a g e r t o p r o d u c e m o r e a n d , a c c o r d i n g t o t h e S u p p l y c u r v e , will d e l i v e r

more

c o m m o d i t i e s to t h e m a r k e t . W e c o n t i n u e this process, o b s e r v i n g t h a t t h e p r i c e a n d r h e a m o u n t o f c o m m o d i t y g r a d u a l l y c o n v e r g e co a c e r t a i n e q u i l i b r i u m s c a t e , w h e n c h e p r i c e will b e j u s t r i g h t for t h e a v a i l a b l e q u a n t i t y s u p p l i e d , a n d t h e q u a n t i t y supp l i e d will m a t c h t h e a m o u n t d e m a n d e d

T h i s is w h a t is c a l l e d " m a r k e t e q u i l i b r i u m . "

E c o n o m i c t h e o r y c o n s i d e r s t h a t m a r k e t s c o m e t o e q u i l i b r i u m in o n e s h o t - i.e. w h e n b o t h p r o d u c e r s and c o n s u m e r s k n o w e x a c t l y t h e equilibrium price and a m o u n t of a m a r k e t g o o d w h i c h will b e sold o n t h i s m a r k e t . H o w e v e r , i n t h e real w o r l d ic a l w a y s t a k e s s o m e t i m e for s u p p l y t o a d a p t t o d e m a n d a n d *'ice versa. H o w c a n w e d e s c r i b e t h i s p r o c e s s in a d y n a m i c m o d e l : C o n s i d e r a s y s t e m

with

t w o v a r i a b l e s : P a n d G . A c c o r d i n g t o t h e S u p p l y L a w , c h e p r o d u c t i o n of t h e c o m modity G

is in p r o p o r t i o n

t o its p r i c e . A c c o r d i n g

to the D e m a n d

s u m p t i o n (if t h e c o m m o d i t y is i n r e v e r s e p r o p o r t i o n t o c h e p r i c e

Law, t h e

con-

T h e r e f o r e , we c a n

a s s u m e t h e e q u a t i o n for g o o d s in t h e f o l l o w i n g f o r m :

• c ,P *

dt

(7.5)

1

cfi2P

B a s e d o n s i m i l a r c o n s i d e r a t i o n s , w e a s s u m e t h a t t h e p r i c e i n c r e a s e s in

reverse

p r o p o r t i o n t o t h e a m o u n t o f t h e c o m m o d i i y a v a i l a b l e for c o n s u m p t i o n a n d d e c r e a s e s in d i r e c t p r o p o r t i o n co t h i s a m o u n t :

dP

V a6)

r c " ^

W e c a n find t h e e q u i l i b r i u m for t h i s m o d e l by a s s u m i n g t h a t t h e r e are n o c h a n g e s in t h e system, s o 0

i..,P

-1>

0

= c

>i(j

T h e f e a s i b l e ( p o s i t i v e ) e q u i l i b r i u m p o i n t is ( P -

l/vM.^), G =

l/Vtc^c^j))

If

we n o w put t h e s e e q u a t i o n s inco a S t e l l a m o d e l a n d run it, we lind t h a t t h e e q u i l i b r i u m p o i n t is IUIC s t a b l e . I n s t e a d , il we start a n y w h e r e away f r o m t h e e q u i l i b r i u m p o i n t we g e n e r a t e a n e c o n o m i c c y c l e t h a t is q u i t e s i m i l a r to c h a t s e e n i n t h e p r i c e - c o m m o d i t y o s c i l l a c i o n s a b o v e , e x c e p t t h a t i h e s e o s c i l l a t i o n s d o n o t d a m p e n ouc ( F i g u r e 7 1 1 ) . For any initial c o n d i t i o n s a n d any c o m b i n a t i o n of p a r a m e t e r s ( e x c e p t t h e o n e s t h a t c i a s h t h e m o d e l , t a k i n g t h e t r a j e c t o r i e s t o t h e n e g a t i v e q u a d r a n g l e s ) , che traj e c t o r y c o n t i n u e s t o c y c l e a r o u n d a n e l l i p s o i d , w i t h n o i n d i c a t i o n ol c o n v e r g e n c e

to

t h e s t a b l e s t a t e . It is noi: q u i t e c l e a r h o w t o m o d i f y t h e m o d e l i n s u c h a w a y t h a t t h e trajectories

lead co e q u i l i b r i u m

Apparently

t h e s u p p l y / d e m a n d c u r v e s t h a t w e are

c h i x i s i n g ( s e e 7 . 5 a n d 7 . 6 ) are s y m m e t r i c a l , s o w e k e e p c y c l i n g a r o u n d t h e e q u i l i b r i u m

266

Systems Scienrf? and Modeling fo' Ecologicst Economics

l Goods v. Price 200 00

100 00

C00

i

150 00

i 300 00

450 00

Goods Figure 7 . 1 1

Cycles in the C o m m o d i t y - P r i c e model. Dynamics in the phase plane IP,G).

1 Goods v. Price 200.00

y £L

100.00-

0.00

200 00

100 0 0

000

Goods Figure 7 . 1 2

Converging cycles in the modified Commodity- Price model Dynamics in the phase

plane (P,G).

w i t h o u t a p p r o a c h i n g it O n e p o s s i b l e m o d i f i c a t i o n o f t h e m o d e l t h a t s e e m s t o m a k e it c o n v e r g e is i f w e d e s c r i b e t h e c o m m o d i t y d y n a m i c s as: dG

= ..

pi 5

:

dt I n t h i s m o d e l , w i t h c ? l = O O I , c e > = 0 . 0 2 , c p i = 0 0 0 5 5 , cp2 -

0 . 0 5 , we can gen-

e r a t e a s l o w l y c o n v e r g i n g t r a j e c t o r y . N o t e t h a t it t o o k 6 , 0 0 0 i t e r a t i o n s t o g e n e r a t e t h e c u r v e s h o w n in F i g u r e 7 1 2 . B e s i d e s , t h i s c o n v e r g i n g m o d e l a p p e a r s t o b e s t r u c t u r a l l y u n s t a b l e , s i n c e e v e n s l i g h t m o d i f i c a t i o n s in t h e f o r m u l a s used o r in t h e p a r a m e t e r s e t result in n o n - c o n v e r g e n c e o r a c r a s h

Add ing Soc io-Econom ics

267

Exercise 7.4 Put t o g e t h e r t h e P r i c e - G o o d s m o d e l in Stella, or d o w n l o a d n f r o m t h e book w e b s i t e Try t o f i n d a n o t h e r f u n c t i o n or set of p a r a m e t e r s that w o u l d m a k e it c o n v e r g e faster.

Let us c o n s i d e r a n o t h e r formulation Instead

for of

the

same

looking

at

system. just

the

price and c o m m o d i t y , let us c o n sider t h r e e

variables;

price

supply ( S ) and d e m a n d (D).

ifd'uifMitij ttjiurfunu. m the

pcuraturttn Perluifx

dots n't heip, chtuttqe

there u/oa

the

wroKy

oMutKjiturtu

(P), The

supply is assumed to be s o m e w h a t identical t o the a m o u n t ol c o m m o d i t y considered earlier. T h e d e m a n d will be treated as t h e reverse ol supply. T h e S t e l l a e q u a t i o n s c a n t h e n be as follows; D e m a n d l i l = D e m a n d t t - dt) + (D_up - D _ d o w n ) * dt INIT D e m a n d = 9 0 INFLOWS: D _ u p = 1/C_d1 /Price J u s t as m the previous m o d e l , the higher the price of t h e c o m m o d i t y g e t s , t h e s l o w e r the demand grows OUTFLOWS. D _ d o w n = C _ d 2 * Price T h e higher the p p c e t h e faster the d e m a n d w i l l actually decrease. Price(t) - Priceft - dt) + (P_change) * dt INIT Price = 100 INFLOWS P._change - C_p " (Demand-Supply) It' t h e d e m a n d e x c e e d s supply, t h e n t h e c o m m o d i t y b e c o m e s s c a r c e a n d the price goes up. It goes down if more of t h e c o m m o d i t y is supplied t h a n is d e m a n d e d . Supplylt) = Supplylt - dt! + (S_up - S _ d o w n ) • dt INIT Supply -

110

INFLOWS S_up = C_s1 * Price There is m o r e incentive t o p r o d u c e a c o m m o d i t y if us p r i c e is high OUTFLOWS: S _ d o w n = 1/C_s2/Price If t h e price is high t h e c o m m o d i t y is less likely t o be c o n s u m e d C_d I = 0 0 0 8 C_d2 = 0 01 C_p = 0.01 C_s1 = 0.01 C_s2 = 0 . 0 0 8 W h e n C _ s l = C _ d 2 ; C _ s 2 = C _ d l we get dynamics, w h i c h are identical to those previous: stable oscillations for all initial conditions. However, if these c o n d i t i o n s do n o t hold t h e n the dynamics are different

W h i l e price is still displaying stable oscillations,

282 Systems Scienrf? and Modeling fo' Ecologicst Economics

1: D e m a n d v. Supply 750.00

g

400.00

OT

50.00 50.00

400.00

750.CO

Demand

|fcj

Oscillating and g r o w m g (or declining) dynamics in the (D.S) phase plane.

Supply and D e m a n d

start t o o s c i l l a t e a l o n g a n i n c r e a s i n g (it' C _ d l < C _ s 2 o r C__

si > C _ d 2 ) or decreasing (if C _ d l > C _ s 2 or C _ s l < C _ d 2 ) trajectory (Figure 7 . 1 3 ) . T h i s is a v e r y c r u d e analysis o f t h e s y s t e m ; h o w e v e r , it already s h o w s t h a t b y a d d i n g a n o t h e r v a r i a b l e t o t h e system w e h a v e m o d i f i e d t h e b e h a v i o r q u i t e s i g n i f i c a n t l y a n d g e n e r a t e d s o m e n e w p r e v i o u s l y u n a v a i l a b l e t r a j e c t o r i e s . W e c a n n o w r e p r e s e n t a situa t i o n w h e n b o t h t h e d e m a n d a n d supply c h a n g e in a s i m i l a r way, e i t h e r g r o w i n g o r decreasing. T h e price dynamics, however, r e m a i n u n c h a n g e d . W h e t h e r this corresponds t o r e a l i t y o r n o t is y e t t o b e figured o u t . W e still c a n n o t m a k e t h e s y s t e m c o n v e r g e t o a n equilibrium state. L e t us f u r t h e r m o d i f y t h e s y s t e m a s s u m i n g t h a t S u p p l y a n d D e m a n d c a n a l s o i n t e r a c t d i r e c t l y , n o t n e c e s s a r i l y o n l y by m e a n s o f P r i c e . F o r t h e o u t f l o w p a r t i n t h e d y n a m i c s o f S a n d D w e will u s e t h e s a m e a s s u m p t i o n a s a b o v e - t h a t is, t h a t t h e p r i c e P will d e f i n e t h e i r v a l u e . H o w e v e r , w e will n o w a s s u m e t h a t t h e g r o w t h o f s u p ply S is d e c i d e d d i r e c t l y f r o m r h e k n o w l e d g e r e g a r d i n g r h e d e m a n d D f o r t h e c o m modity, without t h e price d y n a m i c s being involved. Similarly, t h e growth o f d e m a n d D will b e d i r e c t l y d e t e r m i n e d by t h e s u p p l y o f t h e c o m m o d i t y , a n d w i l l b e in r e v e r s e p r o p o r t i o n t o t h i s supply. A s a r e s u l t , w e will g e t t h e f o l l o w i n g s y s t e m o f e q u a t i o n s :

for t h e m o d e l with direct effects b e t w e e n Supply a n d D e m a n d . W e c a n e i t h e r p u t t o g e t h e r t h i s m o d e l o u r s e l v e s , o r d o w n l o a d it f r o m t h e b o o k website.

HHHBHH^HHBMHHIHiHHHHMi Adding Socio-Economics 269

1 D e m a n d v Supply 60.00 -i

20 00

r

10.00

35 00

60.00

Demand

Oynamics in the (D.S) phase plane for the model w i t h direct effects b e t w e e n Supply and Demand.

B y j u s t p l a y i n g w i t h t h e S t e l l a m o d e l it w o u l d h e h a r d t o find t h e e q u i l i b r i u m in t h i s m o d e l ; h o w e v e r , s o m e s i m p l e c a l c u l a t i o n s w i t h t h e e q u a t i o n s will s h o w t h a t ii CS|CS2 = c d l c d 2 , t h e n t h e r e is a n e q u i l i b r i u m t o r a n y S ~ l / ( C d i Q ; P ) a n d D = S However,

the equilibrium

is u n s t a b l e ,

if t h e i n i t i a l c o n d i t i o n s

are displaced

e v e n s l i g h t l y , w e e m b a r k o n a s p i r a l i n g t r a j e c t o r y l i k e t h e o n e in F i g u r e 7 . 1 4 T h i s e v e n t u a l l y brings o n e o f t h e variables t o zero a n d c r a s h e s t h e model. S o m e

other

i n t e r e s t i n g r e g i m e s c a n b e o b t a i n e d by p l a y i n g w i t h t h e p a r a m e t e r s a n d i n i t i a l c o n d i t i o n s . F o r i n s t a n c e , t h e r e is a t r a j e c t o r y ( F i g u r e 7 . 1 5 ) t h a t s t a r t s o n a g r o w i n g t r e n d but t h e n f o r s o m e r e a s o n r e v e r s e s a n d b r i n g s t h e s y s t e m b a c k d o w n w a r d s

towards

a n i n e v i t a b l e c r a s h , i t is y e t t o b e figured o u t w h e t h e r t h i s k m d o t b e h a v i o r m a y b e f o u n d in a n y r e a l - l i f e e c o n o m i c s y s t e m s . M o s t likely, t h i s is q u i t e i r r e l e v a n t t o a real economy. W e still c a n n o t g e t a n y c l o s e r t o t h e t y p e o l d y n a m i c s t h a t t h e e c o n o m i c t h e o r y assumes for o u r system. W e h a v e already g e n e r a t e d several models that s e e m to c o m ply q u i t e w e l l w i t h o u r a s s u m p t i o n s a b o u t t h e s y s t e m ; t h e y h a v e p r o d u c e d a w i d e v a r i e t y o f d y n a m i c s , b u t w e s t i l l c a n n o t g e t o n t h e c o n v e r g i n g p a t h t h a t we a r e t r y ing t o m o d e l . L e t us g i v e it a n o t h e r cry a n d b u i l d yet a n o t h e r m o d e l . L e t us f u r t h e r s h o r t e n t h e i n f o r m a t i o n l i n k s a n d c o n n e c c S u p p l y a n d D e m a n d directly, with

P r i c e g e n e r a t e d o n l y as a p r o d u c t o f c h e r e l a t i o n s h i p b e t w e e n t h e

t w o . S u p p o s e t h e r e is s o m e d i r e c t i n t e r a c t i o n b e t w e e n S u p p l y a n d D e m a n d t h a t is n o t m e d i a t e d b y p r i c e . I n d e e d , w e k n o w t h a t if w e a r e o f f e r e d o n e glass o f w a t e r it m a y h a v e a v e r y h i g h ( p e r h a p s e v e n i n f i n i t e ) v a l u e f o r us a n d will b e in v e r y h i g h d e m a n d . W h e n w e g e t t h e s e c o n d glass, w e will p r o b a b l y a l s o t a k e it w i t h Afcer

che fourth,

fifth a n d s i x t h

glasses, o u r incerest

thanks.

will q u i c k l y d e c r e a s e a n d

e v e n b e c o m e n e g a t i v e . W e will n o l o n g e r w a n t a n y m o r e w a t e r ; o u r d e m a n d b e c o m e n e g a t i v e ( w e m a y e v e n w a n t t o t h r o w up t h a t w a t e r ) . T h i s is w h a t

will

econ-

o m i s t s c a l l d i m i n i s h i n g m a r g i n a l utility. P e r h a p s w e c a n a s s u m e s o m e t h i n g s i m i l a r

270

Systems Scienrf? and Modeling fo' Ecologicst Economics

l : Demand v Supply 300.00 -|

a 3

200.00 -

100 00

100 00

200 00

300 0 0

Demand

Non-equilibrium dynamics in the iD.S) phase plane for the model with direct effects between Supply and Demand. For particular combinations of parameters and mmal conditions w e may get some weird trajectories. Here S = 0= P = 120; C_d1 = 0 008. C_d2 = 0 01. C_p = 0.1, C. s l = 0 01, C_s2 - 0.01

for t h e whole market scale, a n d formulate a S t e l l a m o d e l with t h e following s e t ot equations: D e m a i d l t ) = DernancKt - d t ! + l O g r o w t h } • cli INIT D e m a n d = 120 INFLOWS D g r o w t h = 1/'C_d1/Supply-C_d2* Supply Pnce(t) = Priced - d t l + (Pgrowth) " dt INIT Price = 100

INFLOWSP g r o w t h = C _ p * (Demand-Supply) Supoiylt! = S u p p l y l t - dt) - (Sgrowth) " dt INIT Supply = 9 0 INFLOWS: S g r o w t h - C_s1 * D e m a n d * ( 1 - S u p p i y / D e m a n d ; C_di = 0 009 C_d2 = 0 . 0 2 C_p = 0.01 C_s1 = 0.01 A s you may see, we h a v e D e m a n d growing in reverse proportion t o Supply, a n d decreasing in proportion to Supply. W e also assume that Supply grows in proportion to D e m a n d as long as Supply is less t h a n D e m a n d

W h e n Supply o v e r s h o o t s and

b e c o m e s larger t h a n D e m a n d , it starts t o decrease. For P u c e , we assumed t h a t it grows if D e m a n d is larger t h a n S u p p l y and vice versa. A quick analysis of t h e model e q u a t i o n s shows that there is an equilibrium S = ljVCdiCji: D = 5 . P will also stabilize, but it is hard t o say where. R u n n i n g t h e S t e l l a

Add ing Soc io-Econom ics

271

1. Demand v. Supply 105 0 0 ^

a a 3 cr>

8 5 00

65 0 0

85.00

45 00

12500

Demand Figure

7.16

Stable focus in the Demand-Supply model.

Dynamics in the ( D - S I phase plane

1 - 5 Demand v. Price 115.00-f

I

CL

80.00

45.00'

50.00

75.00

1 100 00

Demand Figure

7.17

Dynamics in the Demand-Price phase plane.

The equilibrium for demand and supply is independent of initial conditions: hnweve-, the price equilibrium is decided by the initial conditions for price

i m p l e m e n t a t i o n , we see rhat we get a stable focus (Figure 7 . 1 6 ) ; after a n u m b e r of oscillations the trajectories equilibrate at o n e point in the ( S , D ) plane. T h e equilibrium is stable; n o matter how we modify the initial c o n d i t i o n s , we still arrive at t h e same point in the ( S , D ) p l a n e or return to the same line in t h e ( D , P ) plane (Figure 7 . 1 7 ) . T h i s is still not quite a perfect solution, since t h e equilibrium price depends upon t h e initial c o n d i t i o n s that we c h o s e for t h e price.

272

Systems Scienrf? and Modeling fo' Ecologicst Economics S u c h d y n a m i c s s h o u l d p r o b a b l y b e e x p e c t e d , s i n c e we h a v e built in rwo stabilizing f o r m u l a t i o n s in t h e m o d e l e q u a t i o n s . O n e is in t h e P r i c e e q u a t i o n , w h i c h always t e n d s t o return price t o t h e v a l u e t h a t is a c h i e v e d for S = D . T h e o t h e r is in t h e S u p p l y e q u a t i o n , w h i c h looks s o m e w h a t s i m i l a r t o t h e c a r r y i n g c a p a c i t y formalizat i o n we saw earlier. H e r e a g a i n , t h e e q u a t i o n works in s u c h a way t h a t S is always driven back to S = D . W e h a v e finally s u c c e e d e d in r e p r o d u c i n g t h e d y n a m i c s a s s u m e d in t h e system t h a t we are a n a l y z i n g . I t h a s b e e n q u i t e a l o n g process, trying n u m e r o u s d e s c r i p t i o n s , m o d e l scructures a n d p a r a m e t e r sets. W e still d o n o t h a v e a lot o f u n d e r s t a n d i n g o f t h e s y s t e m , a n d t h e r e s e e m s t o b e a lot t h a t still n e e d s t o b e c h e c k e d a n d e x p l o r e d w i t h t h e m o d e l s t h a t we h a v e built. W e may, h o w e v e r , c o n c l u d e t h a t :

•

E c o n o m i c s y s t e m s c a n b e also m o d e l e d w i t h t h e s t o c k - a n d - f l o w f o r m a l i s m used in S t e l l a . H o w e v e r , it m a y be a pretty t i r e s o m e process. M o s t o f c o n v e n t i o n a l e c o n o m i c s is c o n s t r u c t e d a r o u n d t h e a s s u m p t i o n o f e q u i l i b r i u m . T h e e c o n o m i c system is t h o u g h t t o b e a t e q u i l i b r i u m , a n d w h a t e v e r h a p p e n s t o it is " a t t h e m a r g i n , " t h a t is, w e c o n s i d e r s m a l l p e r t u r b a t i o n s from t h e e q u i l i b r i u m . I n c o n t r a s t ,

most

d y n a m i c m o d e l s c o n s i d e r transfer p r o c e s s e s t h a t analyze h o w t o r e a c h e q u i l i b r i u m , or h o w t o j u m p f r o m o n e e q u i l i b r i u m s t a t e t o a n o t h e r . • T h e s y s t e m s d y n a m i c s language is n o t very well suited for c o n v e n t i o n a l e c o n o m i c analysis. T h e language o f e c o n o m i c s m a y he s o m e w h a t difficult t o t r a n s l a t e i n t o t h e s t o c k - a n d - f l o w f o r m a l i s m , e s p e c i a l l y w h e n w e are d e a l i n g w i t h q u a l i t a t i v e t h e o r e t i c a l s y s t e m s w i t h o u t a n y p a r t i c u l a r d a t a sets at h a n d . H o w e v e r , t h i s is p r o b a b l y t h e case w h e n m o d e l i n g a n y q u a l i t a t i v e systems, n o t o n l y e c o n o m i c o n e s . •

A careful analysis o f m o d e l d y n a m i c s may s h e d s o m e light o n t h e system o p e r a t i o n and its p e c u l i a r i t i e s . F o r e x a m p l e , o u r analysis s h o w e d e v i d e n c e o f p r i c e by itself n o t b e i n g a b l e t o bring t h e p r o d u c t i o n system t o e q u i l i b r i u m . W e n e e d e d s o m e a d d i t i o n a l stabilizing m e c h a n i s m s t o b e i n c l u d e d .

•

It is i m p o r t a n t to c o n s i d e r a variety o f structures, p a r a m e t e r s a n d initial c o n d i t i o n s to u n d e r s t a n d t h e system d y n a m i c s b e h a v i o r . P e r f o r m i n g j u s t a few m o d e l runs is i n s u f f i c i e n t t o u n d e r s t a n d h o w t h e system works.

7.3

Corporate rule Let us c o n s i d e r a n o t h e r e c o n o m i c system with s o m e flavor o f social policy in it. S u p p o s e we are l o o k i n g a t t h e d y n a m i c s o f large c o r p o r a t i o n s vs s m a l l businesses. T h e s e will be che t w o m a j o r players ( v a r i a b l e s ) in o u r system. T h e m a i n d i f f e r e n c e in h o w they o p e r a t e is t h a t t h e r e is hardly any c o m p e t i t i o n b e t w e e n t h e c o r p o r a t i o n s , w h i c h m a n a g e t o divide their spheres o f interests w i t h o u t e m p l o y i n g m a r k e t forces. T h e small businesses c o m p e t e with e a c h o t h e r a n d with t h e c o r p o r a t i o n s . T h e y also try to limit t h e g r o w t h o f c o r p o r a t i o n s by legislative m e a n s , w h i c h is also a n o n - m a r k e t m e c h a n i s m . H o w e v e r , c o r p o r a t i o n s also c o m p e t e w i t h t h e small businesses for influe n c e upon t h e legislators. L e t us see h o w s u c h a system c a n d e v e l o p in d y n a m i c terms. T h e variables o f our system are t h e c o r p o r a t i o n s ( w e refer t o them, as Bigs, B ) a n d t h e small businesses ( S m a l l s , S ) . W e suppose t h a t B a n d S are measured in t h e i r t o t a l v a l u e (say, in b i l l i o n s o f dollars). B o t h B a n d S a r e assumed t o grow e x p o n e n t i a l l y , so t h a t t h e larger t h e i r s u e t h e m o r e t h e i r a b s o l u t e g r o w t h will b e . T h e S m a l l s a r e c o n t r o l l e d by s e l f - c o m p e t i t i o n . W e t h i n k t h a t t h e i r total g r o w t h in value is m o s t l y

Add ing Soc io-Econom ics

273

because o f che growth in their numbers. T h e r e f o r e , the larger the number, the h i g h e r the c o m p e t i t i o n will be. Besides, the S m a l l s are suppressed by the Bigs: the larger the size o f the Bigs, t h e more they limit t h e S m a l l s

— = bS - dB dt

cS2

where b is che g r o w t h rate, c is t h e s e l f - l i m i t a t i o n c o e f f i c i e n t and d is the rate o f c o m p e t i t i o n with che Bigs. T h e e q u a t i o n for t h e Bigs will be:

i ® = a B dt

1-

B

MM

H e r e , a is t h e Bigs growth rate and M M is a c e r t a i n carrying capacity, some m a x imal limit set for the t o t a l size o f the Bigs. In such a system with unfair c o m p e t i tion the only result is gradual e l i m i n a t i o n o f the S m a l l s while t h e Bigs reach their carrying capacicy. I f t h e carrying c a p a c i t y M M is set at a high level, t h e S m a l l s are entirely wiped out. If M M is small, t h e n c o e x i s t e n c e is possible. T h i s leads to a possible way t o c o n t r o l t h e Bigs in a d e m o c r a t i c society. T h e M M should be set at such a level that allows the S m a l l s co exisc and develop. T h i s should be d o n e outside of che e c o n o m i c syscem, by a specific political process. T h e allowed size o f the Bigs' develo p m e n t t h e n d e t e r m i n e s the size of b o t h the Bigs' and the S m a l l s ' d e v e l o p m e n t .

N o t e that the so-called s e l f - c o m p e t i t i o n , in m a t h e m a t i c a l l e r m s . is actually identical t o carrying capacity. W e can r e w r i t e t h e e q u a t i o n for Smalls as: — dt

= bS-

dB -cS2

=

bS

1 -

—

-

dB

b/c

Here, w e have carrying capacity equal t o b/c. Similarly, rearranging the e q u a t i o n for the Bigs, w e can have: ^ - - s B dt

(MM

So t;ne w h o l e a s y m m e t r y of t h e s y s t e m is in t h e fact that the Smalls are i m p a c t e d by t h e Bigs {the -dBietm),

w h i l e the Bigs feel no pressure f r o m the Smalls.

T h i s may be just a b o u t che right cime co puc chese e q u a t i o n s i n t o S t e l l a and start e x p e r i m e n t i n g wich the model. Jusc co m a k e sure thac we are on the same page, let us c o m p a r e our S t e l l a e q u a t i o n s : Bigs(t) = Bigs(t - dtj

(B_in - B_out} * dt

INIT B i g s = 1 0 0 INFLOWS' B J n = a*Bigs OUTFLOWS: B„out = a*Bigs*Bigs*m Smalls(t) - Smalls(t - dt) + ( S J n - S_out) * dt INIT Smalls = 3 0 0

288 Systems Scienrf? and Modeling fo' Ecologicst Economics INFLOWS: S J n = b " Smalls OUTFLOWS: S_out = d ' B i g s + c ^ S m a l l s ' S i m a l l s a = 0.2 b = 02 c = 00003 d - 0 001 m = 1/MM M M = 30000 T h e d y n a m i c s b e c o m e m u c h easiei i o understand if we c h e c k out t h e equilibria. R e s o l v i n g t h e e q u i l i b r i u m e q u a t i o n s , we get B = MM b Z y j 4 tic M M 2c T h e r e are t w o points, and o n e o f t h e m s e e m s t o be stable. T h e r e could be a c o e x i s t e n c e of t h e two, but n o t e that t h e equilibrium lor r h e S m a l l s c a n exist o n l y if t h e expression under t h e square root is n o n - n e g a t i v e :

MM <

b— 4dc

W e c a n see t h a t for r h e S m a l l s to exist, they h a v e t o m a k e sure t h a t M M is sufficiently small (Figure 7 . I 8 ) . T h e d e c i s i o n s about such e x t e r n a l c o n t r o l s a r e made in a political process, w h i c h may be assumed to be d e m o c r a t i c . In this case, s i n c e t h e n u m b e r o f S m a l l s is always larger t h a n t h e n u m b e r of Bigs, we m a y hope that t h e c o n t r o l over M M will be successful. H o w e v e r , in reality t h e " d e m o c r a t i c " process is largely influenced by lobbying, w h i c h in turn is defined by t h e a m o u n t ol m o n e y s spent t o i n f l u e n c e t h e p o l i t i c i a n s . L e t us add t h e lobbying process i n t o t h e m o d e l

2. Smalis

1: Bigs 1: 40000.00-1 2 700.00

1 20000.00.

2-

350.00

0 00

125.00

250.00 Time

Crash of Smalls if carrying capacity established lor Bigs is not small enough

Add ing Soc io-Econom ics

275

Suppose t h a t borh t h e Bigs and t h e S m a l l s spend a c e r t a i n portion of t h e i r w e a l t h o n lobbying (e a n d / , respectively). T h e sire of M M will be t h e n d e t e r m i n e d by w h o spends more. T h e e q u a t i o n for S m a l l s will now be:

- = bS dc

d

fS - dB -

cS2

For Bigs, we h a v e : dB dc

aB

B

fS

M M i'D

-

eD

Here, we have added t h e loss of wealth for lobbying (fS and cB) a n d modified the carrying capacity, assuming thac it is now a f u n c t i o n o f D/S - it grows w h e n cD > fS and declines otherwise. T h e model dynamics seem to be more c o m p l e x now. By simply running t h e Stella model, it may be hard to figure out what is going o n . If we have not put together the model ourselves, we c a n download it from t h e brxik website. Playing with t h e model, we find thar there does not seem to be a state of c o e x i s t e n c e any longer, if t h e S m a l l s prevail, in most cases by increasing their spending o n lobbying, che Digs c a n turn around t h e dynamics and wipe nut the S m a l l s encirely, as m Figure 7 . 1 9

A small c o m p a n y called TerraCycle has s t a u e d t o p r o d u c e fertilizers f r o m w o r m

droppings

Organic w a s t e is fed to w o r m s a n d t h e w o r m p o o p c o m p o s t tea is bott>'ed as ready-to-use plant fertilizer, using soda bottles c o l l e c t e d by schools a n d o t h e r chanties. S t a r t e d hy c o l l e g e s t u d e n t s , after five years in business TerraCycle w a s e x p e c t i n g to reach $6 million in sales in 2007. finally m a k i n g s o m e profit. This did n o t look g o o d t o the $ 2 . 2 b'llion giant Scotts Miracle-Gro C o m p a n y w h i c h has 59 p e r c e n t of the plant f o o d m a r k e t S c o n s c l a i m s that t h e t w o c o m p a n i e s ' p r o d u c t s look similar and w i l l c o n f u s e c u s t o m e r s , b e c a u s e s o m e TerraCycle plant f o o d s have a green-and-yellow label w i t h a circle a n d a picture of f l o w e r s a n d v e g e t a b l e s on it. Scotts also o b j e c t s that TerraCycle says its plant f o o d is as g o o d or better than " a leading s y n t h e t i c plant f o o d " Clearly, t h e e x p e c t a t i o n is that a small c o m p a n y w i l ! not be able to s u r v i v e a maior lawsuit arid w i I g o o u t of business. The Bigs c o m p e t e w i t n t h e Smalls

1 Bigs

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S v s t e r - s S c i e n c e and

Ecologies' E c o ^ o ^ i c s

cool-to-liquid producton plants nuarantee n w w r m m p n e * lor big gouemmenl outchases to- I r *

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ike rne amines Peebody Energy t^e world's

coei company. urged m a reconf

advertising c a m p s g n tKsi p e o o * " r n a j ^ a world w^e»e our country runs o n energy from Middle America Instead o> The h ' o c B e E a s t " C o e M n d u s r y lobbying h « reached fever pitch The i r o ^ f t r y i p ® r t S6 rrwlion on

c t K y r g * 5005 and 3006 - three times what it

spent e e l * year from 2O00 through 2004. according to catenations b r Politicaimoneyline com

Peabody, w*«ch h a s t j u ^ o t r t e O • » a n r * * Sobbyng budget to about $ ? m J i o n since

20 W . recently nned

A Geonerr. tie M o w /

Democrat w * w waa H o u s e majority

loader from 1969 to 1995 and a o a - x k a t eto*the Democratic orcs-dennal n o m m a t o n In 1988 and 2004 to help

its c a s e n C o n g r e s s

D o vou think it w j i w *>u bet B a l e d on E d m u n d L A n d r e w s

Push lor B g S u b s i d e s 'or Coal Prccess."

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New >brt Times. M a y 29. 2007

7 A

Sustainability Tlinr

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279

c a t e g o r y - 3 r d t o orovtdo n a monrtor that m a * *

tam$ t v - e s s e r ' a l r e v j u » C » i to. p r e s e n t a n d future g e n e r a t i o n s " "This already implies s c a l e * other m a n g o O e i C o & r z a 1 '957 -v

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t h e s > s l e m s a f r ' . t y t o maintain its itniCture (organisation) a n d function Iwigori over time

n t r e ' s e n o t e x t e r n a t i r e s * i r « i l e n c e ) " S c l o w f t 9 9 1 ) says that t n e s y s t e m is susta-mabin as long <1-- m e tor,, c a p t a l of the s y s t e m 13 e q u a l gr greater m e v e r y next generation

Coe'arm

3 " d Da(y 11992) argue thai t u t t m o b i l i t y o n l y o c c u r s w h e n tt-ere is no d o c - r e n nature' c a p i a t Whatove-t tne flavor of r e different definitions, t h e r e is o n e c o m m o n o o m o o n e n t . at of t h e m W * aoout m e - i e n e n c e

s u s t e n a n c e , continuity of a certain resource, s y s t e m , condition a r c

reltttorahip. and in all c a s e s there is the goal of keeping s o m e t h i n g at a certar\ level o t a w o d r g d e c . r e This is a l t o h o w Gooyln's definition too" d e f i n e s sustainability: a s a s t a t e or p o c e s s l^at c a n be nvuntained Indoti^triy. to Veep in e x i s t e n c e , to m * r t a m or prolong, t o u s e ' n o w c t i

r

a manner that satisfies current n e e o s while a l t o w r g t h e m to persist in tNe long t e r m No wonder tustainabllity h a s o o o o m e a w e l c o m e c o n c e p t m t h e W e & i e m

oevotocec

world People a t e quite f-appv with what t h e y ' v e g o t end it =s a s w e e t d e e t o

Out h o w

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Clearly euch growth i t a n t * » u * t » < n a t t » g o a l b e a r i n g m m r o t h e w n i e d r e s o u r c e s

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294 Systems Scienrf? and Modeling fo' Ecologicst Economics C _ c a p _ g r is che rate c o e f f i c i e n t , w h i c h tells Wow m u c h i n v e s t m e n t c a p i t a l t h e e c o n o m y c a n g e n e r a t e . T h e c a p i t a l will he spenc for t w o purposes. O n e is t o fund f u r t h e r e c o n o m i c g r o w t h , C _ d e v ; t h e o t h e r is to restore t h e resources, C _ e n v . T h e e q u a t i o n to spend c a p i t a l will b e : ~ L

out =

C _ c a p sp^Capital — DT

C _ c a p _ s p is t h e p r o p o r t i o n o f c a p i t a l s p e n t . If C _ c a p _ s p = 1, all c a p i t a l is r e i n v e s t e d . N o t e t h e d i v i s i o n by D T t h a t m a k e s sure t h a t this is i n d e p e n d e n t o n t h e t i m e step. C._out is d i v i d e d b e t w e e n C _ d e v a n d C _ e n v a c c o r d i n g t o t h e c o n t r o l p a r a m e t e r F_dev, 0 < F _ d e v < 1. F_env = 1 -

F _ d e v i n d i c a t e s t h a t w h a t e v e r is left from t h e e c o n o m i c

invest-

m e n t s is s p e n t o n e n v i r o n m e n t a l r e s t o r a t i o n . T h e a m o u n t o f r e s o u r c e s will

then

grow in p r o p o r t i o n t o t h e r e s t o r a t i o n efforts: R _ i n - C _ r e s t o r * C _ e n v + C _ s e l f * R e s o u r c e s , w h e r e C _ r e s t o r is t h e e f f i c i e n c y o f r e s t o r a t i o n a n d C _ s e l f * R e s o u r c e s is t h e process o f s e l f - r e h a b i l i t a t i o n , s e l f - r e s t o r a t i o n . T h e resources are used for e c o n o m i c g r o w t h at a rate o f R_out = C_env_des*D_growth. O f c o u r s e this is a very s i m p l i s t i c m o d e l , very m u c h a l o n g che lines o f n e o c l a s s i c a l e c o n o m i c theory. W e a s s u m e t h a t i c s o u r c e s c a n be always r e g e n e r a t e d o r s u b s t i t u t e d . H o w e v e r , for t h e m o m e n t l e t us assume t h a t this is indeed possible a n d see w h a t b e h a v i o r s u c h a s y s t e m c a n display. W e h a v e already c o n s i d e r e d s o m e p r e l i m i n a r y d y n a m i c s in thus m o d e l , w h e n discussing che d i f f e r e n t i n t e g r a t i o n m e t h o d s ( C h a p t e r 3 ) . W e h a v e o b s e r v e d d y n a m i c s t h a t do n o t s e e m very s u s t a i n a b l e : a f t e r a n i n i t i a l rise in d e v e l o p m e n t , t h e resource base is q u i c k l y d e p l e t e d a n d t h e e c o n o m y c r a s h e s . T h e p o p u l a t i o n c o n t i n u e s to grow, w h i c h is o b v i o u s l y u n r e a l i s t i c a n d begs for s o m e i m p r o v e m e n t . T h e r e are m a n y o b v i o u s a d d i t i o n s t h a t c a n a n d s h o u l d be m a d e to t h e m o d e l , but b e f o r e we go i n t o a n y f u r t h e r d e t a i l s lee us analyze che m o d e l chac we h a v e already put t o g e t h e r . First, let us play with s o m e o f t h e p a r a m e t e r s , W e a s s u m e t h a t the m o d e l h a s b e e n put t o g e t h e r in S t e l l a or a n o t h e r m o d e l i n g p a c k a g e , or d o w n loaded from t h e b o o k webs i re. Firsc, if t h e resources crash, h o w c a n we sustain t h e m ? In che m o d e l we h a v e t h e p a r a m e t e r F_dev, w h i c h defines w h a t f r a c t i o n o f t h e c a p i t a l is s p e n t for d e v e l o p m e n t . W h a t is left is s p e n t o n r e s t o r a t i o n . W e h a d F _ d e v = 0 . 9 . If we decrease t h e c o e f f i c i e n t to F _ d e v = 0 . 6 we will get a perfect g r o w t h p a t t e r n , w h e r e d e v e l o p m e n t is g e n e r a t i n g e n o u g h r e v e n u e co provide for resources recovery - a world vision of a t e c h n o l o g i c optimist (Figure 7-20). H o w e v e r , n o t e t h a t t h e growth t r a j e c t o r i e s are in p l a c e b e c a u s e of a very h i g h e f f i c i e n c y o f o u r r e s t o r a t i o n p r o c e d u r e s ( C _ r e s t o r = 0 . 5 ) . If we d e c r e a s e it t o . say, 0 - 1 , we will be b a c k to t h e r i s e - a n d - c r a s h s c e n a r i o . T h e r e s e e m t o b e o n l y t w o ways t h e syscem c a n possibly d e v e l o p : o n e is runaway g r o w t h , w h e r e ail che e l e m e n t s grow to inhnicy; che o t h e r is rise a n d c r a s h , w h e r e a f t e r a period o f initial fast d e v e l o p m e n t the system v a r i a b l e s d e c l i n e t o zero. Since

there

is n o f e e d b a c k a t this

t i m e from

t h e o t h e r system variables t o

P o p u l a t i o n , let us single it o u t a n d see h o w t h e system b e h a v e s if p o p u l a t i o n is assumed c o n s t a n t a n d ac equilibrium. N o w we find chac che infinite growch b e h a v i o r b e c o m e s

Add ing Soc io-Econom ics

l : Developmenl 2.00e+09-i

2: Capital

i.OC

188 25

281

J Population

3 Resources

375.50

562.75

750 00

A t e c h n o l o g i c optimist w o r l d view

v e r y r o b u s t . E v e n if t h e r e is n o r e s t o r a t i o n a v a i l a b l e , C _ r e s t o r = 0 . , a n d a l m o s t e v e r y t h i n g is r e i n v e s t e d in d e v e l o p m e n t , F _ d e v = 0 . 8 , we still h a v e t h e s y s t e m a l o n g t h e g r o w t h c u r v e . It d o e s c r a s h w h e n

F_dev = 0 9. O n e curious

a l r e a d y e m e r g e s f r o m t h i s : A p p a r e n t l y t h e h u m a n c o m p o n e n t is e x t r e m e l y

evolving

conclusion important

w h e n a n a l y z i n g s u s t a i n a b i l i t y . W i t h a s m a l l a n d fixed n u m b e r of p e o p l e , t h e d e v e l o p m e n t g r o w t h is c o n t r o l l e d o n l y by C a p i t a l a n d R e s o u r c e s . T h i s a l l o w s d e v e l o p m e n t t o grow gradually, b a s e d o n t h e s e l f - r e c o v e r y of r e s o u r c e s . S u s t a i n a b i l i t y is p o s s i b l e w h e n d e v e l o p m e n t is based o n l y o n t h e e x i s t i n g r e s o u r c e base. It is really t h e g r o w i n g h u m a n i n f l u e n c e in p r o d u c t i o n t h a t d e s t a b i l i z e s t h e s y s t e m . If t h e P o p u l a t i o n is s o i m p o r t a n t , l e t us b r i n g it b a c k i n t o c o n s i d e r a t i o n a n d a l s o l o o k at s o m e o f t h e o b v i o u s f e e d b a c k s t h a t t h e rest o f t h e s y s t e m s h o u l d h a v e w i t h r e s p e c t t o t h e p o p u l a t i o n . O n e t h i n g t h a t s e e m e d q u i t e s t r a n g e in t h e o r i g i n a l m o d e l was t h a t P o p u l a t i o n c o n t i n u e d t o g r o w ad in/initum e v e n w h e n all t h e r e s o u r c e s w e r e g o n e a n d t h e e c o n o m i c s y s t e m had c r a s h e d . A c t u a l l y of Population

it was t h i s i n f i n i t e

t h a t b r o k e up t h e m o d e l , b o t h for t h e E u l e r a n d t h e

growth

Runge-Kutta

methods. L e t us a s s u m e t h a t w h e n r e s o u r c e s a r e d e p l e t e d , m o r t a l i t y i n c r e a s e s ( t h i s c a n b e d u e t o , say, a d e c r e a s e in a i r a n d w a t e r q u a l i t y ) :

C_mortality = C_mortality +

I -1- C _ m o r _ e n v * R e s o u r c e s

C _ m o r _ e n v is t h e r a t e of e n v i r o n m e n t a l e f f e c t s o n m o r t a l i t y . N o t e we w r o t e I + mor_env*Resources

to m a k e sure that

w e d o n o t g e t a d i v i s i o n by zero i n

C_ case

R e s o u r c e s b e c o m e v e r y s m a l l . It R e s o u r c e s a t e p l e n t i f u l , t h i s e q u a t i o n r e t u r n s a v a l u e almost equal to the original mortality coefficient

H o w e v e r , as R e s o u r c e s d e c r e a s e ,

m o r t a l i t y r a t e s t a r t s t o grow. A s a result w e get t h e o s c i l l a t i n g " g r o w - a n d - c r a s h " t y p e o f d y n a m i c s , w h e r e a l l t h e e l e m e n t s of* t h e s y s t e m i n i t i a l l y d i s p l a y rapid g r o w t h , f o l l o w e d b y a n e q u a l l y rapid d e c l i n e as r e s o u r c e s b e c o m e s c a r c e ( F i g u r e 7 . 2 1 ) It w e f u r t h e r i n c r e a s e t h e d e v e l o p m e n t b y a l l o c a t i n g m o r e c a p i t a l t o e c o n o m i c growth, the pattern becomes somewhat chaotic, with sudden outbursts of developm e n t f o l l o w e d by e v e n steeper d e c l i n e s (Figure 7 - 2 2 ) .

Systems

282

Scienrf?

1: Development 2000000.00 20000 00 200000.00 30000.00

and

Modeling

fo'

Ecologicst

4. Population

3. Resources

2. Capital

Economics

1: 1000000.00 2 10000.00 3 100000.00 4. 15000.00

Figure

1: 2. 3: 4.

0.00 0.00 0.00 0.00

7.21

The grow and crash pattern of dynamics, F_dev = 0.6.

h3— 1.00

1: Development 2000000.00 9000.00 200000.00 40000 00

4500.25

3000.50

1500.75

2: Capital

6000.00

4 Pooulation

3: Resources

1: 1000000.00 2 4500.00 3: 100000.00 4: 20000.00

1.00 Figure

7.22

1500.75

3000.50

4500.25

l 6000.00

rhe grow and crash partem of dynamics, F dev = 0.9. In any c a s e , this is definitely n o t t h e type of d y n a m i c s we would call sustainable.

Let us try to i n t r o d u c e some s e l f - l i m i t a t i o n s i n t o t h e system t h a t could potentially d a m p e n t h e o s c i l l a t i o n s . W e will m a k e t h e d e c i s i o n about the i n v e s t m e n t s based o n t h e c u r r e n t availability o f Resources. If R e s o u r c e s are plentiful, F _ d c v is u n c h a n g e d . W h e n R e s o u r c e s d e c l i n e , F _ d e v decreases, so that F _ e n v = l - F _ d e v c a n

increase

and more will be reinvested in r e s t o r a t i o n . I lie s-shaped f u n c t i o n , discussed a m o n g o t h e r f u n c t i o n s , s e e m s to be a perfect c h o i c e to provide this type of b e h a v i o r :

F dev

;

C

Resources2

h a l f 2 + Resources^

where C _ h a l f - is tin: hall saturation parameter, w h i c h in this case is the a m o u n t of resources at w h i c h F . d e v is t o be half of t h e original. T h i s is s o m e sort o f an adaptive m a n a g e m e n t t h a t is e m b e d d e d i n t o t b e system. W e are trying to m a k e t h e system react to t h e c h a n g i n g c o n d i t i o n s and adapt accordingly. A s a result, we get a

MM Adding Socio-Economics

). Development

1: 2

2 Capital

3: Resources

4. Population

2: C mortality

1: F dev 0 60 0 04

1.00

Figure 7.24

283

1250.75

2500 50

3750 25

5000 0 0

Changes in morta' ty and investments that stabilize the system

b e h a v i o r of t h e s y s t e m t h a t m a y h e c a l l e d s u s t a i n a b l e . A f t e r a n i n i t i a l p e a k m e c o nomic

development,

the system

returns

to a non-zero condition

which

T h e r e s o u r c e s a r e n o t d e p l e t e d , t h e p o p u l a t i o n is n o t t o o l a r g e , a n d t h e

persists. economic

d e v e l o p m e n t is s u c h t h a t it s u s t a i n s t h e p o p u l a t i o n a n d t h e r e g e n e r a t i o n o f r e > o u r c e s (Figure 7 . 2 3 ) . T h e a d a p t a t i o n is p r o v i d e d by c h a n g e s in t h e m o r t a l i t y race a n d in t h e i n v e s t ment

might

fol-

low, w h e r e a d a p t a t i o n s a n d a d j u s t m e n t s a r c m a d e o n l y w h e n it is t o o l a t e a n d

strategy, as s h o w n

in

Figure

7.24.

Is t h i s t h e s c e n a r i o

humans

che

R e s o u r c e s h a v e d e c l i n e d t o a r e l a t i v e l y low l e v e l ' W e c o u l d c l a i m t h a t w e h a v e b u i l t a m o d e l of a s u s t a i n a b l e s y s t e m , if it w e r e n o t (or t h e tact t h a t t h e m o d e l t u r n s out t o be s t r u c t u r a l l y q u i t e u n s t a b l e . If we s t a r t f r o m a d i f f e r e n t i n i t i a l i n v e s t m e n t s t r a t e g y a n d m a k e F _ d c v = 0 . 8 , we put t h e s y s t e m i n t o s t a b l e o s c i l l a t i o n s , as d i s p l a y e d in F i g u r e 7 . 2 5 . If w e f u r t h e r i n c r e a s e t h e i n i t i a l i n v e s t m e n t i n t o d e v e l o p m e n t , t h e s y s t e m o s c i l l a t i o n s b e c o m e c h a o t i c . Figure 7 . 2 6 p r e s e n t s t h e c y c l e s in t h e p h a s e p l a n e for R e s o u r c e s

284

1: Development 5 00e+07 500000.00 3000000.00 100000 00

3

2: Capital

Resources

4: Popuiahon

2 50e+07 250000 00 1500000 00 50000.00

1 2 3 4

l 00

1250.75

2500.50

1- F d e v

3750 25

5000 00

3750.25

5000 00

2 C mortality

0 80 n

1:

2:

1: 2

1 00

1250.75

2500 50 Years

Figure 7.25

System d y n a m i c s and adaptations w i t h F_dev = 0.8.

a n d P o p u l a t i o n . D e v e l o p m e n t a n d C a p i t a l display s i m i l a r c h a o t i c o s c i l l a t i o n s . It is n o t q u i t e c l e a r w h a t t h e f u t u r e o f t h i s system will b e . II i n s t e a d w e d e c r e a s e F _ d c v a n d m a k e it e q u a l r o 0 S, we g e t yet a n o t h e r e n n r e l v n e w b e h a v i o r : s t e a d y g r o w t h ol t h e e c o n o m i c s u b s y s t e m w i t h a vers' l o w r e s o u r c e b a s e ( F i g j r e 7 . 2 7 ) . A p p a r e n t l y all t h e r e s o u r c e s a r e very e f f i c i e n t l y b e i n g used t o r e c o n o m i c

development, with the population

entirely

c a r e l e s s a b o u t t h e s t a t e o f t h e e n v i r o n m e n t as l o n g as d e v e l o p m e n t is e n s u r e d . If F _ d e v

is f u r t h e r d e c r e a s e d ,

the growth

b e c o m e s so r a p i d t h a t

the

system

q u i c k l y falls i n t o d i s c o n t i n u o u s j u m p s a n d falls, c l e a r l y i n d i c a t i n g t h e i n s u f f i c i e n c y o f t h e n u m e r i c a l a c c u r a c y of t h e c o m p u t e r c a l c u l a t i o n s . In reality, it is s i m p l y b e c a u s e n u m b e r s b e c o m e t o o l a r g e for t h e c o m p u t e r t o h a n d l e p r o j i e r l y . T h i s n u m e r i c a l i n s u f f i c i e n c y d e s e r v e s s o m e f u r t h e r c o n s i d e r a t i o n . In F i g u r e 7 2 8 we p r e s e n t t h e m o d e l t r a j e c t o r i e s a c h i e v e d w h e n , i n s t e a d o f t h e q u a d r a t i c s w i t c h i n g f u n c t i o n for F _ d c v , we use a f u n c t i o n o f t h e M i c h a e l i s - M e n t e n

F dev

=

0.7 * Resources C_halt + Resources

type:

1 Population v Resources 8000000.00

4000000.00 HI

4000000.00

b

000-1 O.OO

150000 00

300000.00

O.OOH 0.00

150000 00

i

—i 300000.00

1 Population v. Resources

1 Population v Resources

0.00

160000.00

300000 00

0 00

1: Population v. Resources

150000 00

300000 00

1 Population v. Resources

BOOOOOO.OO

8000000 00 n

4000000 CO

0 00 O.OO

I 150000 OC

:

4000000.00

I 300000 00

0 00 0 00

150C00 00

Chaotic c y c l e s of Population and Resources w h e n F dev = 0 9

300000.00

286

Systems Scienrf? and Modeling fo' Ecologicst Economics

1: Development 3.00e+09 3.00e+07

2 Capital

3. Resources

4: Population

3 00e+09

1 . 2i 3J

1 50e+09 1 50e->-07.|

4

1 50e-09

1 ;

/

1

1

0 00 0.00

II

0.00

4:

•Tr"

1 0C

r

3

1

1250.75

J

2500.50

3750 25

5000.00

Years Changes n mortality and investments that stabilize the system.

1 2 3 4

1: Development 7.00e-<-09 7 00e+07 6000000 00 5 00e+10

1 2: 3 4

3 50e+09 3.50e+07. 3000000 00 2 5Ce+10

2 Capital

3 Resouices

500.75

1000.50

4- Population

1500.25

2000 00

Years Figure 7 . 2 8

Model crashes caused by computation error. O n o n e h a n d , il we look at t h e averages, we get s o m e sort o f system persistence

P o p u l a t i o n keeps growing in spite o f sharp drops every n o w a n d t h e n , t h e e c o n o m y a l s o grows. R e s o u r c e s are restored after b e i n g depleted. S o m e might call this sustainability. In theory, if t h e c o m p u t a t i o n step was made infinitesimally small t h e n t h e s e c r a s h e s could be removed

However, in real life a d a p t a t i o n s are not m a d e i n s t a n t a -

neously: there is always a time lag b e t w e e n t h e c a u s e a n d t h e effect, a n d it always t a k e s t i m e to m a k e d e c i s i o n s . T h e r e f o r e , it may he argued that t h e real-life system is a l s o discrete, wirh a c e r t a i n t i m e - s t e p , a n d thus such " c r a s h i n g " systems are probably i n e v i t a b l e w h e n growth b e c o m e s t o o iast to track a n d t o master. S o m e o f t h e c o n c l u s i o n s from this study are as follows: •

T h e b e h a v i o r o f a n e c o l o g i c a l - e c o n o m i c system is q u i t e c o m p l e x and hard to c o n t r o l . W e may c r e a t e s o m e b e h a v i o r w h i c h m i g h t r e s e m b l e sustainable develo p m e n t : however, it s e e m s to he very m u c h d e p e n d e n t u p o n r h e particular parameterization of t h e model.

Add ing Soc io-Econom ics

•

287

Ir is i m p o r t a n t ro rest r h e m o d e l with a variety of p a r a m e t e r s and f o r m a l i z a t i o n s ro m a k e sure t h a t we h a v e really c a p t u r e d t h e e s s e n c e of t h e system d y n a m i c s . It is w r o n g to j u m p to c o n c l u s i o n s a b o u t s y s t e m b e h a v i o r based u p o n o n l y o n e m o d e l lealizanon.

• T h e n e o - c l a s s i c a l paradigm q u i t e o f t e n results in s y s t e m b e h a v i o r t h a t is focused o n e c o n o m i c d e v e l o p m e n t , w h e r e e c o l o g i c a l resources are used o n l y to provide for f u r t h e r e c o n o m i c g r o w t h . T h i s may be well in c o n f l i c r with o t h e r h u m a n priorities, s u c h as e n v i r o n m e n t a l q u a l i t y and h u m a n h e a l t h .

5

The

end

of cheap oil

And we ought not at least to delay dispersing a set of plausible fallacies about the economy of fuel, and the discovery of substitutes for coal, which at present obscure the critical nature of the question, and are eagerly passed about among those who like to believe that we have an indefinite period of prosperity before us. W . S . J e v o n s , 1865

W a t e r and e n e r g y are t h e two i c n e w a b l e resources t h a t are e s s e n t i a l lor h u m a n livel i h o o d , W h e r e a s we h a v e b e e n mostly c o n c e r n e d wirh n o n - r e n e w a b l e resources as t h e h u m a n p o p u l a t i o n grows in size and in t e r m s of t h e i m p a c t t h a t it has o n t h e b i o s p h e r e , r e n e w a b l e resources b e c o m e e q u a l l y i m p o r t a n t . R e n e w a b l e resources may b e c o m e l i m i t i n g if t h e rate o f t h e i r r e n e w a l is not fast e n o u g h . R e n e w a l of w a t e r is d e p e n d e n t o n energy. P r o d u c t i o n o f energy, e s p e c i a l l y of r e n e w a b l e e n e r g y ( b i o f u e l and h y d r o ) , is d e p e n d e n t o n water. In b o t h cases, for energy and water, we c o m p e n sate the lack o f flow by digging i n t o t h e s t o c k s . T h e fossil fuels are t h e n o n - r e n e w a b l e reserves t h a t we are q u i c k l y d e p l e t i n g . It is a c t u a l l y t h e s t o c k s t h a t h a v e a l l o w e d h u m a n s to d e v e l o p i n t o a g e o l o g i c a l force ( V e m a d s k i i , 1 9 8 6 ) w h i c h may very well b r i n g itself to e x t i n c t i o n , unless we find a l t e r n a t i v e d e v e l o p m e n t goals and paradigms. A s w i t h energy, we are c o m p e n s a t i n g for a lack o f w a t e r by e x t r a c t i n g from fossil g r o u n d w a t e r reserves. In b o t h cases this is a n u n s u s t a i n a b l e p r a c t i c e t h a i leaves future g e n e r a t i o n s dry, w i t h n o safety n e t to rely u p o n . W e looked at water m s o m e detail in C h a p t e r 6, Let ui now focus o n energy T h e r e has b e e n m u c h discussions lately about the so-called " p e a k o i l . " B a c k in the 1 9 5 0 s , a U S G S geologist, K i n g H u b b e r t , was o b s e r v i n g t h e d y n a m i c s o f o u t p u t from individual oil wells and n o t i c e d that they s e e m e d to follow a pretty similar pattern, A r first t h e i r productivity was low, t h e n ir gradually grew, until it peaked and t h e n followed a p a t t e r n o f steady d e c l i n e . H e has generalized these o b s e r v a t i o n s over multiple oil wells in various regions, and foi the c o n t i g u o u s U S h e c a m e up with a p r o j e c t i o n thar said that oil prod tic t ion across rhe w h o l e c o u n t r y will p e a k . H e e v e n e s t i m a t e d w h e n ir would h a p pen - in t h e early 1970s. It turned out t h a t his p r o j e c t i o n was remarkably close to whar h a p p e n e d in reality (Figure 7 . 2 9 ) . T h e n e x t o b v i o u s step was to apply this s a m e m e t h odology to world oil p r o d u c t i o n . A c c o r d i n g to those p r o j e c t i o n s , the peak is supposed to h a p p e n s o m e t i m e really soon - by s o m e estimates, it has actually already h a p p e n e d . W h y is peak oil such a big issue? Primarily b e c a u s e t h e d e m a n d for oil c o n t i n u e s to grow e x p o n e n t i a l l y , w h i c h m e a n s t h a t as s o o n as oil p r o d u c t i o n p e a k s t h e r e will be a n i n c r e a s i n g gap b e t w e e n d e m a n d and supply. For s u c h an e s s e n t i a l r e s o u r c e as energy, this gap may result in c a t a s t r o p h i c o u t c o m e s .

288

Systems Scienrf? and Modeling fo' Ecologicst Economics US Crude oil production i woo lonoo 8000 6000 4000

2000

^ Total US

Figure 7.29

&

& ^ ^ * Without oKsixtte

j? j? ^ ^ # ^ / n f Without ys and Alaska —Hubfcefl s curve

US Oi! production 1850-2050, as predicted by the peak oil theory of King Hubbert and in

reality. The dashed line is Hubbert's prediction The solid line is the actual extraction. Note that the timing of the peak was predicted almost exactly

B o t h e n e r g y a n d w a t e r b e l o n g to the so-called Critical Natura. Capital category, w h i c h m e a n s that they are essential for h u m a n survival. A s t h e y b e c o m e scarce, t h e y exhibit high pncemelasticity of d e m a n d , s o t h a i a small r e d u c t i o n o l q u a n t i t y leads t o a huge increase in price A small d e c r e a s e in s u p p y w i l l lead t o an e n o r m o u s increase

n price, s o that t o t a l

value (price x quantity) paradoxically increases as total quantity declines. This is true for a n y r e s o u r c e that is essential a n d n o n - s u b s t i t u t a b i e . A s t h e r e s ess w a t e r or e n e r g y available, i h e price quickly increases t o w a r d s infinity. This c r e a t e s havoc w i t h m a r k e t s a n d pretty m u c h p u t s i h e w h o l e s y s t e m o u l o l control - as w e s a w d u r i n g the e n e r g y crisis of t h e 1970s W h i l e e n e r g y a n d w a t e r are abundant, their value is l o w . it may s e e m t h a t w e have an infinite supply, a n d t h e r e is n o t h i n g t o w o r r y a b o u t . However, as d e p l e t i o n accelerates, even small p e r t u r b a t i o n s d u e t o u n f o r e s e e n climatic events or technical m a l f u n c t i o n m a y result in disprop o r t i o n a t e changes in price.

Natural capital stocks

Add ing Soc io-Econom ics

289

A s p o i n t e d c u t in Chapter 6, as long as w e rely u p o n purely r e n e w a b l e e n e r g y and water, t h e y are non-rival and non-excludable

However, as w e n e e d t o d i p into r e s e r v e s of fossil

w a t e r or energy, or even into t h e t e m p o r a r y r e s e r v e s Hakes, reservoirs, or f o r e s t and crop b i o m a s s ! , i m m e d i a t e l y t h e r e s o u r c e s b e c o m e e x c l u d a b l e a n d rival. A s r e s o u r c e s scarcer w e easily c r e a t e c o n f l i c t s i t u a t i o n s i w a t e r a n d e n e r g y w a r s

become

one of w h i c h w e are

w a g i n g right n o w ) (Farley a n d Gaddis

200/1

W h i l e most official sources have b e e n q u i t e r e l u c t a n t to discuss this issue, in 2 0 0 7 several p u b l i c a t i o n s appeared i n d i c a t i n g that there is a growing c o n c e r n e v e n m circles closely related to g o v e r n m e n t s . In July 2 0 0 7 t h e I n t e r n a t i o n a l Energy A g e n c y ( I E A ) , an arm of t b e O r g a n i z a t i o n lor E c o n o m i c C o o p e r a t i o n and

Development

( O E C D ) , published t h e " M e d i u m - T e r m O i l M a r k e t R e p o r t . " T h e report predicts that world e c o n o m i c activity will grow by an average o f 4 . 5 p e r c e n t per year during t h e n e x t several years, driven largely by scrong growth in C h i n a . India, a n d o t h e r A s i a n c o u n t r i e s G l o b a l oil d e m a n d will, as. a result, rise by a b o u t 2 . 2 p e r c e n t per year, pushing world oil c o n s u m p t i o n from an e s t i m a t e d 8 6 . 1 m i l l i o n barrels per day in 2 0 0 7 to 9 5 8 m i l l i o n barrels by 2 0 1 2

If there are n o c a t a s t r o p h e s and there is a m p l e new

i n v e s t m e n t , t h e global oil industry may be able to increase output sufficiently to satisfy this h i g h e r level of d e m a n d - but if so, barely. B e y o n d 2 0 1 2 , t h e p r o d u c t i o n outlook appears far grimmer. A n d r e m e m b e r t h a t this is t h e best-case s c e n a r i o . Let us see what we c a n find out a b o u t t h e future ot oil supplies using s o m e s i m ple d y n a m i c m o d e l i n g

S u p p o s e we h a v e a s t o c k o f oil

S i n c e it is a n o n - r e n e w a b l e

resource, it is safe to assume that it is l i m i t e d . T h e r e will always be oil in the ground, but it is quite c l e a r that e v e n t u a l l y we will run out o f t h e e n e r g e t i c a l l y profitable resource. S o with this stock c o m e s just an outflow, w h i c h we will call E x t r a c t i o n . Let us assume t h a t E x t r a c t i o n is driven by D e m a n d . D e m a n d is e x p o n e n t i a l l y growing, just as it has b e e n over t h e past years: D e m a n d ( t ) = D e m a n d f t - d t ) + ( G r o w t h ) * dt Growth = C_grow * Demand Besides satisfying D e m a n d . E x t r a c t i o n should also produce e n o u g h to power E x t r a c t i o n uself. T h i s is what is k n o w n as the E R O E I (Energy R e r u r n on

Energy

I n v e s t e d ) i n d e x . If e 1K;I is t h e a m o u n t o f energy produced a n d e m is t h e a m o u n t o f energy used in p r o d u c t i o n , t h e n E R O E I , e = e ou ,/e, n . In s o m e cases the net E R O E I index is used, w h i c h is the a m o u n t ot energy we n e e d to produce to deliver a unit ot n e t energy to t h e user e ' - e ( „ „ / ( e 0 u l - e j n ) . O r e ' = e/(e -

1).

T o a c c o u n t for E R O E I , we put: R e s e r v e s ( t ) = R e s e r v e s i t - d t ) + ( - E x t r a c t i o n ) * dt

Extraction = Demand *

I

1+

I eroei,

It also m a k e s sense to assume t h a t E R O E I is n o t c o n s t a n t . In fact, at s o m e p o i n t we had oil f o u n t a i m n g out o f t h e ground, so we just n e e d e d to c o l l e c t and d e l i v e r

290

Systems Scienrf? and Modeling fo' Ecologicst Economics eroei v. R e s e - v e s ; i 100.00

50.00

0.00 - t — 0.00

1

1 500000000.00

1

1 1e+09

Reserves

Figure 7.30

The parabolic d e p e n d e n c y between the EROEI index and the amount of reserves still

available The fewer reserves are left, the more w e need to invest in production.

it; n o w w c n e e d t o drill k i l o m e t e r s d e e p i n t o t h e g r o u n d a n d p u m p t h e o i l o u t , ( h e n p u m p w a t e r o r C C b in t o p u s h s o m e m o r e o i l o u t , a n d s o o n . T h e e n e r g y has declined from over 2000. EROEI

return

1 0 0 : 1 in t h e 1 9 3 0 s t o 3 0 : 1 in t h e 1 9 7 0 s t o a r o u n d 1 0 : 1 i n

is a b a t t l e b e t w e e n t e c h n o l o g y a n d d e p l e t i o n , a n d d e p l e t i o n

is w i n -

n i n g . I n t h e f u t u r e , m o r e e n e r g y i n v e s t m e n t will b e n e e d e d , t a k i n g e n e r g y o u t o f a non-energy socieiy. L e t us a s s u m e t h a t E R O F . I d r o p s w i t h R e s e r v e s d e c r e a s i n g , a c c o r d i n g t o t h e p a r a b o l i c f u n c t i o n s h o w n in Figure 7.30. T h e n

eroei = e m i

assuming that e_ini -

Reserves r mi

1 0 0 is t h e o r i g i n a l E R O E I a n d t h a t r _ i n i =

1 0 0 0 0 0 0 0 0 0 is t h e

o r i g i n a l s t o c k o f oil i n t h e R e s e r v e s . II w e r u n t h i s m o d e l , w e will gei a n e x p e c t e d r e s u l t : t h e g r o w i n g d e m a n d will c e r t a i n l y d e p l e t e t h e r e s o u r c e s ( F i g u r e 7 . 3 1 ) . W h a t is n o t e w o r t h y a b o u t t h i s g r a p h i c is t h e p o w e r o f e x p o n e n t i a l g r o w t h . W h i l e w e h a v e v e r y slow, a l m o s t n e g l i g i b l e c h a n g e o v e r a l o n g i n i t i a l p e r i o d o f t i m e , t h i n g s start t o a c c e l e r a t e t r e m e n d o u s l y by t h e e n d o f t h e s e a s o n . M o s t o f t h e c h a n g e is c o m p r e s s e d i n t o a r a t h e r s h o r t p e r i o d o f t i m e , w h e n a c t i o n is really n e e d e d , b u t t h e r e is v e r v l i m i t e d t i m e t o d n s o m e t h i n g . A l s o , n o t e howm u c h faster we n e e d t o p u m p out our reserves t o supply t h e d e m a n d as reserves b e c o m e depleted

You com. increase

It is a l s o n o t e w o r t h y t h a i t h e v a l u e s o f t h e p a r a m e t e r s t h a t w e used i n this m o d e l d o n o t really m a t t e r .

resutis

your

ii tke snodA

It is hard

coifidettf*

to pterin struexutai

stability,

but that

to search

h a s a vivid t r a c e t h a t s h o w s t h r o u g h

kiwe

erf robustness

any

ntodJicMicns.

modifications

in

parameters.

W e c a n e v e n try a n o t h e r

function

a tjood deal

utod/d stable.

jtrr tncdHs

it K cdwv.YS jood

T h e exponential growth or decline

in

is Urux.tusa.lly

to

structUA'ai

Add ing Soc io-Econom ics

j ® 1: D e m a n d 1: 0 0 0 0 0 0 0 0 0 0 - u 2 1e+09. 3 100.00

291

4 Entraction

3 eroei

4 00000000 00

1 • 50000005.00 ? 5 0 0 0 0 0 0 0 0 003 50 0 0 4 . 5 0 0 0 0 0 0 5 00

10.00 0 00 0 00 i o oo!

I: 2 3. 4

Figure 7.31

387 50

462.50

42500

500 00

System d y n a m i c s shoves very s l o w dynamics at first, fol o w e d by a period of very high

g r o w t h rate and eventual c r a s h of the system due to depletion of resources

Figure 7.32

The s-shaped EROEI function p r o d u c e s very similar results.

W e may argue that the model is s t r u c t j r a ly quite stable. W i t h qualitatively similar assumpt ons about the driving forces a n d processes, the exact f o r m j l a t i o n s and parameter value do not matter that much.

for t h e E R O E I . S u p p o s e we c h o o s e a n s - s h a p e d o n e ( r e m e m b e r t h a r w h i c h w e disc u s s e d in Figure 2 . 2 0 ? ) :

eroei

e_.ru* Reserves2 r_ini 9

Reserves'

F o r t h i s f u n c t i o n we a l s o get a s i m i l a r p a t t e r n , w i t h very s l i g h t c h a n g e s in r h e t r a j e c t o r i e s ( F i g u r e 7 . 3 2 ) . O n c e a g a i n , t h e last d r o p o f o i l is e x t r a c t e d a t a n e x c e e d ingly high rate H o w e v e r , it c o u l d c e r t a i n l y h e a r g u e d t h a t t h e r e is o t h e r e n e r g y o u t t h e r e , a n d t h e r e is r e a l l y n o r e a s o n t o e x p e c t t h a c w e a r c so i g n o r a n t n o r t o r e a l i z e r h e i m m i n e n t c r a s h a n d n o t t o i t a r t e x p l o r i n g a l t e r n a t i v e * . L e t us a d d a l t e r n a t i v e s o u r c e s o f e n e r g y t o o u r m o d e l . L e t us a s s u m e thac t h e i n f r a s t r u c t u r e for a l t e r n a t i v e e n e r g y is

306 Systems Scienrf? and Modeling fo' Ecologicst Economics being produced ac a c e r t a i n slow rate ( a _ g _ c ) with n o big success until t h e E R O E I for oil falls below a c e r t a i n recognized threshold value ( e r o e i _ t ) . A f t e r that we start rapidly investing in a l t e r n a t i v e s , m a k i n g t h e m grow at a rate ol a _ g : A l t e r n a t i v e s ( t ) = A l r e r n a t i v e s ( t - d t ) + ( A l t _ g r ) * dt A l t _ g t — if e r o e i

<

eroei_t then a_g

A l t e r n a t i v e s else a _ g _ c

T h e assumption here is t h a t o n c e we c h a n g e our a t t i t u d e ro A l t e r n a t i v e s we c a n get t h e m built up really fast by c r e a t i n g a positive f e e d b a c k from t h e i r growth. T h i s seems to be q u i t e feasible if we agree t h a t as rhe n e w t e c h n o l o g i e s get d e v e l o p e d they c r e a t e synergies for t h e i r further d e v e l o p m e n t . T h e r e is also t h e E R O E I for A l t e r n a t i v e s , e r o e i _ a . I n this case it will mostly likely grow as new a l t e r n a t i v e infrastructure is put in place. Suppose we use a m o n o d type l u n c t i o n with s a t u r a t i o n :

eroei_a = e_a_min +

e_a_max * Alternatives Alternatives + e_a_hs

where e _ a _ m i n is t h e m i n i m a l starting eroei_ a, w h e n new t e c h n o l o g i e s are only starting to be deployed. It makes sense initially t o h a v e it a t e v e n less t h a n 1, reflecting t h e fact that at first we n e e d to invest a great deal with very little return. e _ a _ m a x is t h e m a x i m a l e r o e i _ a and e _ a _ h s is t h e half-saturation coefficient t h a t tells us at w h i c h level o f d e v e l o p m e n t o f a l t e r n a t i v e energy ( A l t e r n a t i v e s s t o c k ) we get e roe L a equal to h a l f o f t b e m a x i m a l . W e also w a n t to modify t h e e q u a t i o n for E x t r a c t i o n . E x t r a c t i o n = if D e m a n d > A l t e r n a t i v e s t h e n D e m a n d *

— a

eff*Alternatives'1 1

eroei

1 +

1

1

eroei

, else 0

T h e logic here is that if all rhe d e m a n d c a n be c o v e r e d by a l t e r n a t i v e energy (Demand < Alternatives),

then

there is n o need to c o n t i n u e e x t r a c t i o n o f fos-

sil energy, and E x t r a c t i o n = 0 . O t h e r w i s e , we need to e x t r a c t e n o u g h to c o v e r t h e d e m a n d . T h e a l t e r n a t i v e infrastructure chips in wirh t h e efficiency a _ e f f (a n e g a t i v e term in t h e E x t r a c t i o n e q u a t i o n ) , but to produce this a l t e r n a t i v e energy we n e e d to invest l/eroei_a (a positive term in t h e E x t r a c t i o n e q u a t i o n ) . T h e h i g h e r t b e e r o e i _ a , t h e less we n e e d t o run t h e a l t e r n a t i v e infrastructuie. Let us run the m o d e l with t h e following p a r a m e t e r values: a_eff = 10, a_g - 0 . 2 , a_g_c - 1 0 0 , c_grow = 0 . 0 3 eroei_r = 20, e_a_hs = 5 0 0 0 0 0 , e_a_max - 10, e_a_min - 0.5 e_ini - 120, r _ m i-

1000000000

W e will mostly be c o n c e r n e d with general q u a l i t a t i v e behavior, a n d will n o t try to figure o u t w h a t t h e real values for these parameters are ( w h i c h is also a very worthwhile effort). F o r now, let us explore what t h e overall system d y n a m i c s are. W i t h t h e

293

Add ing Soc io-Econom ics * 1 Alternative* ZX&XQC 2CC8 )•» '00

5 5 0 OC

Figure 7.33

If w e start investing in Alternatives too late, w e only a c c e l e r a t e the crash of the system.

& I AII-.-OrttlV.il iOCe-0? Ii 00

i 2

3

1 moo 2 t i».0» 3 5 00" IK CD 560 00

ooo C.00 C.C0+'

Figure 7.34

•IO-JCO

5CT00

iSOOO

SSGOO

Early investment in Alternatives, while there is still ample supply ol conventional energy,

allows for a smooth transition to r e n e w a b l e energy.

values a b o v e . A l t e r n a t i v e s make almost n o c h a n g e to rhe system (Figure 7 . 3 3 ) .

On

t h e c o n t r a r y , i n v e s t i n g in t h e a l t e r n a t i v e s e c t o r w h e n we are a l r e a d y p u m p i n g o u t t h e s e c o n d half of o u r r e s e r v e s o n l y a c c e l e r a t e s t h e c r a s h . C h a n g i n g d i f f e r e n t p a r a m e t e r s r e l a t e d t o A l t e r n a t i v e s e f f i c i e n c y d o e s n o t s e e m t o h e l p . T n e s y s t e m still c r a s h e s What

does help

is c h a n g i n g

parameters related

t o t h e timing of the

switch

t o a l t e r n a t i v e s . I f we start d e v e l o p i n g a l t e r n a t i v e s w h e n t h e E R O E 1 o f t r a d i t i o n a l e n e r g y is still as h i g h as a b o u t 6 0 o r m o r e ( e r o e i _ t > 5 8 ) , w e get a c o m p l e t e l y d i f f e r ent picture (Figure 7 . 3 4 )

T h e s a m e o p p o r t u n i t y e x i s t s if we h a v e b e e n s l o w l y d e v e l -

o p i n g a l t e r n a t i v e s s i n c e t h e v e r y b e g i n n i n g ( a _ g _ c = 1 0 0 0 ) . In t h e s e c a s e s w e h a v e a pretty s m o o t h t r a n s i t i o n from fossil-based e n e r g y to a l t e r n a t i v e energy, with e x t r a c t i o n g o i n g d o w n t o z e r o w h i l e t h e r e is still p l e n t y o f oil left in r h e g r o u n d . O b v i o u s l y o t h e r f a c t o r s will k i c k in, s u c h as l i m i t e d l a n d r e s o u r c e s , s o it is a m a j o r s i m p l i f i c a t i o n t o t h i n k t h a t i n d e e d we will b e a l w a y s a b l e t o p r o v i d e for t h e e x p o n e n t i a l l y g r o w i n g d e m a n d . H o w e v e r , in t e r m s o f e n e r g y we c a n d o it ( o r , m o r e likely, c o u l d h a v e d o n e i t ) . T h a t is il we h a d s t a r r e d r h e t r a n s i t i o n e a r l y e n o u g h p r o v i d e for r h e n e w a l t e r n a t i v e

i n f r a s t r u c t u r e . T o find o u t m o r e e x a c t l y h o w

to late

w e a r e a r r i v i n g a t r h e s h o w , w e will n e e d t o find s o m e m o r e r e a l i s t i c v a l u e s for t h e p a r a m e r e r s . In t h i s c a s e , t h e r e a r e s e v e r a l p a r a m e t e r s t h a t d o m a t t e r . H o w e v e r , q u a l i t a t i v e l y it s e e m s t h a t t h e h a l f d e p l e t i o n t h r e s h o l d is a n i m p o r t a n t f a c t o r in dynamics.

Ii we s t a r t r h e t r a n s i t i o n

t o t h e a l t e r n a t i v e s well b e f o r e

we have

these half

d e p l e t e d t h e r e s o u r c e , t h e r e is e n o u g h t o f u n d r h e d e v e l o p m e n t o f t h e n e w a l t e r n a tive infrastructure

If w e p r o c r a s t i n a t e a n y l o n g e r , t h e c r a s h is i m m i n e n t .

>ystsms Science nr.d Modeling -for Ecoiog'ca 1 Ecoron->»cs

294

it is b e c o m i n g m c r e a s n g . v d e a r thnt. i n

loriQ run, h u m a n i t y can s u r v i v e <><% '.vino w i t h "

i r e limits o t resources t h a t it has I n thin c o n t e x t , fossil f u e l s u p p e a ' o s w i r w v n g » « . s*>/.

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e r « r g y g e e i t h a t c n m « w i t h toa-.il f u e l s w « » o u r c h a n c e t o e a r n h o w bettor to h o w * o t . uotor erergy ouzo

a n d w t v t r e v e r t e c h n o l o g i e s w e b u i l d w h i l e w e still h a v e a r e a t a t o fatal

f u t f s * v^-at w * w J i b e i « * w i t h n t h e l o n g years t o f o l l o w w h e n t h e y w i l l b o n o

m o t e A p s w e c e r i a Jy vs*; RUN out 0» t h e r v » ? o n e f or later

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Adding Socio-Economics

295

h a v e supply, and co a level chac c a n b e susrained. Ironically, in many places we h a v e e x a c t l y t h e reverse: population is growing mosc rapidly where watei a n d energy are least available. C o n v e y i n g energy c r e a t e s more losses: currently up ro rwo-thirds of e l e c t r i c energy is lost in transmission. C o n v e y i n g water requires m u c h energy, and also results in significant losses due t o e v a p o r a t i o n and seepage. T h e r e is a c l e a r c o r r e l a t i o n b e t w e e n energy c o n s u m p t i o n and e c o n o m i c develo p m e n t (Figure 7 J 5 ) . A t t h e s a m e t i m e , there is n o o b v i o u s c o r r e l a t i o n

between

G D F and such indicators as hie s a t i s f a c t i o n or life e x p e c t a n c y ( F i g u r e 7 . ^ 6 ) . W e c a n s e e rhat with n o sacrifice t o life q u a l i t y indices we c a n at least h a l v e the per

capita

G D F , and t h e r e f o r e energy c o n s u m p t i o n . It is really a m a t t e r o f c h o i c e , social a t t r a c t i v e n e s s , and cultural priorities. T h e s e c a n be c h a n g e d only with a strong leadership t h a t should he a d v a n c e d and p r o m o t e d by t h e federal g o v e r n m e n t . D e c r e a s i n g consumption,

may be an unpopular

measure

that

makes

federal

i n v o l v e m e n t especially i m p o r t a n t . S o far, most of t h e advertising industry is working towaids increasing c o n s u m p t i o n , buying things t h a t we do n o t n e e d , wasting m o r e energy and water. O n l y federal a c t i o n c a n stop that and help to shift awareness of t h e p o p u l a t i o n towards c o n s e r v a t i o n and efficiency. I n c r e a s i n g e f f i c i e n c y in all areas

-

industiial, residential and agricultural - is a n o t h e r c l e a r l o c u s begging for a c t i o n .

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5000 10000 15000 ?oooo 25000 3C000 35000
B • Figure 7.36

A. Relationship between GDP per capita and life expectancy based on 2005 World Bank

data for 184 countries (http-y/devdata.worldbank org/query/default.htm). B. GNP and Life Satisfaction Index for 2000 (Veenhoven, 2004; another great source of this kind of information is http://www gapminder.org/world/)

296

Systems Scienrf? and Modeling fo' Ecologicst Economics

C o m p a r e p r o d u c t i v i t y per capita

in d i e U S A a n d J a p a n

T h e y fire yit a c o m p a r a b l e

level, and arg actually the best in the world. Yer J a p a n needs only hair o f [he energy t h a t t h e 1 J S A needs! Japan etruts 9 5 t o n s C O ; per cafntrt, whereas rhe U S A e m i t s 19.7 cons C O i per capita

- roughly proportional ro t h e energy c o n s u m p t i o n varies of

t h e two c o u n t r i e s . T h a t shows an obvious way t o cut U H G emissions.

7,6

The World S o far we h a v e been crying t o focus o n s o m e very simple models, t h e d y n a m i c s o f w h i c h we c a n carefully e x p l o r e to reveal s o m e o ! t h e e m e r g e n t properties a n d surprises in systems' b e h a v i o r T h e s e models easily t e n d ro b e c o m e more a n d more c o m plex. A s we find more c o n n e c t i o n s , processes, factors a n d parameters that seem to be i m p o r t a n t tor t h e overall system's d y n a m i c s , the e n t i c e m e n t is very strong to add t h e m t o t h e model, because, indeed, t h e y s e e m important a n d the m o d e l would n o t look relevant w-ithout rhem. S o m e t i m e s we promise ourselves t h a t we will try to simplify rhe model later o n , after running sensitivity analysis, a n d finding parameters and processes that are n o t really m a k i n g m u c h of a d i f f e r e n c e . Q u i t e o f t e n we forget about t h a t , especially if we are happy with t h e results that we are g e t t i n g , and we t e n d t o c a r e less about t h e more e l a b o r a t e model analysts thar would b e n i c e t o perform if t h e model were simpler A s a result we tend to build models t h a t may be classified as knowledge bases, since they c o n t a i n a huge a m o u n t o f i n f o r m a t i o n , a n d probably present t b e best stateof-rhe-art knowledge about particular systems T h e y are c e r t a i n l y way more a d v a n c e d t h a n simply databases, since in these models we have data sets linked rogeiher: there are casual links t h a t i n d i c a t e how o n e process atfecrs a n o t h e r one, what the feedbacks in t h e system are, a n d h o w o n e data set is c o n n e c t e d to a n o t h e r o n e . T h i s a b u n d a n c e of i n f o r m a t i o n that is e m b o d i e d in t h e model c o m e s ar a price we c a n n o longer dig i n t o t h e details o f systems d y n a m i c s and we have to keep t h e processes quite simple, otherwise we will n o t be able to run t h e model. A n d we still n e e d to be a b l e t o run these models, at least to m a k e sure t h a t the i n f o r m a t i o n t h e y c o n t a i n is consistent; that t h e logic ot t h e links a n d r e l a t i o n s h i p works, and that we get a meaningful, c o h e r e n t picture or t h e modeled system. M o s t o f t h e models that h a v e b e e n developed to describe rhe d y n a m i c s in t h e global scale belong to this category. O n e o f t b e first a n d probably b e s t - k n o w n m o d els o f t h e world is t h e W o r l d 3 model by DoneMa a n d D e n n i s M e a d o w s a n d their colleagues ( M e a d o w s et ill., 1 9 7 9 ) . ( W o r l d 2 was built earlier hy Jay Forrester.) T h e m o d e ! brought t o g e t h e r i n f o r m a t i o n about several m a m subsystems: • T h e food system, dealing with agriculture and food p r o d u c t i o n • T h e industrial system • T h e population system • T h e n o n - r e n e w a b l e resources system • T h e p o l l u t i o n system. T h e n o n - r e n e w a b l e resources have probably caused the most controversy and d e b a t e , since in t h e model there is a finite limit t o t h e a m o u n t o f n o n - r e n e w a b l e resources thar c a n b e extracted Besides, all n o n renewable resources h a v e been lumped i n t o o n e . T h i s allowed immediate and costless substitution of o n e n o n - r e n e w a b l e resource ( c o a l ) for a n o t h e r (say, gas), but excludes the substitution by o t h e r resources through n e w t e c h n o l o g y that science and e n g i n e e r i n g are yet to discover

Adding Rooo-Fcoi-omic.*

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c o n s t r u c t i o n b e c a m e c h e a p e n a n d m o r e e f f o e i t , w e g o t Digger h o u s e s . E v e n n o w . w h e n a n e w s u p e r o n e r g y - o l l i c i o n l r e f r i g e r a t o r is i n s t a t e d m o u r k i t c h e n , t h e 0«d o n e is m o v e d 10 t h e UdrtKie, Ho »ve h a v e t w o r e f r i g e r a t o r s , with t h e c o n s e q u e n t h i f p e r e n e r g y c o n s u m p t i c n

298

S y s t e m s Science and Modeling for Ecological Economics

: population 1 l.20e+10" 6.00e+12 4 00e+12 32 00 2 OOei12.—

2 : food 87

3

induslrial onto/

4

pooli index 143 5

nr resources 1/

6 00e+09 3.00eH2. 2.00e+l2 16.00 I.00e+12

000

3

1900 00

Graph 1 : Page 2

Figure 7.37

1950.00

1 2000.00 Years

1 2050.00 10:39 AM

1 2100.00 96.7.6

A typical output from the World3 model. The system crashes when non-renewable

resources are consumed

T h e m a i n result of this m o d e l was t h a t it s t i m u l a t e d m u c h d i s c u s s i o n o n several g l o b a l p r o b l e m s , s u c h as p o p u l a t i o n g r o w t h , d e p l e t i o n of n a t u r a l c a p i t a l , p o l l u t i o n , e t c . A c c o r d i n g to s o m e e s t i m a t e s , t h e n u m b e r o f lines o f t e x t c o n t r i b u t e d ro t h e s e d e b a t e s has e x c e e d e d t h e Size o f " T h e L i m i t s t o G r o w t h " hy two or m o r e orders of m a g n i t u d e . T h e m a r k e t i n g o f t h e W o r l d 3 m o d e l h a s d r a w n m u c h a t t e n t i o n to applic a t i o n s ol m o d e l s in p o l i t i c s a n d p o l i c y - m a k i n g . T h e u n f o r t u n a t e o u t c o m e is t h a t n o t h i n g or very little has b e e n a c t u a l l y a c c o m p l i s h e d ro s o l v e o r m i t i g a t e t h e p r o b l e m s t h a t were b r o u g h t t o light by t h e m o d e l . A m o r e r e c e n t r e i n c a r n a t i o n of a world systems d y n a m i c s m o d e l 15 t h e G l o b a l U n i f i e d M e t a m o d e l of t h e B i o s p h e r e ( G U M B O ) , d e v e l o p e d hy R o e l o f B o u m a n s a n d o t h e r s c i e n t i s t s at t h e G u n d I n s t i t u t e for E c o l o g i c a l E c o n o m i c s t o s i m u l a t e t h e i n t e grated Earth s y s t e m with t h e implicit goal ol assessing t h e d y n a m i c s and values of e c o s y s t e m s e r v i c e s . T h e m o d e l is p r e s e n t e d as a s y n t h e s i s a n d s i m p l i f i c a t i o n o f s e v e r a l e x i s t i n g d y n a m i c g l o b a l m o d e l s in b o t h t h e natural and s o c i a l s c i e n c e s , a n d c l a i m s t o a i m for t h e i n t e r m e d i a t e l e v e l o f c o m p l e x i t y . W i t h 2 3 4 state v a r i a b l e s , 9 3 0 v a r i a b l e s in t o t a l , a n d 1 7 1 5 p a r a m e t e r s , this m a y be a bit ol a s t r e t c h . W e are c e r t a i n l y d e a l ing w i t h a beast o f a d i f f e r e n t kind t h a n t h a t we h a v e s e e n in o t h e r c h a p t e r s o f this h o o k . If s o m e b o d y t h o u g h t t h a t s o m e of t h o s e m o d e l s were c o m p l e x - t h i n k a g a i n . H o w e v e r , i n d e e d , t h e r e are c e r t a i n l y m o t e c o m p l e x m o d e l s a v a i l a b l e GUMBO

is rhe first g l o b a l m o d e l t o i n c l u d e t h e d y n a m i c f e e d b a c k s

among

h u m a n t e c h n o l o g y , e c o n o m i c p r o d u c t i o n and welfare, a n d e c o s y s t e m g o o d s and serv i c e s w i t h i n t h e d y n a m i c e a r t h system. G U M B O i n c l u d e s m o d u l e s t o s i m u l a t e carb o n , water, and n u t r i e n t fluxes t h r o u g h t h e A t m o s p h e r e , L i r h o s p h e r e , H y d r o s p h e r e , a n d B i o s p h e r e of r h e g l o b a l s y s t e m . S o c i a l a n d e c o n o m i c d y n a m i c s are s i m u l a t e d w i t h i n t h e A n t h r o p o s p h e r e ( F i g u r e 7 . 3 8 ) . G U M B O links t h e s e five s p h e r e s across

Adding Socio-Economics

Figure 7.38

299

Overall structure cf the GUMBO model. Using the Stella array functionality, all the r i a n

"spheres" are replicated over the 11 biomes assumed in the mode

e l e v e n b i o m e s ( O p e n o c e a n , C o a s t a l o c e a n , Forests, G r a s s l a n d s , W e t l a n d s , Lakes/ R i v e r s , D e s e r t s , T u n d r a . Ice/rock, C r o p l a n d s , a n d U r b a n ) , w h i c h t o g e t h e r c o v e r t h e e n t i r e surface of t h e p l a n e t . T h e S t e l l a v e r s i o n o f t h e m o d e l c a n be d o w n l o a d e d from hctp://ecoinformaticsuvm.edu/GUMBO/GUMBO.zip.

P e r h a p s it would be most useful t o d o w n l o a d t h e

m o d e l a n d d o s o m e c l i c k i n g o n t h e diagram ro u n d e r s t a n d w h a t it looks like and w h a t it is d o i n g . T h e d y n a m i c s of I I m a j o r e c o s y s t e m goods a n d s e r v i c e s for e a c h o f t h e b i o m e s are s i m u l a t e d a n d e v a l u a t e d . H i s t o r i c a l c a l i b r a t i o n s f r o m 1 9 0 0 t o 2 0 0 0 for 14 key vari ables for w h i c h q u a n t i t a t i v e t i m e - s e r i e s d a t a were a v a i l a b l e p r o d u c e d a n average R " of 0 . 9 2 2 . F o r a m o d e l o f this level of c o m p l e x i t y , this level of c o r r e l a t i o n w i t h data is very unusual and q u i t e a s t o u n d i n g . T h e o n l y possible e x p l a n a t i o n is t h a t we arc w o r k i n g at a very aggregated level a n d t h e r e is n o t m u c h variability
300

Systems Science and Modeling for Ecological Economics

Biophysical Variables Atrnospneric carbon

Global temperature

Waste

Alternative energy

Fossil fuel extraction

Fossil fuel market share

Total energy

2050

2000

Year

Year _ Observations Scenarios —— Base Case Star Trek (ST) Big Government (BG) Mao Max (MM) EcoTcpia (ET)

Figure 7.39

Calibration results for GUMBO, and runs cf future development scenarios-

•

C o s t a n z a d e s c r i b e d t h e s e four p o s s i b l e f u t u r e s >n C o s t a n z a ( 2 0 0 0 ) . B r i e f l y h e r e is w h a t t h e y a r e about

1. StarTrek. The Default Technological Optimist Vision " W a r m f u s i o n " w a s d i s c o v e r e d a n d p o w e r e d h u m a n i t y t o t h e s t a r s . B y 2 0 1 2 , natural r e s o u r c e s w e r e v e r y s t r a i n e d . T h e w a r m f u s i o n a l l o w e d a rapid r e d u c t i o n of g l o b a

foss;

f u e l burning,

w i t h e v e n t u a l r e v e r s a l of t h e g r e e n h o u s e e f f e c t T h e air pollution p r o b l e m w a s e s s e n t i a l l y elimin a t e d o v e r t h e p e r i o d f r o m a b o u t 2C-15 to 2 0 5 0 . Electricity c a m e i n c r e a s i n g l y f r o m w a r m f u s i o n , nuclear fission reactois w e i e d e c o m m i s s i o n e d and s o m e h y a r o p o w e r stations w e r e eliminated. T h e w o r l d w a s still getting pretty c r o w d e d . T h e solution w a s s p a c e c o l o n i e s , built w i t h materia l s t a k e n f i o m the m o o n a n d a s t e i o i d s a n d e n e r g y f r o m t h e n e w w a r m f u s i o n r e a c t o r s

Since

f o o d p r o d u c t i o n a n d m a n u f a c t u r i n g a r e mainly a u t o m a t e d a n d p o w e r e d by c h e a p w a i m f u s i o n e n e r g y , only a b o u t o n e - t e n t h of The population actually n e e d s to w o r k f o r a living. M o s t a r e f r e e to p u r s u e w h a t e v e r interests t h e m

Often the b i g g e s t technological and social

breakthroughs

h a v e c o m e f r o m this h u g e population of "les t h e n o r m

2. Mad MaxtThe Skeptic's Mightmare T h e turning point c a m e in 2 0 1 2 , w h e n t h e w o r l d ' s oil p r o d u c t i o n finally p e a k e d , a n d t h e long slide d o w n s t a r t e d . T h e r e w e r e no c h e a p e r a l t e r n a t i v e s for oil, only m o r e e x p e n s i v e o n e s . Oil w a s s o i m p o r t a n t in t h e e c o n o m y that t h e price of e v e r y t h i n g e l s e w a s tied to it a n d t h e altern a t i v e s iust kept getting m o r e e x p e n s i v e at the s a m e rate. T h e g i e e n h o u s e e f f e c t w a s really kicking in a n d the e a r t h ' s c l i m a t e a n d e c o l o g i c a l s y s t e m s w e r e m a c o m p l e t e s h a m b l e s . T h e pollution crisis c a m e n e x t . Rising s e a level i n u n d a t e d all low-lying c o a s t a l a r e a s by a b o u t 2 0 5 0 . The financial b u b b l e really burst

B o t h t h e physical i n f r a s t r u c t u r e a n d t h e social i n f r a s t r u c t u r e

n a v e b e e n gradually d e t e r i o r a t i n g , a l o n g with the natural e n v i r o n m e n t . T h e h u m a n population w a s declining s i n c e t h e global e p i d e m i c killed a l m o s t 2 5 p e r c e n t in 2 0 2 5 - 2 0 2 6 . T h e population w a s a l r e a d y w e a k e n e d by regional f a m i n e s a n d w a r s o v e r w a t e r a n d other natural r e s o u r c e s S m c e t h e n d e a t h r a t e s h a v e e x c e e d e d birth r a t e s a l m o s t e v e r y w h e r e , and t h e c u r r e n t population of 4 billion is still d e c r e a s i n g by a b o u t 2 p e r c e n t p e i y e a r

National g o v e r n m e n t s h a v e b e c o m e

weak., a l m o s t s y m b o l i c , relics. Transnational c o r p o r a t i o n s run t h e world, m a k i n g t h e distribution of w e a l t h e v e n m o r e s k e w e d . T h o s e w h o w o r k for global c o r p o r a t i o n s lead c o m f o r t a b l e a n d prot e c t e d lives in highly fortified e n c l a v e s . T h e s e p e o p l e w o r k 90- or 1 0 0 - h o u r w e e k s w i t h n o vacation. T h e r e s t of the population s u i v i v e s in a b a n d o n e e buildings or m a k e s h i f t s h e l t e r s built f r o m s c r a p s T h e r e is no s c h o o l , little f o o d , a n d a c o n s t a n t s t r u g g l e just tc s u r v i v e . T h e a l m o s t c o n s t a n t social u p h e a v a l s a n d ' e v o l u t i o n s 3 r e put d o w n w i t h brutal e f f i c i e n c y by the c o r p o r a t e s e c u r i t y f o r c e s ( g o v e r n m e n t s are t o o b r o k e to maintain a r m i e s a n y m o r e )

3. Big Government: Reagan's Worst Nightmare T h e turning point c a m e in 2 0 1 2 , w h e n t h e c o r p o r a t e c h a r t e r of G e n e r a l M o t o r s w a s by t h e U S F e d e r a l G o v e r n m e n t for fai ing to p u r s u e t h e public i n t e r e s t . E v e n t h o u g h

revoked "warm

f u s i o n " had b e e n d i s c o v e r e d in 2 0 1 5 . strict g o v e r n m e n t r e g u l a t i o n s h a d kept its d e v e l o p m e n t s l o w w h i e t h e s a f e t y i s s u e s w e r e b e i n g fully e x p l o r e d . W a r m f u s i o n ' s s l o w n e s s in c o m i n g on line w a s b a l a n c e d w i t h high t a x e s on f o s s i l e n e r g y t o c o u n t e r a c t t h e g r e e n h o u s e e f f e c t a n d s t i m u l a t e r e n e w a b l e e n e r g y t e c h n o l o g i e s . Global C 0 2 e m i s s i o n s w e r e b r o u g h t to 1 S 9 0 l e v e l s by 2 0 0 5 , a n d kept t h e r e t h r o u g h 2 0 3 0 w i t h c o n c e r t e d g o v e r n m e n t e f f o r t a n d high t a x e s , a f t e r w h i c h t h e n e w f u s i o n r e a c t o r s e l i m i n a t e d t h e n e e d for fossil fuels. The w o r s t ptedicted climate-change e f f e c t s w e r e thus averted

G o v e r n m e n t popula-

tion p o l i c i e s that e m p h a s i z e d f e m a l e e d u c a t i o n , universal a c c e s s to c o n t r a c e p t i o n , and f a m i l y planning m a n a g e d to stabilize the global h u m a n population at a r o u n d 8 billion, w h e r e it r e m a i n e d .

302

Systems Science and Modeling for Ecological Economics

T h e i n c o m e distribution h a s b e c o m e m u c h m o r e e q u i t a b l e w o r l d w i d e

G o v e r n m e n t s h a v e explic-

itly a d v o c a t e d s l o w or n o - g r o w t h policies, preferring t o c o n c e n t r a t e i n s t e a d on a s s u r i n g e c o l o g i cal sustainability a n d m o r e e q u i t a b l e distribution of w e a l t h

S t a b l e h u m a n population a l s o took

m u c h of t h e p r e s s u r e off o t h e r s p e c i e s . 4. E c o t o p i a : T h e L o w C o n s u m p t i o n S u s t a i n a b l e V i s i o n T h e turning point c a m e in 2 0 1 2 , w h e n ecological t a x r e f o r m finally w a s e n a c t e d a l m o s t simultan e o u s l y in t h e U S , t h e EU, J a p a n and Australia. Coincidentaliy, it w a s t h e s a m e y e a r that H e r m a n Daly w o n t h e Nobel Prize lor H u m a n S t e w a r d s h i p (formerly t h e prize f o r E c o n o m i c s )

P e o p l e real-

ized that g o v e r n m e n t s h a d to take t h e initiative back f r o m transnational corporations a n d r e d e f i n e t h e b a s i c rules o l t h e g a m e . T h e public had f o r m e d a p o w e r f u l j u d g m e n t a g a i n s t the c o n s u m e r lifes t y l e and for a s u s t a i n a b l e lifestyle A coalition of H o l l y w o o d celebrities and p r o d u c e r s g o t behind t h e idea a n d b e g a n making a s e r i e s of m o v i e s and T V s i t - c o m s that e m b o d i e d t h e " s u s t a i n a b l e vision." It s u d d e n l y b e c a m e " c o o l " to b e s u s t a i n a b l e , a n d un-cooi to c o n t i n u e to p u r s u e t h e m a t e rialistic, c o n s u m e r lifestyle All d e p l e t i o n of natural capital w a s t a x e d at t h e b e s t e s t i m a t e of t h e full social c o s t of that depletion, with additional a s s u r a n c e b o n d s t o c o v e r t h e uncertainty a b o u t social c o s t s . T a x e s on labor and i n c o m e w e r e r e d u c e d for middle- a n d l o w e r - i n c o m e p e o p l e , with a " n e g a t i v e i n c o m e t a x " or b a s i c life s u p p o r t for t h o s e b e l o w t h e p o v e r t y level T h e QLI (Quality of Life Index} c a m e to r e p l a c e t h e G N P a s t h e primary m e a s u r e of national p e r f o r m a n c e

Fossil f u e l s b e c a m e m u c h

m o r e e x p e n s i v e , a n d this both limited travel and t r a n s p o r t of g o o d s a n d e n c o u r a g e d t h e u s e of r e n e w a b l e alternative e n e r g i e s . M a s s transit, b i c y c l e s a n d car-sharing b e c a m e t h e n o r m . H u m a n habitation c a m e to b e s t r u c t u r e d a r o u n d small v i l l a g e s of roughly 2 0 0 p e o p l e T h e village provided m o s t o l t h e n e c e s s i t i e s of life, including s c h o o l s , clinics and s h o p p i n g , all within e a s y walking distance

P e o p l e r e c o g n i z e d that G N P w a s really t h e " g r o s s national cost," w h i c h n e e d e d to b e

minimized w h i l e t h e QLI w a s being m a x i m i z e d . B y 2 0 5 0 t h e w o r k w e e k h 3 d s h o r t e n e d in m o s t c o u n t r i e s to 2 0 h o u r s or l e s s , a n d m o s t "full-time" j o b s b e c a m e s h a r e d b e t w e e n t w o or t h r e e p e o p l e . P e o p l e c o u l d d e v o t e m u c h m o r e of their t i m e to leisure, but rather than c o n s u m p t i v e v a c a tions taken far f r o m h o m e , t h e y b e g a n t o p u r s u e m o r e c o m m u n i t y activities (such a s participatory m u s i c and s p o r t s ) and public s e r v i c e (such a s day c a r e and elder c a r e ) U n e m p l o y m e n t b e c a m e an a l m o s t o b s o l e t e t e r m , a s did t h e distinction b e t w e e n w o r k and l e i s u r e T h e distribution of i n c o m e b e c a m e an a l m o s t u n n e c e s s a r y statistic, s i n c e i n c o m e w a s not e q u a t e d w j t h w e l f a r e or p o w e r , and t h e quality of a l m o s t e v e r y o n e ' s life w a s relatively high. With electronic c o m m u n i c a t i o n s , t h e truly global c o m m u n i t y could b e m a i n t a i n e d w i t h o u t t h e u s e of c o n s u m p t i v e physical travel.

G U M B O c o u l d h a n d l e t h e s e s c e n a r i o s t o p r o d u c e che results in Figures 7 . 3 9 a n d 7 . 4 0 . A g a i n , t h e e x a c t n u m b e r s o n t h o s e graphs are hardly i m p o r t a n t , a n d m a y b e difficult t o justify. W h a t really m a t t e r s is t h a t t h e m o d e l t o o k i n t o a c c o u n t m u c h o f t h e e x i s t i n g k n o w l e d g e a b o u t g l o b a l processes a n d translated t h a t k n o w l e d g e

into

m e a n i n g f u l t r e n d s t h a t c a n b e discussed, c o m p a r e d a n d e v a l u a t e d . T b e f u r t h e r d e v e l o p m e n t o f t h e m o d e l was for v a l u a t i o n of e c o s y s t e m services. E c o s y s t e m services m G U M B O are aggregated to 10 m a j o r types. T h e s e are: gas regulation, c l i m a t e regulation, d i s t u r b a n c e regulation, water use, soil f o r m a t i o n , n u c r i e n t c y c l i n g , waste t r e a t m e n t , food p r o d u c t i o n , raw materials, a n d recreation/cultural. T h e s e 10 services t o g e t h e r represent t h e c o n t r i b u t i o n o f natural capital t o t h e e c o n o m i c prod u c t i o n process. T h e y c o m b i n e with r e n e w a b l e a n d n o n - r e n e w a b l e fuels, built capital, h u m a n c a p i t a l ( l a b o r and k n o w l e d g e ) , a n d social c a p i t a l to produce e c o n o m i c goods

Adding Socio-Economics

303

Landuse changes

Ice and rocks

2100

2000 Year a

Observations

Scenarios

Figure 7.40

Base

Star Trek (ST)

- - Big Governmeni(BG)

EcoTopia (ET)

- • - Wad Max

Change in landuse composition under various future development scenarios

and serv ices. T h e y also c o n t r i b u t e directly to h u m a n welfare. Several different m e t h o d s to value ecosystem services are implemented in t h e model, allowing users to observe all of t h e m and c o m p a r e t h e results. Historical data o n landuse, C O ? c o n c e n t r a t i o n in t h e a t m o s p h e r e , global m e a n temperature, e c o n o m i c production, population and several o t h e r variables are used t o calibrate t h e model

304

Systems Science and Modeling for Ecological Economics W i t h special locus o n e c o s y s t e m services, G U M B O h a s r e c e n t l y m o r p h e d into the Multi-scale translated

Integrated M o d e l s o f Ecosystem S e r v i c e s ( M I M E S ) .

It has b e e n

i n t o S i m i l e , a n d is now available a t http://www.uvm.edu/giee/mimes/

d o w n l o a d s . h t m l . T h e model has n o t b e c o m e any simpler; actually, m o r e a n d more c o m p o n e n t s a n d processes h a v e b e e n added t o it. T h e promise ts t o be able t o g o t o an i n t e r f a c e like G o o g l e E a r t h , c h o o s e a n area a n y w h e r e o n t h e globe, a n d i m m e d i ately e i t h e r get a n e s t i m a t e o f e c o s y s t e m services for t h a t area, o r d o w n l o a d a model t h a t c a n be used t o m a k e t h e s e estimates. A n o t h e r somewhat similar effort in modeling and quantifying ecosystem services is the Natural Capital Project that is currently underway at Stanford University, with collaboration with t h e T h e Nature C o n s e r v a n c y and World Wildlife Fund. T h e project also aims at developing a full suite o f tools that will allow landuse decision makers and investors to weigh t h e full value of ecosystem services that nature provides for h u m a n liie (http://www.naturalcapitalproject.org/).

T h e i r toolbox is called

InVEST

- Integrated Valuation of Ecosystem S e r v i c e s and Tradeoffs - and is supposed to model and map t h e delivery, distribution and e c o n o m i c value of life-support systems (ecosystem services) well into t h e future. T h e life-support systems that will be analyzed and t h e ecosystem services they provide include c a r b o n sequestration, drinking water, irrigation water, hydropower, flood mitigation, n a t i v e pollination, agricultural crop production, c o m m e r c i a l timber production, n o n - t i m b e r forest products, real-estate value, recreation and tourism, a n d cultural and esthetic values. It is yet to be seen how these models will work together, and how c o m p l e x a knowledge base model will c o m e out of this effort.

For y e a r s , ecological e c o n o m i c s h a s b e e n distinguishing itself from environmental e c o n o m i c s by denying m o n e t a r y evaluation a s an ultimate m e a n s for making d e c i s i o n s . With t h e e c o s y s t e m s e r v i c e s c o n c e p t it s e e m s to c a v e in, at least to a certain extent. The dollar value still a p p e a r s to b e a very p o w e r f u l c o m m u n i c a t i o n tool, a n d in m a n y c a s e s it s e e m s helpful t o b e able to s h o w that the e c o s y s t e m s around us d o deliver s o m e crucial life-supporting s e r v i c e s , which w e normally take a s granted but which actually m a y c o s t a lot This is probably OK a s long a s w e r e m e m b e r ihat all the e c o s y s i e m s e r v i c e s m o n e t a r y e s t i m a t e s a r e on t h e very l o w sioe. 3 n d that actually m m a n y c a s e s w e s h o u d realize that w e a r e dealing with infinite values which it is i m p o s s i b l e to c o m p a r e and meaningfully quamify. For e x a m p l e , a s w e have s e e n on p a g e 2 8 8 , t h e value of critical natural capital i n c r e a s e s asymptotically to infinity a s t h e suppiy of this capital a p p r o a c h e s critical v a l u e s What is the " p n c e " of the bottle of w a t e r if it is a matter of survival? Similarly, w h a t is the " v a l u e ' of a s p e c i e s that is b e c o m i n g extinct, if w e d o not k n o w w h a t b e n e f i t s it c a n potentially provide, and, say. h o w many p e o p l e m a y be cured with drugs extracted f r o m t h e tissue of that s p e c i e s ' It m a k e s s e n s e to put a dollar v a l u e on a b u n d a n t natural capital w h e n it is u s e d for leisure and recreation, w h e n n o b o d y is at risk of irreversible transitions.

Further reading Ehrlich, P.R. (2000) Population Bomb. Random House. 202 pp - First p u M i s M in 1969. the book gives an excellent accuunt of what exponential

grou ch of population memis fur this ptonei. A

Icier book by Ehrlich P.R. and A n n e H Ehrlich, 1 9 9 1 . The Population Explosion. Touchstone Books, 320 pp. - Digs deepei into causes and consequences of imputation growth Kodikoff, L.J. and Bums, S. ( 2 0 0 5 ) The Coming Generational Storm: What Vrra Need to Know aboui America's Economic rent trends ir. US population,

Future. T h e M I T Press. 302 pp. - Gives a vivid account of the curexplores the future of a country with an increasingly older

and with a welfare and social security system on the verge, of collapse

population

•

Adding Socio-Economics

For scrupulous mathematical

analysis of population

Logofei D O . ( 1995). Matrices e.rid Graphs:

models, including models with age struciure

see

Stability Problems in Maihemanraf Ecology, C R C

Press. For even more on matrix modeling of population papulation

305

dynamics

see Caswell, H . 2 0 0 1 . M a f m

models: construction, nrujl-y.sis, and inter/iretaiton, 2nd ed. Sinauer, Sunderland, M A ,

722 pp. Harrmann, T. (2004). Unequal

Protection:

The Rise of Corporate

Human Rights. Rodale Books. 360 pp. - A Jtmory of corporate globally

Dominance takeover

and the Theft of

in the U S A and now

!i shows how very small, insignificant events in the past (like a clencal error) can result in

tremendous consequences

for all See lntp77www.rciiacYcle.net/ and Krtp://suedbyscotts.com/

for more in/onnanon about the TerraCycie

story. Another ver\ well known book on this subject is

David Konen. 2 0 0 1 . When Corporations Rule the World. Bcrrett-Koehler Publishers, 585 pp. Much of the sustamnbiliiy talk started after the famous report of the Bniniianci Commission: W C E D (World Commission on Environment and Development), 1987. Our Common Future. Oxford University Press, Oxford, 400 pp. For various definitions of sustainability and t'i
see Vomov. A . , 2007. Understanding and communicating sustainability: global

versus regional perspectives. Em-iron. Dev. and Sustain. (hup://w\vw.springerlink.com/conu.'ni/ c7737766lp8j2786/). The definitions quoted here are uiken from: Wimberly, R. C , 1995. Policy perspectives on social, agricultural and niral sustainability. Rural Sociology 58: 1 - 2 9 : Custan:a, R

1992 Toward an operational definition of c-cosystcm health. In: Gistanza, R., Haskell B D..

Norton B.C.. (Editors). Ecosystem Health: new t>oab for environmental

management. Wand Press,

Washington, D C , pp. 239 256; Cost an zn, R „ Daly. H. E. 1992. Natural capital and sustainable development. Conservation Biology. 6, 37-46: Sotaw, R M , 1 9 9 1 Suiwmabilny an economist's perspective. Marine Policy Center, W H O I , Woods Hole, Massachusetts. U S A . For further analysis of sustairiabtiuy and its various economic

implications see Ncumavcr, E.. 1999. Weak versus Strong

Sustainability. Edward Elgar, Cheltenham, U K , 294 pp Our observation on management problems with rapidly growing and comp/e.v socio-economic systems echoes with a bo ok by Joseph Tamtei, 1990. The Collapse

of Complex Societies (New Studies in

Archaeology). Cambridge University Press, 260 pp. VWimir Verrwdsfcir is a Russian scientist of great importance, who unfortunately is still very little known in the West One of his major findings was rhe theory of the noosphere, the new state of the biosphere characiemed by the dominance of intellectual and moral j/Owers of humans. Vernadskii. V I , 1 9 8 6 Biosphere. Syncrgetic Press, 86 pp. - This is a reprint of the 1929 paper thai contains some of these ideas. There are several classic books on die Peal Oil issue that couid be recommended, such as Kenneth Deffeycs, 2 0 0 1 . Hubbert's Peak: The Impending World Oil Shortage, Princeton University Press. 224 pp; Richard Heinberg, 2005. The Parry's Over, Temple Lodge, 320 pp; David GooJscein, 2005. Qui of Gas: The End of the Age of Oil, W W Notion & Company. 1 4 8 pp. Paul, Robert (2005). The End of Oil: On the Edge of a Perilous New World. Mariner Books. 4 1 6 pp Perhaps the best latest account of the situation with oil and hou far we are from its "peak" can be found on T h e Oil Drum blog at htip://www.theoildrum com/. The "Medium-Term Oil Market Report" from the International chased at http://omrpublic.iea.org/mtomr.htm.

Energy Agency (IEA) can be pur-

For an interesting discussion on how humans make decisions, and why there is a disjoint between ecological and economic reasoning see Wallerstein, I., 2003. T h e Ecology and the Economy: What is Rational' Paper delivered at Keynote Session of Conference, WorW System History and Global Environmental Change, Lund, Sweden, 1 9 - 2 2 September 2003. http://www.binghamton.edu/ fbc/iwecoratl.htm The Critical Natural Gipital concept is discussed by Farley, J. and E. Gaddis, 2007 A n ecological economic assessment of restoration. In J. Aronson, S. Milton and J. Blignaur (Eds). Restoring Natural Capital: Science, Business and Practice. Island Press: Washington, D C The perfect inelasticity

306

Systems Science and Modeling for Ecological Economics nf cssennal gpods and services when they become increasingly scarce is analyzed in Daly, H , Farley. J , 2C04 Ecological Economics. Island Press (p 1 9 7 ) The World! model is described in Donella H. Meadows. Dennis L. Meadows, Jorgen Randers, William W. Rehrens III, 1979. The Limits io Growth. Mncmillan, 208 pp. For a more recent analysis vf ihe model and ihe various discussions and controversies thac followed, sec Donella H. Meadows, lorgen Randers, Dennis L. Meadows, 2004 J.i'mits to Grou/rh: The 30-Year Update. Chelsea Green, 3 6 3 pp (For Wodd2 mode! see: Jay W. Forresrer, 1 9 7 2 . World Dynamics. Cambridge, M A : Wnghr A l l e n Press, Inc.) The Jefon's paradox is well presented in the visionary book by W S )evons, The Coal

Question;

An Inquiry Concerning ihe Progress of the Nation, and the Probable Exhaustion of Our Coalmines, 1 3 6 5 . U R L of an E-Book: http://oll.libertyfund.org/ERx>l
in R. Boumans, R. Custanr.i, J. Farley, M. A . Wilson, R.

Portela, J. Rotmans, F Villa a n J M Grasso, 2002 Modeling the dynamics of the integrated earth system and rhe value ol global ecosystem services using the G U M B O model. Ecological Economics, Volume 4 1 , Issue 3. p 5 2 9 - 5 6 0 To learn more atom the value of ecosystem services, see ihe special issue of Ecological

Economics,

Volume 25, Issue 1, April 1998. in particular, the famous paper by R Costanza, R. d'Arge, R de Groot. S . Farber, M Grasso, B. Hannon, K. Limlxirg, S . Naeem, R, V. O'Neill, J. Paruelo, et al. The value of the world's ecosystem services and natural capital For more analysis, 1997

including philosophical

p. 3 - 1 5 .

and economic issues oj valuation

Natures Services. Societal Dependence

on Natural

Ecosystems.

see: G . Daily (Ed.),

Island Press, 4 1 2 pp. For

more case studies see G Daily. K. Ellison, 2002. The N e w Economy oj Nature. The Quest to Muke Conservation Profitable

Island Press, 250 pp

8. O p t i m i z a t i o n 8.1

Introduction

8.2

Resource management

8.3

Fishpond

8.4

Landscape optimization

8.5

Optimality principles

SUMMARY R u n n i n g a m o d e l , w e get a glimp.se o l t h e system b e h a v i o r for a g i v e n set of p a r a m e t e r s a n d f o r c i n g f u n c t i o n s . T h i s set is c a l l e d a s c e n a r i o . W e l u n a s c e n a r i o a n d learn h o w t h e system may b e h a v e u n d e r c e r t a i n c o n d i t i o n s . S u p p o s e we k n o w h o w we w a n t i h e system t o l i e h a v e . C a n we m a k e t h e c o m p u t e r sort out t h r o u g h various scen a r i o s t o find t h e o n e t h a t w o u l d b r i n g t h e system as c l o s e as possible t o t h e desired b e h a v i o r ? T h a i is e x a c t l y w h a t o p t i m i z a t i o n d o e s for us. It we h a v e s o m e p a r a m e t e r s t h a t w c c a n c o n t r o l , t h e c o m p u t e r will look a t various c o m b i n a t i o n s o f values t h a t c a n m a k e t h e result as c l o s e as p o s s i b l e t o t h e desired o n e . T h e software t h a t c a n h e l p us d o it is M a d o n n a . W e will look at a c o u p l e of simple m o d e l s t o learn h o w o p t i m i z a t i o n c a n he p e r f o r m e d . For m o r e c o m p l e x systems, e s p e c i a l l y if t h e y are spatially e x p l i c i t , s i m p l e m e t h o d s d o not work. W e will n e e d t o i n v e n t s o m e i n e k s t o solve t h e o p t i m i z a t i o n tasks. F u r t h e r m o r e , in s o m e systems it s e e m s as t h o u g h t h e system itself i n v o l v e s a n o p t i m i z a t i o n process t h a t is d r i v i n g t h e system, as if t h e system ts s e e k i n g a c e r t a i n b e h a v i o r that is o p t i m a l , in a s e n s e . W h e n m o d e l i n g s u c h systems, it m a k e s s e n s e t o e m b e d this o p t i m i z a t i o n p r o c e s s in the model.

Keywords O b j e c t i v e function, control parameter, constraints. M a d o n n a software, global and local o p t i m u m , M o n t e C a r l o m e t h o d , o p t i m a l i t y p r i n c i p l e .

8.1

Introduction In m a n y cases, we w a n t t o d o m o r e t h a n u n d e r s t a n d how a system works. W e w a n t t o figure out h o w t o i m p r o v e its p e r f o r m a n c e , or. ideally, find t h e best way it c a n possibly p e r f o r m . In t h e s e c a s e s we will be t a l k i n g a b o u t optimisation. W e huefly c a m e across this c o n c e p t tn C h a p t e r 4 , w h e n we were e x p l o r i n g m o d e l c a l i b r a t i o n . R e m e m b e r , in t h a t c a s e we a l s o w a n t e d t h e m o d e l t o b e h a v e in a c e r t a i n way. T h e r e were t h e d a t a

307

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Optimization

309

U s i n g m o d e l s of ; e a l systems to find o p t i m a l r e g i m e s m a k e s a lot o f sense. If we h a v e a good m o d e l ot a system, we f a n d e f i n e till s o n s o f sets til c o n t r o l s a n d s u b j e c t t h e system t o all sorts o f e x p e r i m e n t s at n o risk. W e j u s t n e e d t o m a k e sure t h a t t h e m o d e l is still a d e q u a t e w i t h i n t h e w h o l e d o m a i n of c h a n g i n g c o n t r o l f a c t o r s A n o p t i m i s a t i o n task in a g e n e r a l form c a n be f o r m u l a t e d as follows S u p p o s e we h a v e a m o d e l o f a system; -

w h e r e 3C, = ( x ( ( t ) , a n d P = ( p j , fh,

F(X,.P.[)

(8.1}

--•) is t h e v e c t o r of s t a t e v a r i a b l e s ut t h e system ac t i m e t,

. . . ) is t h e v e c t o r o f p a r a m e t e r s W c a s s u m e a d y n a m i c m o d e l t h a t

d e s c r i b e s t h e system in t i m e . T h e r e f o r e , we d e f i n e e a c h n e x t s t a t e ot t h e s y s t e m , a s a f u n c t i o n , P, of its previous s t a t e , X , , a v e c t o r o f p a r a m e t e r s , P, a n d t i m e . t. S u p p o s e we h a v e identified a goal o r a n ubjc'Ctuv f u n c t i o n , w h i c h tells us w h e r e we w a n t t h e s y s t e m t o b e . T h e o b j e c t i v e f u n c t i o n is f o r m u l a t e d as a f u n c t i o n o f m o d e l p a r a m e t e r s a n d s t a t e variables. It is f o r m u l a t e d m s u c h a way that we c a n t h e n iry t o m i n i m i z e o r m a x i m i z e it F o r e x a m p l e , we c a n be studying an agricultural system that p r o d u c e s grass, \|(t) a n d s h e e p , ^ ( O - O u r goal c o u l d be t o m a x i m i z e t h e o u t p u t o f goods p r o d u c e d by t h e s y s t e m . T h e o b j e c t i v e f u n c t i o n would t h e n be based u p o n t h e sum of b i o m a s s e s ol s h e e p a n d grass, X ] ( r ) + x.(t,V H o w e v e r , a sum o f t h e s e two v a r i a b l e ; does n o t give us a v a l u e t o m a x i m i z e . W e s h o u l d e i t h e r d e c i d e thai we w a n t t o track r h e total hiuma.ss o v e r t h e w h o l e t i m e period [ 0 , T ] ; T

G = f i x ( t ) - x . {£));:• i -0 or agree that it is o n l y t h e final b i o m a s s that we are i n t e r e s t e d i n , b e c a u s e t h a t is w h e n we take t h e p r o d u c t s t o r h e m a r k e t :

C; = x ; ( T ! +

It s h e e p are t h e o n l y p r o d u c t we are c o n c e r n e d w i t h , we m a y n o t c a r e a b o u t grass, let s h e e p e a t as m u c h grass as t h e y wish a n d m a x i m i z e o n l y t h e b i o m a s s o f sheep. T h e n G = I , {TJ

A l t e r n a t i v e l y , if we are m a x i m i z i n g f o r t h e farm profits, we m a y be g e t t i n g m u r e r e v e n u e from grass t h a n from s h e e p a n d we t h u s w a n t t o i n c l u d e b o t h , but w i t h w e i g h t s t h a t will r e p r e s e n t t b e m a r k e t values ot b o t h goods at t h e t i m e we sell t h e m : G = />;*|(T) + g , X ; ( T ) w h e r e p, a n d

a t e t h e p r i c e s o f grass a n d sheep, respectively. C l e a r l y ,

defining

the right o b j e c t i v e f u n c t i o n is a very i m p o r t a n t part ot t h e o p t i m i z a t i o n task. If we d o n o t d o a good j o b d e s c r i b i n g w h a t we want co o p t i m i z e , t h e results will be useless.

324 Systems Science and Modeling for Ecological Economics In C h a p c e r 4, as you may r e m e m b e i , t h e o b j e c t i v e f u n c t i o n was t h e difference b e t w e e n m o d e l trajectories and t h e observed dynamics given in t h e data available. W e were t h e n trying to minimize this function by c h o o s i n g t h e right set o f parameters. Similarly, with sheep and grass, t h e total biomass produced is a f u n c t i o n o f c l i m a t i c c o n d i t i o n s , soil properties, fertilizers applied, grazing strategies, e t c . S o m e o f these parameters c a n be c h a n g e d while others c a n n o t . F o r i n s t a n c e , we will n o t be able to c h a n g e c l i m a t i c c o n d i t i o n s ; however, we c a n c h a n g e t h e a m o u n t o f fertilizers used. T h e parameters thar are at our disposal, t h a t we c a n c h a n g e , are called control

factors.

T h o s e are t h e ones that we c a n c o n t r o l w h e n trying to maximize (or m i n i m i z e ) t h e objective function. A g a i n , in C h a p t e r 4 we h a d c e r t a i n parameters t h a t were measured in experim e n t s a n d we k n e w their values quite well. W e did n o t want to c h a n g e those when t w e a k i n g t h e model output. T h e r e were o t h e r parameters that were only e s t i m a t e d , and those were our c o n t r o l factors - t h o s e we could c h a n g e a n y w h e r e w i t h i n reasonable d o m a i n s to bring t h e o b j e c t i v e f u n c t i o n to a m i n i m u m . Let us put some more formalism i n t o these descriptions. W e h a v e a m o d e l ( 8 . 1 ) , and define a n o b j e c t i v e f u n c t i o n G ( X , P ) , where X is t h e v e c t o r o f state variables X = ( x ] , . . ., x n ) , and P is t h e v e c t o r ol" parameters, P = (f>|, . . ., pO- F o r this o b j e c tive f u n c t i o n we t h e n find a mm G(X,P) REP

(q 2 )

s u b j e c t to S ( X , P ) = 0 , and Q ( X , P ) ^ 0 . T h i s should be read as follows: we minimize t h e objective

function

P, w h i c h are t h e control

G ( X , P ) over a subset R = (pjo . . ., p () . . . ) o f parameters parameters,

provided t h a t t h e constraints ( o r r e s t r i c t i o n s ) o n

X hold. If t h e c o n t r o l s a r e scalars and c o n s t a n t , they are also called d e c i s i o n variables. If they are f u n c t i o n s a n d allowed to c h a n g e in time, they are k n o w n as c o n t r o l variables. A s we will see below, i n many cases we may w a n t to describe our c o n t r o l varia b l e in terms of s o m e analytical f u n c t i o n with parameters - say, a p o l y n o m i a l or a t r i g o n o m e t r i c f u n c t i o n . T h e n your t i m e - d e p e n d e n t c o n t r o l variable b e c o m e s formulated in terms of c o n s t a n t parameters, and we c a n say that a c o n t r o l variable is expressed in terms o f several decision variables. T h e constraints bound the space where the model variables can change. T h e r e may be two types of constraints: equality type ( S ( X , P ) - 0 ) and inequality type ( Q ( X , P) 2 s 0 ) . N o t e t h a t it really does n o t m a t t e r w h e t h e r we are minimizing or m a x i m i z i n g the o b j e c t i v e f u n c t i o n . If we have a n o b j e c t i v e f u n c t i o n G , w h i c h we need to m a x i mize, we c a n always substitute ir with a f u n c t i o n G * — 1/G, o r G * * = — G , w h i c h you c a n now safely minimize ro get t h e same result. S i n c e t h e m o d e l t r a j e c t o r i e s are a result o f r u n n i n g t h e m o d e l ( 8 . 1 ) , t h e m o d e l b e c o m e s part o f t h e o p t i m i z a t i o n task ( 8 . 2 ) . H e r e is h o w it works. First, we c h o o s e an o b j e c t i v e f u n c t i o n , w h i c h is formulated in terms o f a c e r t a i n m o d e l trajectory. W e identify t h e parameters t h a t we c a n c o n t r o l to optimize our syst e m . For a c o m b i n a t i o n of these c o n t r o l parameters, we run t h e m o d e l and figure o u t a trajectory. F o r this t r a j e c t o r y we c a l c u l a t e t h e o b j e c t i v e f u n c t i o n , t h e n we c h o o s e a n o t h e r c o m b i n a t i o n o f c o n t r o l parameters, c a l c u l a t e a new value for rhe o b j e c t i v e f u n c t i o n , a n d c o m p a r e it with t h e previous o n e . If it is smaller, t h e n we are o n t h e right track, and c a n try to figure out t h e n e x t c o m b i n a t i o n o f parameters such t h a t it will take us further d o w n t h e m i n i m i z a t i o n path.

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Build model

I Define conirol parameters

I Define objective function

I

Guess first combination ol control parameters

I

| Yes Stop

Figure 8.1

The optimization process.

W h e n solving an optimization task, w e normally go through all these steps. The real challenge that is solved by optimization methods is where to make the next step, how to find the next combination of control parameters.

If F ( x ) is a l i n e a r f u n c t i o n (Figure 8 . Z A ) , o b v i o u s l y we gee a m i n i m u m o n o n e o f che e n d s o f t h e [a, b] i n t e r v a l . E i t h e r x = a , o r x = b d e l i v e r s a m i n i m u m t o t h e f u n c t i o n . W h e n generalized t o several i n d e p e n d e n t v a r i a b l e s , we find o u r s e l v e s in t h e r e a l m o f a s p e c i a l b r a n c h o f o p t i m i z a t i o n c a l l e d l i n e a r p r o g r a m m i n g . S i n c e for l i n e a r f u n c t i o n s t h e m i n i m u m is always o n t h e boundary, t h a t is w h e r e t h e linear p r o g r a m m i n g m e t h o d s s e a r c h . T h e y a r e d e s i g n e d in such a way t h a t they go o v e r all t h e possibly c o m p l e x b o u n d a r i e s o f che f u n c t i o n d o m a i n in t h e m u l t i v a r i a t e s p a c e . I f F ( x ) is n o n - l i n e a r b u t n i c e a n d s m o o t h

(Figure 8 . 2 B ) ,

the most

common

m i n i m i z a t i o n t e c h n i q u e is t h e s o - c a l l e d g r a d i e n t m e t h o d , or t h e " s t e e p e s t d e s c e n t 1 ' m e t h o d . I n o u r case o f o n e i n d e p e n d e n t v a r i a b l e , we c a n c h o o s e a p o i n t , X], c a l c u late t h e value F ( x , ) > a n d t h e n c h o o s e t h e n e x t p o i n t , x'2, such t h a c F ( x i ) < F(-*i)W e m a y n e e d t o try several d i r e c t i o n s b e f o r e we find s u c h a p o i n t . If we h a v e several v a r i a b l e s , w e will m o v e in t h e d i r e c t i o n o f t h e v a r i a b l e t h a c delivers t h e lowest v a l u e co t h e f u n c t i o n in t h e v i c i n i t y o f X[. W e t h e n m o v e t o this n e x t p o i n t , x 2 ; a n d repeat t h e s a m e p r o c e d u r e t o find Xy A n d so o n , u n t i l we realize t h a t , w h i c h e v e r d i r e c t i o n we go, we a r e o n l y i n c r e a s i n g t h e v a l u e o f che f u n c t i o n . T h i s a l g o r i t h m works really well unless r h e f u n c t i o n we are d e a l i n g with h a s s e v eral l o c a l m i n i m a , like t h e f u n c t i o n in Figure 8 . 2 C , w h i c h h a s a m i n i m u m o n t h e

Optimization

313

Fix)

b

Fix)

Figure 8.2

a

*

F(x)

X, X?

r

Searching the minimum in some simple functions.

We can see that a local minimum can be quite different from the global one.

boundary, o r t h e f u n c t i o n in Figure 8 . 2 D , w h i c h h a s t w o local m i n i m a inside t h e d o m a i n for x. If we f o l l o w t h e a l g o r i t h m a b o v e , c h a n c e s a r e t h a t we will find t h e o t h e r m i n i m u m a n d will stay t h e r e , n e v e r realizing t h a t t h e r e is y e t a n o t h e r m i n i m u m w h i c h is e v e n s m a l l e r . T h e s o l u t i o n in this c a s e c o u l d b e t o try several s t a r t i n g p o i n t s f o r t h e g r a d i e n t s e a r c h a l g o r i t h m a n d s e e w h e r e we e n d up g o i n g d o w n h i l l . 1 h e n w e c a n c o m p a r e t h e v a l u e s we g e t a n d c h o o s e t h e m i n i m a l o n e . A s t h e f u n c t i o n F ( x ) b e c o m e s m o r e poorly b e h a v e d , with strong n o n - l i n e a r i t i e s as in Figure 8 . 2 E , rhe gradient search b e c o m e s almost impossible. T h e c h a n c e s that we will h i t t h e global m i n i m u m are b e c o m i n g very low. In this case we m i g h t as well d o a r a n d o m search across t h e w h o l e i n t e r v a l |a, fr], p i c k i n g a value f o r x, x., finding t h e value o f F ( x , ) , t h e n p i c k i n g t h e n e x t v a l u e x,,,

again at random a n d c o m p a r i n g F(x.)

314

Systems Science and Modeling for Ecological Economics

Objective functions may have very unusual forms, w h i c h makes it only harder to find good methods for optimization It is especially hard to do global optimization, yet that is the kind of optimization w h i c h is usually most desired

and F ( . \ * | ) , k e e p i n g t h e lowest value loi further c o m p a r i s o n s . If we a r e lucky playing this g a m e o f r o u l e t t e , we m a y e v e n t u a l l y g e t pretty d o s e t o t h e real global m i n i m u m lor o u r f u n c t i o n . L i k e o t h e r m e t h o d s based o n random s e a r c h this m e t h o d is k n o w n as t h e M o n t e C a r l o m e t h o d o f o p t i m i z a t i o n , a f t e r t h e famous c a s i n o t o w n in Europe, lust as w h e n playing roulette we d o n o t k n o w w h a t t h e result will h e ( e x c e p t thar most likely we will l o s e ! ) , h e r e t o o we k e e p randomly p i c k i n g a set o f c o n t r o l parameters from their d o m a i n of c h a n g e , h o p i n g that e v e n t u a l l y we will a i m s o m e w h e r e c l o s e e n o u g h t o r h e global m i n i m u m . It also may b e helpful t o c o m h i n c r h e r a n d o m walk a l g o r i t h m with t h e gradient s e a r c h , w h e n for e a c h r a n d o m l y c h o s e n value o f x we also m a k e a few steps tn t h e d i r e c t i o n o f t h e steepest d e c e n t . In this way we a v o i d t h e u n p l e a s a n t possibility o f b e i n g lucky e n o u g h to pick a p o i n t s o m e w h e r e really c l o s e ro t h e global m i n i m u m a n d t h e n m o v i n g away from it, o n l y because we were n o t close e n o u g h . T h e gradient s e a r c h is entirely i n a p p r o p r i a t e l o r piecewi.se l i n e a r o r c a t e g o r i c a l f u n c t i o n , like r h e o n e in Figure 8.2F. In this c a s e we c a n n o t e v e n define r h e d i r e c t i o n nf the- s t e e p e s t d e c e n t by e x p l o r i n g t h e v i c i n i t y ol a point of o u r c h o i c e - w e

Optimization

get t h e same results tor F(.v) unless we j u m p over t o t h e next s e g m e n t

315

Foi such func-

tions, it is only t h e random walk or some variations of it t h a t are appropriate. T h e r e arc numerous o t h e r o p t i m i s a t i o n m e t h o d s available these days

Among

t h e m are t h e G c n e t i c A l g o r i t h m s and o t h e r e v o l u t i o n strategy m e t h o d s that try to m i m i c t h e way genes m u t a t e in search tor an optimal configuration

T h e r e is

t h e S i m u l a t e d A n n e a l i n g algorithm, w h i c h , by analogy with t h e physical

process

of a n n e a l i n g in metallurgy, o n e a c h step replaces t h e c u r r e n t solution by a random " n e a r b y " solution, c h o s e n with a c e r t a i n probability. T h e optimization problem is not an easy o n e ; t h e o b j e c t i v e function c a n b e c o m e very c o m p l e x , especially w h e n it involves multiple variables, a n d it b e c o m e s quite hard to find t h e m i n i m u m in functions like t h e o n e s shown in Figure 8 . 3 . It takes a lot of m a t h e m a t i c a l c r e a t i v i t y and computet power t o hnd t h e s e o p t i m a . S t i l l , in many cases this kind of c o m p u t e r simulation is a m u c h safer a n d c h e a p e r a l t e r n a t i v e t h a n m a n y o t h e r kinds of o p t i m i z a t i o n .

8.2

Resource management Let us c o n s i d e r a simple e x a m p l e of a system where o p t i m i z a t i o n c a n help u.s find t h e best way t o m a n a g e it. S u p p o s e there is a natural resource that we wish t o m i n e t o sell t h e product t o g e n e r a t e revenue. T h e resource is limited; there is only a c e r t a i n a m o u n t of this resource chat t h e m i n e h a s b e e n e s t i m a t e d t o c o n t a i n . H o w do we e x t r a c t t h e resource in order t o g e n e r a t e t h e most profit? T h e r e a r e also
which

li>ok almost like clouds. A s in Stella, if we double click o n any ol the icons, we open up a dialogue b o x that allows us to specify t h e formulas

price

or parameter values to use. For example, for t h e pricc

Required Inputs:

variable you get: A s you draw t h e diagram you put t o g e t h e r the equations o f t h e model. Below are t h e M a d o n n a equations this

generated for

model

with c o m -

m e n t s added in brackets

(T)|_8j|T)[oj

• Lock Position

0000

( Create GrdpiT|

0

0

0

0

O

0

0

t d i t Graph

0

Graph

price s|a/(l • m i n i n g ) • I) Cancel

OK

j

316

Systems Science and Modeling for Ecological Economics (Top m o d e l ) (Reservoirs) d/dt ( R e s o u r c e ) = - m i n i n g INIT R e s o u r c e = 1 0 0 0 0 d/dt (Profit} = — to_proiit INIT Profit - 0 (Flowsl mining - if (qq < = Oi t h e n 0 e l s e if i R e s o u r c e > qq) t h e n q q e l s e R e s o u r c e ( W e a r e checKing that t h e r e is e n o u g h r e s o u r c e t o e x t r a c t ) to_prolu = pnce*mining-costs ( P r o c e e d s f r o m s a l e s of p r o d u c t s m i n u s c o s t s of o p e r a t i o n s ) (Functions) qq = d " T I M E A 2

e*TIME - f

{This is i h e a m o u n t e x t r a c t e d . W e d e f i n e it a s a f u n c t i o n of t i m e to b e a b e to find t h e optimal e x t r a c t i o n s t r a t e g y , a s e x p l a i n e d b e l o w ) e = 0.1 f = 20 d = 10 ( P a r a m e t e r s of t h e e x t r a c t i o n f u n c t i o n ) price = a/(l +• mining) + b (Price of g o o d s m o d i f i e d by t h e a m o u n t s of g o o d s p r o d u c e d

Price c a n

i n c r e a s e s u b s t a n t i a l l y if t h e r e is v e r y little supply, mining is smail) costs = c c " mining/Resource { C o s t s of m m i n g a r e n proportion t o t h e v o l u m e s e x t r a c t e d . C o s t s g r o w a s the lesource b e c o m e s scarcer)

Fl

e l8B. 4 B Fi iSg u Hr U i l

Vnmo - rinwfh*rt'=='-

'

PI H

A model built in Madonna. The similarities with the Stella interface are quite clear;

however there is much more power "under the hood" in a Madonna implementation

Optimization

317

a - 100 b - 10 cc - 0.3 in addition to t h e e q u a t i o n s , M a d o n n a assemble* t h e parameters window, which is c o n v e n i e n t to manage all the parameters u i t h e m o d e l : This

window

contains

all the

parameter values from t h e equations, as well as t h e initial c o n d i t i o n s a n d the

simulation

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to

stop; D T ~ t h e time-step for t h e numerical method; a n d D T O U ~ f - t h e timestep for output. In this window wc c a n also chno.st' t h e n u m e r i c a l m e t h o d t o solve p i e e q u a t i o n s T h e ( S T O P T I M E START'TIME)

m our model

actually

tells us what t h e lifetime is for the mine that we have in mind - t h a t is, for h o w long we plan ro operate it.

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— ~

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STARTT! ME 0 * SO STOPTIME DT 0.02 DTOLIT 0 10000 I NIT Resource INIT Profit 0 e 0 1 f 20 10 ij IS 100 Q 10 cc 0.3-

O n c e i h e model is defined we c a n set up a c o u p l e o f graphics tor output and run the m o d e l . N o t e t h a t t h e p a t ' tern of e x t r a c t i o n is defined hy a parab o l i c f u n c t i o n described ui qq = d * T I M E A 2 + e * T I M E + f

(.5.3)

If d = e = C, we get c o n s t a n t : a t e o l e x t r a c t i o n t h r o u g h o u t t h e lifetime o f t h e m i n e . By c h a n g i n g f, we c a n specify h o w fast we wish co e x t r a c t t h e resource. B y c h a n g i n g d and e, w e c a n configure t h e rate of e x t r a c t i o n over time, making it different at different times

W e c a n set up some sliders a n d start our o p t i m i s a t i o n by

m a n u a l l y c h a n g i n g t h e values for the parameters.

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A s noted a b o v e , every t i m e we m o v e a slider. M a d o n n a will c a l c u l a t e a n e w set of t r a j e c t o r i e s , so we c a n get s o m e idea ol h o w c h a n g e s in p a r a m e t e r values i m p a c t t h e mode I d y n a m i c s . Figure 8 . 5 gives a sample of model output for the p a r a m e t e r \ allies defined tn this slider window. N o t e t h e spike o f price around year 5, when m i n ing was very low a n d supply of t h e resource p l u m m e t e d . T h e Profit at t h e e n d o f t h e s i m u l a t i o n 1; around $ 1 0 1 8 1 7 , w h i c h is quite high, as we c a n easily see by trying t o adjust t h e parameters in t b e slider window. T h e question is, c a n we further increase ir by finding t h e very best C o m b i n a t i o n o f c o n t r o l parameters?

318

S y s t e m s Science and Modeling for Ecological Economics

• •F'l l l : l ' n " B

A model for a combination of parameters defined in the slider window. Madonna will

redraw the graph once a parameter is changed on any of the sliders

T h e reason we c h o s e M a d o n n a for this analysis is b e c a u s e ii c a n run optimizat i o n a l g o r i t h m s a u t o m a t i c a l l y Indeed, in t h e " P a r a m e t e r s " m e n u let us c h o o s e t h e "Optimize" option; ^

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Flowchart

Model

Compute

Graph

Parameters ^ ^ ^ ^ Help ParameterWindow

O86P

Define S l i d e r s Hide Sliders Batch Runs... Repeal Batch Runs

seM 096M

Curve F i t Optimize... Parameter P l o t Sensitivity...

*

1

A n o t h e r dialogue b o x o p e n s up, where we are advised t o c h o o s e t h e p a r a m e t e r s that are allowed t o c h a n g e ( c o n t r o l p a r a m e t e r s ) , and t o specify t h e f u n c t i o n t h a t is t o be m i n i m i z e d ( t h e o b j e c t i v e f u n c t i o n ) : Optimize

Parameters:

Available:

Minimize Cxprenion

•Profit

Tolerance: Cancel

O.I OK

In o u r case, t h e c o n t r o l p a r a m e t e r s w.ll he t h e d, e, a n d f in ( 8 . 3 ) ; these define the pattern o f resource e x t r a c t i o n o v e r time. M a d o n n a c a n o n l y m i n i m i z e , so o u r previous c o n s i d e r a t i o n o f m a x i m i z a t i o n as t h e reverse of m i n i m i z a t i o n c o m e s in

Optimization

319

q u i t e handy. T h e o b j e c t i v e f u n c t i o n t o minimize, c a n lie t h e n e g a t i v e of t h e Profit, s i n c e o u r goal is t o m a x i m i z e Profit- O b v i o u s l y , m i n i m i z i n g

Profit is t h e s a m e as

m a x i m i z i n g Profit. T h i s is o n e o f t h e ways we c a n c o n v e r t a m a x i m i z a t i o n task i n i o a minimization o n e Note that M a d o n n a , when doing the optimization, conveniently uses t h e e n d i n g values o f t h e v a r i a b l e s in t h e o b j e c t i v e f u n c t i o n

S o in our c a s e we

will b e i n d e e d o p t i m i z i n g for t h e profit at t h e e n d o f t h e s i m u l a t i o n , a n d n o t at s o m e i n t e r m e d i a t e steps. F o r e a c h o f t h e c o n t r o l p a r a m e t e r s we a r e asked t o set t h e l i m i t s of t h e i r allowable c h a n g e . T h e s m a l l e r t h e i n t e r v a l s w e c h o o s e , t h e easier a n d q u i c k c r it will b e for t h e a l g o r i t h m t o find a s o l u t i o n . O n t h e o t h e r h a n d , we n e e d t o k e e p t h e r a n g e s broad e n o u g h t o a c c o u n t for a v a r i e t y o t d i f f e r e n t s c e n a r i o s o f r e s o u r c e e x t r a c t i o n . W h e n c h o o s i n g t h e s e r e s t r i c t i o n s o n p a r a m e t e r s , it is e s s e n t i a l t o t a k e

into

a c c o u n t t h e e c o l o g i c a l m e a n i n g ot t h e p a r a m e t e r s we specify F o r e x a m p l e , it s o m e of t h e c o n t r o l p a r a m e t e r s were p r e s e n t i n g , say, rate* of c h a n g e , o r values r e l a t e d t o b i o m a s s o r o t h e r s t o c k s , it would be c l e a r t h a t t h e y n e e d co he c l a m p e d t o be positive. T h e r e is n o n e e d t o s e a r c h t o r o p t i m a l s o l u t i o n s t h a t Wuuld i n c l u d e n e g a t i v e g r o w t h r a t e s o f p o p u l a t i o n s . T h e s e c o n t r o l s a r e n o t possible, s o t h e r e is n o n e e d to c o n s i d e r t h e m as o p t i o n s In t h e c a s e o f ( 8 . 3 ) t h e r e a r e n o o b v i o u s e c o l o g i c a l c o n d i t i o n s for d,

a n d f,

e x c e p t t h a t we d o w a n t t o m a k e sure t h a t t h e resulting s c e n a r i o will p r o d u c e a positive flow o f r e s o u r c e , qq s 0 , for 0 < T I M E < 5 0 , W c m a y t a k e a closer look at ( 8 . 3 ) a n d c o m e up w i t h s o m e r e l a t i o n s h i p s b e t w e e n p a r a m e t e r s i h a t would k e e p q q » 0 , o r we m i g h t play w i t h t h e sliders in M a d o n n a a n d s e e whac c o m b i n a t i o n s ot p a r a m e t e r s m a k e q q become- n e g a t i v e a n d t h e n try t o e x c l u d e t h e m from t h e o p t i m i z a t i o n . H o w e v e r , in o u r c a s e t h i s may n o t b e so i m p o r t a n t , b e c a u s e we h a v e built t h e c o n d i tion of q q b e i n g p o s i t i v e i n t o t h e m o d e l f o r m u l a t i o n . i n d e e d , w h e n d e s c r i b i n g t h e flow f o r " m i n i n g " we h a v e p u t : m i n i n g = if ( q q < = 0 ) t h e n 0 E f f e c t i v e l y , we h a v e i m p l e m e n t e d a c o n s t r a i n t ( h a t is usually part o f a g e n e r a l o p t i m i z a t i o n task ( s e e t h e d e f i n i t i o n in s e c t i o n 8 . 1 ) , but w h i c h h a s n o special p l a c e i n t h e o p t i m i z a t i o n p r o c e d u r e in M a d o n n a . In o u r p a r t i c u l a r m o d e l it m e a n s t h a t we d o n o t n e c e s s a r i l y h a v e t o l i m i t t h e c o n t r o l p a r a m e t e r s t o s u c h values t h a t would g u a r a n t e e t h a t q q > 0 . T h i s will be t a k e n c a r e o f by t h e m o d e l . A n y w a y , we still w a n t t o sec s o m e l i m i t s t o these p a r a m e t e r s , m a k i n g sure t h a t t h e rate of e x t r a c t i o n is n o t o v e r l y h i g h . F o r d we c h o o s e - I < d < 4 T h i s is b e c a u s e la rye i v a l u e s o f d c a u s e very big d i f f e r e n c e s i n t h e e x t r a c t i o n rate o v e r t h e l i f e t i m e o f t h e m i n e - s o m e t h i n g we p r o b a b l y w a n t t o avoid

F o r e , we c h o o s e t h e i n t e r v a l

- 2 0 < e < 3 0 . T h e q q is n o t very s e n s i t i v e t o c h a n g e s in e, as we c a n see from playing w i t h t h e sliders. Foi f, let us c h o o s e a larger i n t e r v a l o f 0 < f < 3 0 0 . T h i s is t o a l l o w high e n o u g h rates of e x t r a c t i o n it t h e a l g o r i t h m c h o o s e s a c o n s t a n t rate. N e x t , t o r u n o p t i m i z a t i o n w e a r e required t o specify a c o u p l e of " g u e s s e d " values for e a c h p a r a m e t e r . R e m e m b e r , w h e n discussing h o w most o p t i m i z a t i o n a l g o r i t h m s w o r k , we m e n t i o n e d t h a t in most c a s e s w c s o l v e t h e e q u a t i o n s for a c o u p l e o f fixed p a r a m e t e r v a l u e s a n d t h e n c o m p a r e w h i c h o f t h e s o l u t i o n * brings a Inwer value t o t h e o b j e c t i v e f u n c t i o n . T h a t h e l p s us t o d e t e r m i n e in w h i c h way t o g o in s e a r c h o f t h e o p t i m u m . W h i l e we d o n o t k n o w h o w e x a c t l y che o p t i m i z a t i o n a l g o r i t h m works in M a d o n n a , m o s t likely it also needs s o m e values t o i n i t i a l i z e t h e p r o c e s s , a n d probably t h o s e are t h e guesses t h a t w e n e e d t o specify. C e r t a i n l y t h e guesses n e e d t o b e w i t h i n t h e p a r a m e t e r d o m a i n s ( t h a t is, larger t h a n t h e M i n i m u m v a l u e a n d less t h a n t h e M a x i m u m ) . T l u - y s h o u l d also b e d i f f e r e n t . O t h e r t h a n t h a t , t h e y c a n really he

320

Systems Science and Modeling for Ecological Economics q u i t e arbitrarily set. H o w e v e r , it may h e l p t o rerun t h e o p t i m i z a t i o n with a set of diflerent guesses. T i n s may h e l p us step away from a local m i n i m u m and find a b e t t e r s o l u t i o n and a different c o m b i n a t i o n ol c o n t r o l parameters. Finally, we c a n press t h e " O K " b u t t o n , sit b a c k a n d w a t c h t h e optimization m a g i c happen. T h e model will he rerun multiple times with different c o m b i n a t i o n s ol parameters in s e a r c h for t h e o n e that will make t h e o b j e c t i v e f u n c t i o n t h e smallest. W h i l e optimizing, a lx>x will appear reporting how many model runs h a v e b e e n made and what t h e c u r r e n t value o f t h e o b j e c t i v e f u n c t i o n is:

;

Running j

Run: 317 Finished

Done

-597439

W e will also s e e t h a t apparently t h e r e a r e several a l g o r i t h m s involved in t b e o p t i m i z a t i o n process and a c o m b i n a t i o n o f t h e m is used. R u n n i n g o p t i m i z a t i o n with t h e c h o s e n settings returns a set o f c o n t r o l parameters, d = - 2 . 2 6 0 2 9 e - 6 , e = 1 . 1 4 0 9 3 e - 4 , f " 2 1 4 - 5 3 6 , w i t h w h i c h t h e e n d i n g profit is $ 1 0 4 9 1 8

If we try o t h e r c o m b i n a t i o n s o f p a r a m e t e r values, we d o n o t s e e m t o get

a n y w h e r e b e t t e r t h a n t h a t . W e c a n round up these p a r a m e t e r values t o d = 0 , e = 0 and f = 2 1 4 . 5 4 , a n d see t h a t we are t a l k i n g a b o u t a c o n s t a n t e x t r a c t i o n rate as an o p t i m a l strategy of m i n i n g (Figure 8 . 6 ) . N o t e t h a t there is yet a n o t h e r p a r a m e t e r in t h e O p t i m i z e dialogue b o x , w h i c h is c a l l e d " T o l e r a n c e . " T h i s specifies t h e a c c u r a c y o f t h e o p t i m i z a t i o n , telling us w h e n t o s t o p looking for a b e t t e r c o m b i n a t i o n o f p a r a m e t e r values. By default T o l e r a n c e is set ro 0 . 0 0 1 , a n d that was t h e value we used in our c o m p u t a t i o n s a b o v e . If we try

3

~

Mine - Run l: qq, Profit vs. TIME

S ^ S ^ i l l B S I E Q l i E f f l l ]

K

:

-". -

EH E

Run 1: £500 steps in0.0167 seconds

TIME

B

H

u ri ie^ 8 • FaiigM - K. -6M extraction scenario

ri Model output for the optimized set of parameters that turns out to stand for a constant

Optimization

321

t o m a k e T o l e r a n c e large e n o u g h , say 0 . 1 , we will n o t i c e t h a t it t a k e s less t i m e t o r u n t h e o p t i m i z a t i o n , fewer m o d e l runs will b e r e q u i r e d , h u t t h e results will b e c o m e w a y m o r e s e n s i t i v e t o t h e i n i t i a l guesses t h a t we m a d e f o r t h e p a r a m e t e r s . S u p p o s e w e t a k e parainecer d , - 2 < d < 4 , a n d use t h e f o l l o w i n g t w o guesses, d , = — l , a n d d 2 = 4 . If we n o w h i t t h e O K b u t t o n , we will g e t back t h e f o l l o w i n g values t o r t h e c o n t r o l parameters: d — 0 2 7 , e — - 6 . 6 9 , I -

1 8 0 . 5 3 . T h e resulting p a t t e r n o r t h e

r e s o u r c e e x t r a c t i o n ( F i g u r e 8 7 ) l o o k s q u i t e d i f f e r e n t t h a n w h a t we h a d a b o v e , while t h e Profit with t h e s e p a r a m e t e r s is a l m o s t t h e s a m e a s b e f o r e : Profit = $ 1 0 4 9 1 8 . N o w , with

t h e s a m e t o l e r a n c e , l e t us m a k e a n o t h e r guess tor p a r a m e t e r d:

d| = - 1, a n d d? = 3 N o w t h e s o l u t i o n t o t h e o p t i m i z a t i o n p r o c e s s c o m e s w i t h c o n trol p a r a m e t e r s ; d = - 0 1 2 , e = 6 . 1 7 , f = 1 5 5 . 0 8

A g a i n t h e r e is y e t a n o t h e r differ-

e n t p a t t e r n for e x t r a c t i o n ( F i g u r e 8 . 8 ) , a n d w h a t is most surprising t h e Profit is a g a i n $ 1 0 4 9 1 8 , w h i c h is t h e s a m e as w i t h a c o n s t a n t e x t r a c t i o n rate. S o w h a t is g o i n g o n ? A p p a r e n t l y t h e r e are several local m i n i m a t h a t a r e q u i t e c l o s c co t h e g l o b a l o n e . T o s t o p t h e o p t i m i z a t i o n process M a d o n n a c a l c u l a t e s t h e d e v i a t i o n o f t h e o b j e c t i v e f u n c t i o n b e t w e e n d i f f e r e n t m o d e l tuns, a n d o n c e che c h a n g c b c c o m e s less t h a n t h e T o l e r a n c e it stops. W h e n t h e t o l e r a n c e is large e n o u g h it is m o r e likely t o s t o p at a l o c a l m i n i m u m , instead o f c o n t i n u i n g t o s e a r c h l o r a b e t t e r s o l u t i o n e l s e w h e r e . T h a t is h o w we g e t i n t o r h e Figure 8 . 7 or Figure 8 . S s o l u t i o n s

T h e Figure 8 . 8 s o l u t i o n is

p r o b a b l y still a l i t t l e worse t h a n w h a t we g e n e r a t e d w h e n r u n n i n g t h e m o d e l w i t h t h e s m a l l e r t o l e r a n c e ( F i g u r e 8 6 ) , b u t we c a n n o t s e e it b e c a u s e profit is r e p o r t e d w i t h n o d e c i m a l s , i n a n y c a s e t h e d i f f e r e n c e is p r o b a b l y n e g l i g i b l e , but n is g o o d co k n o w t h a t t h e very best s o l u t i o n is very s i m p l e : just k e e p m i n i n g a t a c o n s t a n t race, a n d c h e m o d e l c a n cell us w h a t chat race s h o u l d b e . H o w e v e r , if t h e c o n s t a n t e x t r a c t i o n is n o t a n a c c e p t a b l e s o l u t i o n fur s o m e o t h e r reasons, we c a n still c o m e u p w i t h a l t e r n a t i v e s t r a t e g i e s ( F i g u r e s 8 7 a n d 8 . 8 ) w h i c h will p r o d u c e results q u i t e i d e n t i c a l t o t h e o p t i m a l strategy. A c t u a l l y , ii we c o m p a r e t h e p a t t e r n of r e s o u r c e d e p l e t i o n f o r all t h e s e strategies, it is c l e a r thac t h e d i f f e r e n c e s a r e q u i t e s m a l l .

Mine - Run 1: qq, Profit vs. TIME ^ ( M o j ^ S E H E E E I E H a SSO

•

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0 B

Run 1:2500 siep-sinO.0167 seconds

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Optimization results for a different tolerance parameter

o

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S y s t e m s Science and Modeling for Ecological Economics

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Mine - Run I: qq, Profit vs. TIME z s ^ s i i s

0 B

Run t 2500 steps m 0 seconds 1 2e-S

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200

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20

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H i S

Yet another quasi-optimal strategy p r o d u c e d from a different initial guess of parameters.

L e t us n o w slight Iv m o d i f y o u r s y s t e m a n d i n t r o d u c e a d i s c o u n t i n g r a t e i n t o o u r c a l c u l a t i o n s . T h e w a v t h e e c o n o m i c svsrem w o r k s t o d a y is t h a t S i

t o d a y is w o r t h

m o r e t h a n $ 1 t o m o r r o w . W h e n we h a v e a g r o w i n g e c o n o m y , t h e i d e a is t h a t if we h a v e t h i s $ 1 t o d a y w e c a n a l w a y s i n v e s t it i n t o s o m e t h i n g r h a t will h a v e a p o s i t i v e r e t u r n a n d t h e t e f o r e t o m o r r o w w e will h a v e $1 TJI. A S a r e s u l t , in o u r e c o n o m i c c a l c u l a t i o n s w e h a v e t o t a k e t h i s d i s c o u n t i n g in a c c o u n t w h e n w e c a l c u l a t e o u r f u t u r e p r o h t s : t h e m o n e y t h a t will he c o m i n g in l a t e r o n will h e w o r t h less. T h i s c a n b e easily t a k e n i n t o a c c o u n t if w e add a s m a l l m o d i f i c a t i o n t o o u r m o d e l : to_profit = {price*mining-cosrs)*(l -

disc)ATIMF

w h e r e d i s c is t h e d i s c o u n t r a t e , usually v a r y i n g b e t w e e n

1 and

10 p e i c e n t .

Hence

0 . 0 1 < d i s c < 0 . 1 . H o w will t h i s s m a l l c h a n g e a f f e c t t h e o p t i m a l s t r a t e g y of r e s o u r c e c o n s u m p t i o n in o u r s y s t e m ' L e t us a s s u m e t h a t d i s c = 5 p e r c e n t a n d r u n t h e o p t i m i z a t i o n a l g o r i t h m

The

r e s u l t s we g e t for t h e c o n t r o l p a r a m e t e r s a r e d = 3 . 9 9 9 , c — 2 9 . 6 5 6 , f = 2 9 9 9 9 8 , a n d r h e o p t i m a l s t r a t e g y c o m e s our q u i t e d i f f e r e n t f r o m t h a t w h i c h w e h a d b e f o r e ( F i g u r e 8 . 9 ) . If w e t a k e a c l o s e r l o o k at t h e v a l u e s o f c o n t r o l p a r a m e t e r s , w e m a y n o t i c e t h a t t h e y a r e a t t h e u p p e r b o u n d a r y c h o s e n for t h e i r c h a n g e . L e t us m o v e t h i s b o u n d a r y f u r t h e r up a n d s e t d m j x = 2 0 a n d f,, u v = 4 0 0 0 . W h e n w e r u n t h e o p t i m i z a t i o n , o n c e a g a i n w e g e t t h e v a l u e s for t h e c o n t r o l p a r a m e t e r s a t r h e u p p e r b o u n d a r y . A r r h e s a m e t i m e , t h e profit j u m p s f r o m $ 6 7 0 9 0 . 1

in t h e p r e v i o u s run r o $ 9 3 7 7 4 - S o it w o u l d b e

r e a s o n a b l e t o a s s u m e t h a t if we f u r t h e r i n c r e a s e t h e m a x i m a l a l l o w e d v a l u e s , t h e o p t i m u m will m o v e there. G o i n g b a c k t o o u r m i n e , t h i s s i m p l y m e a n s t h a t we n e e d t o e x t r a c t as fast as w e p o s s i b l y c a n . N e v e r m i n d t h a t w c w i l l b e s e l l i n g t h e r e s o u r c e at a l o w e r p r i c e s i n c e w e will b e s a t u r a t i n g t h e m a r k e t - srill j u s t m i n e it as fast as j x j s s i b l e a n d s e l l . Q u i t e a s t r a t e g y ! T h e sad n e w s is t h a t in m a n y c a s e s t h a r is e x a c t l y h o w c o n v e n t i o n a l e c o n o m i c s deals with natural resources. T h e e c o n o m i c

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p o n d thai is e n t i r e l y d r i v e n by a r t i f i c i a l f e e d . A M a d o n n a m o d e l c a n he put t o g e t h e r as in F i g u r e 8 . 1 0 . " F e e d " is a v a r i a b l e chat p r e s e n t s t h e a c c u m u l a t i o n of a r t i f i c i a l f e e d in the p o n d . " F i s h ' ' is t h e b i o m a s s ol fish. N o t e that t h e r e is yet a n o t h e r state v s i r f a b l e , " D e t r i t u s " A n i m p o r t a n t c o n d i t i o n of fish s u r v i v a l is t h a t there is a s u f f i c i e n t l e v e l of d i s s o l v e d o x y g e n in the p o n d . O x y g e n is c o n s u m e d for fish r e s p i r a t i o n , bin is also, v e r y i m p o r t a n t l y , utilized f o r d e c o m p o s i t i o n o f d e a d o r g a n i c m a t e r i a l - d e t r i t u s D e t r i t u s is f o r m e d f r o m t h e p r o d u c t s o f hsh m e t a b o l i s m , e x c r e t e d by fish, as w e l l as f r o m che r e m a i n s o f t h e f e e d t h a t are n o t utilized by fish a n d stay in t h e p o n d . D e t r i t u s is a n i m p o r t a n t f a c t o r in che p u n d e c o s y s t e m b e c a u s e as its c o n c e n t r a t i o n g r o w s , a n o x i a or a n a e r o b i c c o n d i t i o n s are most likely. A s t h e c o n c e n t r a t i o n ot o x y g e n falls N ' l o w a c e r t a i n t h r e s h o l d , fish die o f f . if w e a s s u m e t h a t t h e o x y g e n c o n s u m p t i o n i n c r e a s e s as d e t r i t u s c o n c e n t r a t i o n g r o w s , t h e n p e r h a p s we m a y get a w a y w i t h o u t an a d d i t i o n a l v a r i a b l e to crack o x y g e n a n d s i m p l y a s s u m e chat t h e fish d i e o f f is triggered by h i g h d e t n e u s c o n c e n t r a t i o n s . A l l t h e s e p r o c e s s e s are d e s c r i b e d by t h e f o l l o w i n g e q u a t i o n s in M a d o n n a :

(Reservoirs) d/d1 (Fish) = + G r o w t h - Mortality INIT F i s h = 0 . 1 d/dl (Feed) = - G r o w t h + F e e d i n g - L o s s INIT F e e d = 0 d/dt (Detritus) = + A c c u m - D e c o m p INIT Detritus = 0 . 1 I Flows) G r o w t h = if F e e d > 0 t h e n C _ g r o w t h * F e e d * F i s h / ( F e e d + C _ H s } e l s e 0 (We u s e the M o n o f u n c t i o n w i l h saturation for I is h g r o w t h ) F e e O m g = if ( C _ f e e d > 0) t h e n C _ f e e d e l s e 0 (There m a y b e m a n y w a y s w e plan to f e e d t h e f i s h - 'et u s m a k e s u r e that t h e s c e n a r i o n e v e r g o e s to n e g a t i v e v a l u e s ) Mortality = ( C _ m o r 1 + D e t n i u s A 4 / i C _ r r , o r t _ d A 4 + D e t r i t u s A 4 ) ) * F i s h (Mortality is m a d e of t w o p a n s

first is the l o s s ol b i o m a s s d u e to m e t a b o l i s m

a n d respiration, s e c o n d iS d>e oft d u e to anoxia d e s c r i b e d by a s t e p f u n c t i o n that kick m w h e n c o n c e n t r a t i o n of detritus b e c o m e s m o r e than a certain threshold.) Loss = C J c s s ' F e e d + Growth*0 |A part of t e e d that is not c o n s u m e d by fish turns into detritus A trick to m a k e S u r e that this f l o w is c a l c u l a t e d A F T E R fish G r o w t h is t a k e n c a r e oft In o r d e r t o c a l c u l a t e this f l o w w e e n f o r c e lhai fish g r o w t h s h o u l d b e a i r e a d y c a l c u l a t e d I Accum -

L o s s + Mortality

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0.2

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0.6

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20

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C rnort d

C growth

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Growth C mort Detritus Feeding C decomp

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Figure 8.10

A fishpond model formulated in M a d o n n a .

N o t e t h a r in a way s i m i l a r r o t h e p r e v i o u s m o d e l , w e h a v e d e s c r i b e d t h e s c e n a r i o o f f e e d i n g as a s e c o n d - o r d e r p o l y n o m i a l - e x c e p t t h a t t h i s time, w e h a v e f o r m u l a t e d i h e p o l y n o m i a l in a s o m e w h a t d i f f e r e n t way. I n s t e a d o f A * T ' + B * T + C , w e u s e t h e formula A * ( T + B ) ' + C - By rearranging t h e coefficients, w e get a m u c h b e t t e r h a n d l e o n h o w w e c o n t r o l t h e f o r m o f t h e f e e d i n g c u r v e ( F i g u r e 6 . 1 1 ) . W h i l e in t h e original form o f t h e p o l y n o m i a l t h e role o f p a r a m e t e r s A a n d B was s o m e w h a t u n c l e a r , in t h e n e w f o r m u l a t h e y h a v e a d i s t i n c t i m p a c t o n w h a t k i n d o f f e e d i n g s t r a t e g y w e g e n e r a t e . T h i s is o n e o f t h e e x a m p l e s o f h o w b y u s i n g t h e right a p p r o x i m a t i o n w e g e t a b e t t e r c o n t r o l o f t h e c h a n g e s t h a t w e try t o i n t r o d u c e t o o u r s y s t e m . O f c o u r s e , w e c o u l d a l w a y s use t h e g r a p h i c f u n c t i o n t o i n p u t t h e f e e d i n g s c e nario. Like S t e l l a , M a d o n n a allows input o f a g r a p h i c o f a n i n d e p e n d e n t

variable

( F i g u r e 8 . 1 . 2 ) . W e just d r a w i h e l i n e hy d r a g g i n g t h e c u r s o r o r by i n s e r t i n g v a l u e s i n t o t h e t a b l e . T h i s will c e r t a i n l y g i v e t h e u l t i m a t e f l e x i b i l i t y ; n t e r m s o f t h e f o r m o f t h e f u n c t i o n we c a n c r e a t e ; however, every t i m e we wish to modify this f u n c t i o n , w e w i l l n e e d t o d o it m a n u a l l y . T h i s i n v o l v e s o p e n i n g t h e g r a p h i c , c h a n g i n g r h e f u n c t i o n , c l o s i n g t h e g r a p h i c , r u n n i n g r h e m o d e l , s e e i n g t h e results - t h e n r e p e a t i n g t h e w h o l e t h i n g a g a i n . T h i s is n i c e f o r m a n u a l o p e r a t i o n s , b u t t h e r e will b e n o c h a n c e t o use a n y o f t h e o p t i m i s a t i o n a l g o r i t h m s a v a i l a b l e . By d e s c r i b i n g t h e input in a m a t h e m a t i c a l f o r m as a f u n c t i o n , w e h a v e s e v e r a l p a r a m e t e r s t h a t c o n t r o l t h e f o r m o f t h e i n p u t , a n d t h a t c a n b e c h a n g e d a u t o m a t i c a l l y w h e n r u n n i n g o p t i m i z a t i o n . T h i s is c l e a r l y a n a d v a n t a g e , w h i c h c o m e s a t a c o s t - w e will n e e d t o stay w i t h i n a c l a s s o f c u r v e s t h a t will b e a l l o w e d a s i n p u t t o r h e m o d e l . F o r e x a m p l e , n o m a t t e r h o w w e

Optimization

327

Changes in t h e feeding straiegy resulting from variations of the three parameters in the equation The role of e a c h c o e f f i c i e n t is clearly seen: A makes the curve either convex or concave, B shifts the graphic either horizontally. :o the right or left. C shifts if vertically, u p or down.

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c h a n g e t h e p a r a m e t e r s in F i g u r e 8 l l w e w i l l n o i b e a b l e t o g e n e r a t e t h e i n p u t as in Figure 8 . 1 2 . T h i s does n o t m e a n t h a t t h e r e ? r c n o o t h e r f u n c t i o n s o u t t h e r e that c a n b e used t o r e p r o d u c e r h e c u r v e in F i g u r e 8 . I 2 . F o r e x a m p l e , by g o i n g t o a p o l y n o m i a l o f h i g h e r o r d e r - say, t h r e e or a b o v e - we will h e a b l e t o g e t p r e t t y c l o s e t o t h e f o r m o f t h e i n p u t t h a t i.s in Figure S . 1 2 . H o w e v e r , t h i s will n o w c o m e at a c o s t o f more parameters, m o r e complexity, lunger c o m p u t a t i o n times, e t c . T h e r e are always trade-offs. S o l e t us hrst c h o o s e s o m e f e e d i n g s t r a t e g y a n d m a k e t h e m o d e l p r o d u c e s o m e r e a s o n a b l e results. W i t h A = 0 . 0 0 1 , B = - ] 0 , C

0 . 2 , a n d t h e rest o f t h e p a r a m e -

ters d e f i n e d a b o v e in t h e e q u a t i o n s , w e g e t t h e f e e d i n g s c e n a r i o a n d t h e fish d y n a m i c s

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^ 0

S

5000 sl«ps in 0.066? si

A feeding scenario and the system dynamics that it produces

Bewate of the population crash that occurs at low oxygen concentrations.

as s h o w n in Figure 8 . 1 3 . W e s e e t h a r t h e fish p o p u l a t i o n gradually grows until a c e r t a i n point w h e r e t h e a m o u n t o( detritus that is e x c r e t e d a n d p r o d u c e d hy t h e d e c o m p o s i t i o n o f feed e x c e e d s a t h r e s h o l d a n d causes massive die-off o f fish. T h e fish p o p u l a t i o n c r a s h e s , further a d d i n g t o r h e detritus pool. Clearly, t h i s is a c o n d i t i o n we w a n t to avoid. S o w e s h o u l d use c a u t i o n w h e n supplying t h e teed i n t o t h e p o n d : t h e r e is always a risk of a fish-kill if we let it grow t o o fast. W i t h t h e e x i s t i n g s c e n a r i o we c a n see t h a t t h e r e is n o h u g e a c c u m u l a t i o n o f unused feed w h i l e t h e fish are srill present, so we c a n c o n c l u d e that in rhis c a s e t h e fish-kill is really caused by p r o d u c t s of fish m e t a h o l i s m . n o t o v e r f e e d i n g .

Optimization

329

Feed price

Mortality

Number Figure 8.14

The economic pan of the model thai calculates the profits from fish sales and the

expenses of fish feeding It t h e growth coefficient for fish were smaller, more feed would he c h a n n e l e d to detritus and its a c c u m u l a t i o n c o u l d o c c u r e v e n faster. T h e feeding process is clearly an i m p o r t a n t factor in this e c o s y s t e m p e r f o r m a n c e . If we were t o optimize this syst e m we would seek a feeding s c e n a r i o t h a t would produce t h e highest fish biomass at a c e r t a i n p o i n t , so t h a t fish c o u l d be harvested at that t i m e a n d sold for a profit. However, ii could be a little more c o m p l e x t h a n t h a t , s i n c e leed is also n o t c h e a p and h a s t o be purchased at a cost. S o it is more likely t h a t we would be optimizing for t h e n e t profit rather t h a n just rhe total fish biomass. Let us build an e c o n o m i c submodel that will take c a r e o f all t h e s e additional processes and flows of money. T h e M a d o n n a diagram is shown in Figure 8 . 1 4 N o t e that in addition to t h e ghost state variable for fish, we h a v e two s t a t e variables: o n e to track t h e n u m b e r o f fish in t h e pond and t h e o t h e r o n e t o c a l c u l a t e t h e total profit from t h e p o n d o p e r a t i o n . T h e n u m b e r ol fish is needed to k e e p track o f t h e average weight o f t h e fish. W e want t o take i n t o a c c o u n t t h e fact t h a t larger hsh with higher weight are more likely to cost more o n t h e market A n o t h e r important parameter t h a t we c a n introduce a n d use as a c o n t r o l is t h e T i m e _ o f _ s a l e parameter, w h i c h tells us w h e n e x a c t l y we will harvest t h e fish and sell t h e m . T h e new M a d o n n a e q u a t i o n s are: {Reservoirs}

d/dt (Tolal_protil) =-- + Profit INITTotal_profit = 0 d/dt (Number) = - J 3 INIT N u m b e r = 1 0 0 (Flows) Profit = R e v e n u e - C o s t J 3 = if W e i g h t > 0 then Mortality/Weight e l s e N u m b e r / D T ( L o s s m n u m b e r ot f i s h w h e n they diel

" 330

' - ' ^ • p M W in im «• Systems Science and Modeling for Ecological Economics (Functions} Fish„Price Feed_price

1 0 + 0.2* Weight • 2

R e v e n u e — if Time > T i m e _ o t _ s a l e A N D T i m e < T i m e _ o f _ s a l e + 2 t h e n P s h _ PriceFISH else 0

Time_of_sale - 1 0 0 W e i g h t = if N u m b e r > \ t h e n F i s h / N u m b e r e l s e 0 Cost -

Feed_pnce*Feeding

By i n t r o d u c i n g

t h e T i m e _ o t _ s a l e p a r a m e t e r , we h a v e also modified t w o ear-

lier e q u a t i o n s t o d e s c r i b e t h e h a r v e s t o f frsb a n d t o stop f e e d i n g after all t h e fish is harvested:

F e e d i n g - if (Time < T i m e _ o f _ s s l e and C _ f e e d > 0) t h e n C _ f e e d e l s e 0 Mortality = if (TIME > T i m e _ o f _ s a l e + i ) t h e n Fish/DT e l s e (C_mort - Detritus A 4/(C m a r t _ d 4 -+ D e c r i t u s A 4 ) r F i s h

N o w we are ready t o s e t up t h e O p t i m i z a t i o n process. In a d d i t i o n t o che t h r e e c o e f f i c i e n t s in t h e f e e d i n g s c e n a r i o ( A , B , a n d C ) , let us also use t h e T i m e _ r i f _ s a l e p a r a m e t e r as a c o n t r o l . W e h a v e already realized that a d d i n g m u c h teed is hardly a g o o d strategy, so p r o b a b l y A is n o t g o i n g to be large - o t h e r w i s e o v e r t h e 1 0 0 - d a y t i m e period we m a y get rjuite h i g h c o n c e n t r a t i o n s ol feed, w h i c h will b e d a m a g i n g to t h e system. L e t us s e t t h e l i m i t s f o r A as 0 < A •< 0 . 0 0 1 . N e g a t i v e n u m b e r s are also e x c l u d e d , s i n c e we already u n d e r s t a n d t h a t it m a k e s l i t t l e s e n s e to add m o r e feed at first, w h e n r h e fish b i o m a s s is l;iw, t h a n later o n , w h e n i h e i e is m o r e fish to c o n s u m e t h e feed. S o most likely t h e w i n n i n g strategy wili start low a n d t h e n i n c r e a s e t o m a t c h t h e d e m a n d s of t h e growing fish. L e t us l e a v e s o m e r o o m for B: - 3 0 < B < 2 0 . A s we l e m e m h e r , B p l a c e s t h e m i n i m u m p o i n t o n t h e c u r v e r e l a t i v e to T I M E — 0 . W e d o n o t n e e d a very large inteiv.il lor C , w h i c h d e s i g n a t e s t h e m i n i mal v a l u e ( i f A is p o s i t i v e ) o r t h e m a x i m u m ( o t h e r w i s e ) . L e t us set 0 < C < 4Nuw, by c h o o s i n g t h e guess values s o m e w h e r e w i t h i n t h e s e ranges, s e t t i n g t h e goal f u n c t i o n t o be m i n i m i s e d t o

T o t a L p r o t u , a n d pressing che O K b u t t o n , we c a n

scare che o p t i m i z a t i o n a l g o r i t h m

This will return a v a l u e of T n t a L p J O t i t = 2 4 6 5 4 2

a f t e r s o m e 4 0 5 i t e r a t i o n s o f che m o d e l . T h e o p t i m i z e d c o n t r o l p a r a m e c e r v a l u e s a r e T i m e _ o f _ s a l e - 8 0 2 8 9 5 , A = 0 . 0 0 1 , B = - 2 6 . 5 , ami C = 0 . 1 2 2 . T h e dynamics o f Total

profit is are s h o w n in F i g u i e 8 15. W e see t h a t at first we g e t a loss, b e c a u s e we

o n l y Spend m o n e y o n feed p u r c h a s e s , b u t t h e n at t h e e n d , w h e n we finally sell t h e fish, wc e n d up with a profit o f 2 6 4 . U s i n g sliders, we c a n e x p l o r e t h e v i c i n i t y o f t h e o p t i m i z e d c o n t r o l p a r a m e t e r s a n d see thac a p p a r e n t l y , i n d e e d , t h e values identified are d e l i v e r i n g a m i n i m u m t o t h e o b j e c t i v e f u n c t i o n , so t h e r e is n o reason t o e x p e c t t h a t we c a n find a b e t t e r s o l u t i o n . In s o m e c a s e s it is w o r t h while e x p l o r i n g s o m e very d i f f e r e n t areas of t h e c o n t r o l d o m a i n , just t o m a k e sure thac che o p t i m u m we a r e d e a l i n g with is i n d e e d a global a n d n o t a local o n e . It d o e s l o o k , t o r this g i v e n c o m b i n a t i o n o f p a r a m e t e r s , like t h e o p t i m u m d e s c r i b e d a b o v e is g l o b a l . Let

us c h e c k

o u t h o w t h e weight

f a c t o r in fish p u c e a f f e c t s

the optimiza-

t i o n results- W i l l we g e t s i g n i f i c a n t l y d i f f e r e n t results If t h e r e is a h u g e p r e f e r e n c e for really big fish a n d t h e price o f s u c h fish is c o n s i d e r a b l y larger t h a n che price foi small fish:

Optimization

331

parameter

C l e a r l y t h e r e is s o m e d i f f e r e n c e . L e t u s c h a n g e i h e i m p a c t ( h a t w e i g h t h a s o n p r i c e , using t h e f o r m u l a : F i s h _ P r i c e = 1 0 + 2 * W e i g h t . H e r e we h a v e i n c r e a s e d t h e e f f e c t o f W e i g h t hy a n o r d e r of m a g n i t u d e , c h a n g i n g il f r o m 0 . 2 t o 2 . T h e o p t i m i z e d parameters are somewhat different: A = 0 . 0 0 0 7 , B = — 3 0 , C = I 0 7 5 , and T i m e _ o f _ s a l e = 6 7 . T h e T o t a L p r o h t w i t h t h e s e p a r a m e t e r s is 5 7 9 ( f i g u r e s 8 . 1 6 A , 8 . 1 6 B ) . N o t e , h o w e v e r , t h a t t h e o p t i m a l v a l u e r e p o r t e d f o r B is o n t h e l o w e r limit c h o s e n for t h i s c o n t r o l p a r a m e t e r . T h i s s h o u l d c a u s e s o m e c o n c e r n , s i n c e v e r y l i k e l y it m e a n s t h a t t h e a l g o r i t h m w o u l d r a t h e r u s e a y e t s m a l l e r v a l u e f o r B , hut w a s n o t a l l o w e d t o g o t h e r e . L e t us r e l e a s e t h i s c o n s t r a i n t a n d s e t B : - 5 0 < B < 2 0 . Rerunning

the optimization

A = 0.000347, B = - 4 4 ,C -

procedure,

we tind

a different

set o f c o n t r o l s :

l . a n d T i m e _ o f _ s a l e = 8 3 . T h e T o t a L p r o f i t with these

p a r a m e t e r s is 1 5 3 7 . 7 7 - a n a l m o s t t h r e e f o l d i n c r e a s e in c o m p a r i s o n w i t h t h e p r e v i o u s e x p e r i m e n t (Figure 8 1 6 C ) If w e c o m p a r e t h e p e r f o r m a n c e o f this system with w h a t we c a n get from a system w h e r e F i s h _ P r i c e = 1 0 + 0 . 2 * W e i g h t , we will s e e t h a t i n d e e d t h e n e w f e e d i n g s t r a t egy results in fish t h a t a r e fewer b u t larger, so w e c a n take advantage o f t h e higher market

prices

for bigger

fish.

Also

t h a t if w e r e r u n t h a t 0 . 2 * W e i g h t with

note model

if the (rpivm.iuu iijvuid at the IroiwJary OF a. fkiriu-iteftr's CUMWMI or clc,e to it, maLe sure tkal this cwAtralnt is real arid uHpertaKt You iKay ire aMe to rsitMt it atui jind a much Iretter ophnuil uriutunt

t h e larger p a r a m e t e r i n t e r v a l for B ,

- 5 0 < B < 2 0 , we will g e n e r a t e a s l i g h t l y higher

TotaLprofit — 2 4 6 5 7 4

T h e control

parameters

w e e n d up w i t h

will

be

A = 0 . 0 0 1 , B = - 2 7 , C = 0 . 1 2 , a n d Time_of_sale = 8 1 . T h e differences are small, but n o t e w o r t h y . A p p a r e n t l y t h e v a l u e o f B = - 2 6 . 5 w a s still a l i t t l e t o o c l o s e t o t h e b o u n d a r y f o r t h e a l g o r i t h m t o m o v e f u r t h e r b e l o w t o - 2 7 , w h i c h g i v e s a b e t t e r result.

332

S y s t e m s Science and M o d e l i n g for Ecological E c o n o m i c s

Bs02£ZiflULl

Experiments with the fish weight as a factor in the fish price and therefore the total

profit. A. Optimized results 'or low importance of weight in fish pricing Fish_Price = 10 ^ 0.2 * Weight; B. Resu'ts for a higher preference for larger fish Fish_Price = 10 + 2 * Weight. For optimal results we get higher fish weight, while total biomass is actually lower. The optimization process hits a constraint for B. w h i c h is set to B > - 3 0 ; C Removing the constraint allows a better optimal solution. We get a smaller number o f f i s h but with much higher weight, which is rewarded by the objective function.

Optimization

333

S n hy r e l e a s i n g t h e c o n s t r a i n t s we L.an e n d up with B = — 2 7 a n d slightly d i f f e r e n t values f o r rhe o t h e r p a r a m e t e r s , with an o v e r a l l g a i n in t h e o b j e c t i v e f u n c t i o n

-

Total., profit. A l l this w o u l d m a k e s e n s e , a s s u m i n g t h a t t h e m o d e ! is c o r r e c t

Unfortunately,

ii we rake a c l o s e r l o o k at t h e way we p r e s e n t e d t h e trsh n u m b e r s a n d weight in this m o d e l , we m a y very well start w o n d e r i n g . T h e n u m b e r o f fish s h o u l d be an i n t e g e r ; o t h e r w i s e it d o e s n o t m a k e sense, A fish is e i t h e r a l i v e o r dead; we c a n n o i h a v e 8 5 . 6 fish in t h e p o n d . S o far we h a v e ignored t h a t . A t t h e s a m e t i m e , t h e w e i g h t is c a l c u l a t e d as t h e total fish b i o m a s s d i v i d e d by t h e n u m b e r o f fish. T i n s m a k e s sense at the b e g i n n i n g , w h e n w e a r e s t o c k i n g t h e p o n d , but later o n as o n e fish dies it c e r t a i n l y does n o t m e a n t h a r t h e rest o f t h e fish are g a i n i n g w e i g h : . T h e fact that t h e n u m b e r o f fish d e c r e a s e s d o e s n o c imply t h a t t h e r e m a i n i n g fish grow tatter 1

Does

this m e a n t h a t t h e w h o l e m o d e l s h o u l d b e trashed, o r s o m e parts o f it at least c a n be salvaged? First, we s h o u l d realize that a c t u a l l y i f t h e n u m b e r o f fish d e c r e a s e s r h e r e is stili s o m e p o t e n t i a l f o r w e i g h t i n c r e a s e , b e c a u s e t h e r e will b e less c o m p e t i t i o n f o r feed a n d t h e r e f o r e e a c h individual fish will b e e a t i n g m o r e a n d g r o w i n g faster. O n e q u i c k fix t h a t we c a n i n c o r p o r a t e i n t o t h e m o d e l e q u a t i o n s is t o m a k e sure t h a t N u m b e r s are integers. T h i s c a n be a c h i e v e d by using a b u i l t - i n f u n c t i o n I N T . I N T ( x ) returns t h e largest i n t e g e r that, is less o r equal t h a n >;, S o il we write

.3

w e will

r

if

, ._ W e i g h t >0 b

be really s u b t r a c t i n g

, then

INI

something

Mortality Weight

from

else

the variable

Number DT Number

only

when

( M o r t a l i t y / W e i g h t ) is larger t h a n I, a n d irr this c a s e we will be s u b t r a c t i n g I If it is larger t h a n 2 , we will b e s u b t r a c t i n g 2 - a n d so o n . M a k i n g i h i s c h a n g e we d o n o t see a very big d i f f e r e n c e in m o d e ! p e r f o r m a n c e , b u t at least we c a n feel good that we d o n o t h a v e any half-fishes s w i m m i n g a r o u n d in our pin id. S e c o n d , we m a y also n o t e t h a t a c t u a l l y t h i n g s are n o t so b a d w i t h t h e w e i g h t n u m b e r c o n t r o v e r s y . I n d e e d , t h e n u m b e r s in o u r m o d e l d e c l i n e only w h e n t h e total fish b i o m a s s also d e c l i n e s : M o r t a l i t y is c a l c u l a t e d as a n outflow t o r t h e Fish able. S o t h e situation described above, when W e i g h t

is t o i n c r e a s e with

vari-

Number

d e c r e a s i n g , is hardly possible; f i s h will h a v e tu d e c l i n e first, so t h e r e m a i n i n g Fish will be divided by t h e r e m a i n i n g N u m b e r , p r o d u c i n g t h e s a m e r e a s o n a b l e e s t i m a t e for W e i g h t . T h e o n l y p r o b l e m is w h e n t h e fish p o p u l a t i o n is losing w e i g h t but n o t dying. T h i s S i t u a t i o n is riot t r a c k e d by o u r m o d e l , a n d c a n c a u s e us s o m e

trouble.

I n d e e d , d e c r e a s i n g w e i g h t ot t h e p o p u l a t i o n , say d u e t o m a l n u t r i t i o n , in o u r formalism will result in i h e d e c l i n e of t h e N u m b e r i n s t e a d o f W e i g h t , Let us t a k e a c l o s e r look at t h e results o f t h e r e c e n t o p t i m i z a t i o n . O n e of t h e r e a s o n s that B n e e d e d tu b e m a d e s m a l l e r a n d s m a l l e r w a s t o push t h e f e e d i n g c u r v e f u r t h e r t o t h e right, s o t h a r at first w e h a d a pretty long period with a l m o s t n o feed added to t h e p o n d a n d t h e fish p o p u l a t i o n gradually s t a r v i n g a n d , under t h e c h o s e n f o r m a l i s m , d e c r e a s i n g in n u m b e r s

If we plot t h e N u m b e r we will s e e thac o v e r t h e

first 5 0 days o r so it was gradually d e c r e a s i n g f r o m 1 0 0 t o a b o u t 5 0 . O n l y a f t e r t h a t f e e d i n g was started. S o a p p a r e n t l y t h e o p t i m a l s t i a t e g y t h a t was f o u n d was relying o n a s m a l l e r n u m b e r o f fish ir. t h e pond. Let us test this d i r e c t l y a n d add t h e initial

fish

n u m b e r I N I T N u m b e r to t h e list c f c o n t r o l variables t h a r we o p t i m i z e for. N o w we will be o p t i m i z i n g for t h e n u m b e r ol fish t h a t w e s t o c k in t h e p o n d ( I N I T N u m b e r ) , t h e f e e d i n g strategy ( A , B , a n d C ) a n d t h e t i m e o f h a r v e s t ( T i m e _ o f _ s a l e ) .

334

Systems Science and Modeling for Ecological Economics T h e m o r e c o n t r o l p a r a m e t e r s we h a v e , t h e l o n g e r t h e a l g o r i t h m runs it still c o n v e r g e s w i t h a s o m e w h a t a s t o n i s h i n g result: [ N I T N u m b e r

However,

= 1. If we k e e p

o n l y o n e fish in t h e p o n d , b e a r i n g in m i n d t h e e x t r e m e l y high value t h a t we a t t r i b u t e ro fish w e i g h t , we will lit' g r o w i n g this o n e i n d i v i d u a l t o s o m e g i g a n t i c si^es a n d reaping a huge profit ot 2 4 2 0 . W e have c e r t a i n l y learned s o m e things about t h e system a n d a b o u t o p t i m i z a t i o n W e h a v e also identified s o m e areas w h e r e she m o d e l c a n use s o m e

improvements

T h e r e is a p o t e n t i a l p i o b l e m with b o w we m o d e l fish weight. If we want a m o r e realistic m o d e l o f this system, we p r o b a b l y n e e d t o d o it o n a n individual basis, describing rhe lilecycle o f a n individual fish a n d t h e n l o o k i n g at t h e w h o l e pond as an aggregate o f these individuals. O t h e r w i s e , w h e n total fish biomass goes down ;t will be always difficult to attribute, this e i t h e r t o t b e d e a t h of o n e or m o t e individuals in t h e Stock ( i n w h i c h case t h e weight o f o t h e r individuals does n o t c h a n g e ) o r ro a gradual leaning o f the w h o l e p o p u l a t i o n ( w h e n obviously t h e average w e i g h t o f all individuals d e c l i n e s ) T h e s i m p l e s t way t o fix t h e m o d e l will be t o use t h e Fish v a r i a b l e as r h e m e a n weight of fish in t h e p o n d , a n d t h e n t o h a v e t h e N u m b e r v a r i a b l e r e p r e s e n t i n g t h e t o t a l n u m b e r o f fish. T h e n we will b e d o i n g t h e r e v e r s e c a l c u l a t i o n t o get t h e t o t a l fish biomass: we will t a k e r h e Fish a n d m u l t i p l y it by N u m b e r . S i n c e n o w s o m e o f t h e v a r i a b l e s will b e d e f i n e d in units different t h a n c o n c e n t r a t i o n s , we also n e e d t o m a k e c.eitain a s s u m p t i o n s a b o u t t h e size o f d i e p o n d S u p p o s e we are d e a l i n g with l O t n X 1 0 m p o n d , I m deep, so l h e total v o l u m e is 1 0 0 ITI 1 . L e i us see w h a t t h e m o d e l will look like in this c a s e . . h e n e w m o d e l e q u a t i o n s with c o m m e n t s are as follows: {Reservoirs} d/dt I F i s h _ W } = - G r o w t h -

Metabolism

( F i s h _ W is n o w t h e b i o m a s s of a n individual f i s h in kg} INIT F i s h _ W = 0 , 0 1 I W e s t o c k t h e p o n d with

fishes,

d/dt F e e d ! = - G r o w t h - F e e d i n g -

I 0 g each.}

Loss

(The F e e d is t h e c o n c e n t r a t i o n of f e e d in t h e p o n d , kg/m 3 } INIT F e e d - 0 d/dt (Detritus) = t A c c u m -

Decomp

( D e t n t u s is a l s o t h e total c o n c e n t r a t i o n in t h e p o n d , kg/rri 3 } INIT Detritus - 0 . 0 1 (Let u s a s s u m e that at first t h e p o n d is realty c l e a n , s o w e h a v e only I Q g of d e t r i t u s in e a c h rrifj d/dt (TotaLprofit) -

-Profit

INlTToial_profir - 0 d/dt ( N u m c e r l = - M o n ( N u m o e r is t h e n u m b e r of f i s h s t o c k e d in t h e p o n d

A s f i s h m a y die, their

n u m b e r m a y d e c r e a s e . W e a s s u m e that d e a d f i s h a r e picked up a n d do not add t o t h e D e m t u s p o o l } INIT N u m b e r - 1 0 0 (Flows! G r o w t h - if

Number > 0.5

then

(1-Fish_W/10i*C_growth"Feed*Fish_W/(Feed

-

C_Hs) else 0 (There is a limit l o h o w big a f i s h c a n g r o w W e a s s u m e that this s p e c i e s d u e s not g e t B i g g e r than 1 0 k g . T h e condition on N u m b e r died at l e a s t t h e y d o n o i c o n t i n u e t o g r o w in s>ze.}

to m a k e s u r e that r al fish

Optimization

335

F e e d i n g = if (Time < T i m e _ o f _ s a l e a n d C J e e d > 0} then C J e e d e l s e 0 ( S a m e c l a m p i n g on t h e f e e d i n g s c e n a r i o t o m a k e s u r e it n e v e r s t a r t s t o extract feed from p o n d ) Methabolism = C_mort*Fishj/V Loss = C J o s s ' F e e d + Growth* 0 Accurn - Loss + M e t a b o l i s m * N u m b e r / 1 0 0 {The fish m e t a b o l i s m p r o d u c e s detritus. T h e r e a r e " N u m b e r " of fish, s o w e multiply by N u m b e r . The size of t h e p o n d is 1 0 0 m 3 . s o w e divide by 1 0 0 t o g e t concentration.} Decomp = C_decomp*Detritus Profit = R e v e n u e - C o s t Mort = INTiif (TIME > T i m e _ o l _ s a l e + 1 ) then Number/DT e l s e •!Detntus A 4/(C_ m o r t _ d A 4 + Detritus A 4 j } * N u m b e r ) {Functions} C_growth = 0.5 C_m = 0.02 CJoss = 0 1 C j e e d = A * n i M E + B l A 2 + C, C_mort_d - 2 C_dec:omp = 0.2 Fish.Price = 1 0 + 2*Fish_W Feed^ptice = 2 R e v e n u e = if

Time > T i m e _ o f _ s a l e

A N D Time < T i m e _ o f _ s a l e + 2

then

Fish_

Pnce'Number else 0

Time_of_sale = 100 A = 0

B - 0.04 C = -1 C_Hs - 0.3 Cost = Feed_price "Feeding A c t u a l l y , this model turns o u t t o be m u c h b e t t e r b e h a v e d and seems t o p r o d u c e e v e n more r e a s o n a b l e results. You may n o t i c e in t h e future, w h e n building many more m o d e l s of your o w n , t h a t t h e b e t t e r your m o d e l gets, t h e more

reasonable

b e h a v i o r it produces. In a way, t h e first i n d i c a t o r t h a t most likely t h e r e is s o m e t h i n g wrong e i t h e r w i t h t h e logic o r t h e formalism in your m o d e l is w h e n you start g e t t i n g s o m e t h i n g totally u n e x p e c t e d a n d hard t o interpret. Running produces

the

optimization maximum

in this

model

TotaLprotit =

2 5 7 2 with A = 0 . 0 0 0 3 2 , B = - 1 9 . 4 3 , C = 0 . 0 0 0 3 4 , and T i m e _ o f _ s a l e = 6 7 (see Figure 8 . 1 7 ) . T h e time o f harvest is picked carefully to c a t c h

t h e m o m e n t when

detritus

approaches t h e threshold and starts t o put the

fish

population

'The better

THE

-w>deiyov. VVJUA, five,

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at risk o f e x t i n c t i o n .

A possible gain o f a few grams m hsh body weight is offset by more and more fish dying, a n d the size o f the stock rapidly decreasing. T h e feeding s c e n a r i o is q u i t e sensitive to t h e m e t a b o l i s m rate used in t h e m o d e l , t h e C _ m parameter. If we c h a n g e C _ m from 0 . 0 2 t o 0 . 0 1 , t h e o p t i m i z a t i o n results

336

Systems Science and Modeling for Ecological Economics

•

F i s h _ p o n d _ n e w 2 - R u n 1: C . f e e d , N u m b e r v s . TIME "1=3 0 @ Ri.nl 5000 SIMS in 0.0S33 seconds

2.5

TIME

•\C0 T - ,

'rrfffl o ~ h ;

Run i 5000 steps in 0 OS33 seconds 9

Figure 8.17

Result lor a modified model that tracks the biomass of an individual fish.

c h a n g e q u i t e d r a m a t i c a l l y ( F i g u r e 8 . 1 8 } N o w , w i t h a lower rate of m e t a b o l i c

loss,

t h e a c c u m u l a t i o n o f d e t r i t u s o c c u r s m o r e slowly a n d it n e v e r r e a c h e s t h e c r i t i c a l c o n d i t i o n s t h a t m a y c a u s e a ftsh die-off. T h e r e f o r e , t h e o p t i m i z a t i o n works o n l y t o try to get t h e fish weight t o as h i g h a value as possible, s p e n d i n g t h e least o n feed. N o t i c e t h a t t h e f e e d i n g strategy n o w is s i g n i f i c a n t l y d i f f e r e n t from what wc h a v e b e e n g e t t i n g before. W e e n d up w i t h a h i g h e r T o t a L p r o f i t = 2 9 4 7 w i t h A = - C . 0 0 0 0 6 4 ,

B =

- 3 2 . 4 , C = 0 . 2 4 8 , and T i m e _ o f _ s a l e = 7 4 . C e r t a i n l y , if w e were t o apply t h e s e m o d e l i n g and o p t i m i z a t i o n t o o l s t o s o m e reallife system we w o u l d b e c o n s t r a i n e d by a c t u a l m o n i t o r i n g d a t a , a n d t h e m o d e l parameters w o u l d he m e a s u r e d in s o m e e x p e r i m e n t s . T h e m a i n purpose o f t h i s e x e r c i s e

Optimization

337

3 i S ^ F i s h _ p o m l j i e w 2 - Run »:C_feed, Number vs. TIME I B S

^SHiSBBfiBlBlEr

Run 1 5000 steps in 0 I seconds 100

-0 OS TIME |[CJ««d||Nuini)«r1 Run

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r

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1

f

40

50

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r

, 70

: r

i

r ^ i

80

90 100

L

10

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TIME |Feedj|Fi$h_W

Figure 8.18

To>(J_preli!

I

A very different feeding slraiegy works heifer when l h e metabolic rate is low. The

optimum is reached w h e n w e get lhe fish weight to as high a value as possible, spending the least on feed.

h a s b e e n t o s h o w h o w o p t i m i z a t i o n works a n d h o w it c a n b e used t o d e r i v e possibly t h e best s t r a t e g i e s for m a n a g i n g systems. T h e r e is h a r d l y a n y o t h e r way in w h i c h , by m e a n s o f r e a s o n i n g o r e x p e r i m e n t , we c o u l d m a t c h t h e e f f i c i e n c y o f r h e o p t i m i z a t i o n m a g i c , w h e n in a m a t t e r o f s e c o n d s o r m i n u t e s h u n d r e d s a n d t h o u s a n d s o f s c e n a r i o s a t e c o m p a r e d a n d t h e best o n e s a r e c h o s e n . W e h a v e also s e e n that t h e r e a r e always c a v e a t s a n d u n c e r t a i n t i e s that n e e d ro b e carefully analyzed a n d realized w h e n m a k ing t h e real m a n a g e m e n t d e c i s i o n s .

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339

100.00 90.00

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scenarios Figure 8 . 1 9

Nitrogen loading and concentration of nitrogen in the Patuxsnt River under different

scenarios o! landuse. The different patterns o l landuse result in dilferent loading factors for nitrogen and, as a result, produce different leve's ot dissolved nitrogen in t h e esluary

c e l l t o m a k e t h e c h a n g e 7 I! it is just u n e c e l l t h a t we m o d i f y , w h e r e s h o u l d t h i s e e l he l o c a t e d ? Besides, while parameters c a n be c h a n g e d continuously, maps - especially landuse m a p s - a r e c a t e g o r i c a l . I t m e a n s t h a t n u m b e r s o n a l a n d u s e m a p s t a n d for d i f f e r e n t l a n d u s e t y p e s - f o r e x a m p l e , l is f i n e s t . I is c o r n . 3 is w h e a t , e t c B u t it c o u l d b e a l s o a n y o t h e r w a y r o u n d - 5 is forest, I is c o r n , 2 is w h e a t , e t c . It r e a l l y d o e s n o t m a t t e r h o w WE c o d e t h e d i l f e r e n t l a n d u s e t y p e s . T h e r e f o r e , a " l i t t l e " c h a n g e OR t h e map does n o t m e a n much. W e c a n change

I co 2 o r t o } o r t o 9 9 , b u t in e f f e c t we

will b e c h a n g i n g o n l y o n e l a n d u s e type o n t h e m a p . I n s t e a d ot c o n t i n u o u s c h a n g e s , we g e t d i s c r e t e variations ot o u r c o n t r o l m a p . T h e r e a r e n o g o o d o p t i m i z a t i o n m e t h o d s i o r t h i s k i n d o f t a s k . L e t us l o o k a t t h e e x a m p l e o f a m o d e l b u d t f o r t h e H u n t i n g C r e e k w a t e r s h e d , w h i c h is l o c a t e d witLt.fi C a l v m C o u n t y in M a r y l a n d , U S A . T h e Z 2 . 5 - k m " w a t e r s h e d b e l o n g s t o t h e d r a i n a g e b a s i n o f t h e P a r u x c n t R i v e r ( 2 , 3 5 6 k m " ) , w h i c h is o n e o f t h e m a j o r t r i b u t a r i e s o f C h e s a p e a k e B a y , S o i l t y p e s a r e w e l l d r a i n e d , m o s t l y s e v e r e l y e r o d e d soils t h a t h a v e a d o m i n a n t l y s a n d y - c l a y l o a m t o fine s a n d y l o a m s u b s o i l . T h e a n n u a l r a i n f a l l v a r i e s b e t w e e n 4 0 0 a n d 6 0 0 m m . M a i n l a n d u s e s of t h e w a t e r s h e d a r e f o r e s t a n d a g r i c u l t u r a l habitats

R a p i d p o p u l a t i o n g r o w t h , d e v e l o p m e n t a n d c h a n g e in l a n d u s e a n d land

c o v e r h a v e bet. u n i t o b v i o u s f e a t u r e s of t b e l a n d s c a p e .

340

Systems Science and Modeling for Ecological Economics T h e ecosystem model we will look ac is an i m p l e m e n t a t i o n o f the P a t u x e n t m o d e l ( P L M ) for t h e smaller sub watershed. It covers t h e hydrologic processes ( a b o v e ground, in t h e unsaturated soil zone and in groundwater), plant growth a n d n u t r i e n t cycling. A n important featuie o f the model is that it is grid-based. M a n y regional

models

assume spatial aggregation to larger units, called e l e m e n t a r y landscapes, e l e m e n t a r y watersheds, e l e m e n t a r y areas o f pollution or hill-slopes. T h e s e units are considered to he h o m o g e n e o u s , and form t h e basis for t h e hydrologic flow network. I f we are to c o n sider scenarios of landuse c h a n g e , generated by che e c o n o m i c considerations, w h i c h were n o t e n v i s i o n e d in t h e design o f t h e e l e m e n t a r y spatial units, this approach may be inappropriate. T h e boundaries b e t w e e n spatial units are fixed and c a n n o t be modified during t h e course o f the simulation, w h i c h may be s o m e w h a t restrictive. In t h e model we use, t h e landscape is p a r t i t i o n e d i n t o a spatial grid o f square unit cells. T h e landscape is described as a grid o f relatively small h o m o g e n e o u s cells, and simulations are run for e a c h cell with relatively simple rules for material fluxing b e t w e e n t h e cells. T h i s a p p r o a c h requires e x t e n s i v e spatial data sets and high c o m p u tational capability in terms o f b o t h storage and speed. However, t h e approach allows quasi-continuous

modifications

o f t h e landscape,

where

habitat

boundaries may

c h a n g e in response to s o c i o - e c o n o m i c transformations. T h i s is o n e o f the prerequisites for spatial optimization analysis, since it allows modification o f t h e spatial arrangem e n t o f the model endogenously, o n t h e fly, during t h e s i m u l a t i o n procedures. A s described in C h a p t e r 6, with t h e S M E a p p r o a c h t h e m o d e l is designed to s i m u l a t e a variety o f ecosystem types using a fixed model structure for e a c h h a b i t a t type. T h e m o d e l captures t h e response o f p l a n t c o m m u n i t i e s to n u t r i e n t c o n c e n t r a tions, water and e n v i r o n m e n t a l inpucs. Ic explicitly incorporates e c o l o g i c a l processes t h a t d e t e r m i n e water levels o r t h e c o n t e n t o f surface water a n d t h e saturated a n d unsaturated soil zone, p l a n t p r o d u c t i o n , n u t r i e n t c y c l i n g associated with

organic

m a t t e r d e c o m p o s i t i o n , a n d c o n s u m e r d y n a m i c s . T h e r e f o r e , t h e s i m u l a t i o n model for a h a b i t a t consists o f a system o f coupled n o n - l i n e a r ordinary differential e q u a t i o n s , solved with a 1-day t i m e - s t e p . Let us n o w formulate t h e o p t i m i z a t i o n task. T h e study area c a n lie described as a set o f discrete grid points R = {(t, ; ) , 0 < nf < i < N, < N ; 0 < m; < j < M < M } (Figure 8 . 2 0 ) . N is t h e n u m b e r o f cells in t h e row, and M is t h e n u m b e r o f cells in t h e c o l u m n . N o t all o f these cells are in che study area. A cell that belongs co che study area is d e n o t e d by ? G R. S i x different landuse types are e n c o u n t e r e d in t h e study area: soybeans, winter wheat, c o r n , fallow, forest, and residential. W e will assume that che residential areas a r e fixed, but otherwise landowners are free to decide what type o f crop to grow in a cell, or w h e t h e r to keep it forested or in fallow. L e t c ( ? ) be t h e landuse ( o r h a b i t a t type) in cell z- T h e c o n t r o l parameters in our case are t h e landuse types thar. are c h o s e n for e a c h cell. T h e set o f landuse types will be L = {soybeans, winter wheat, c o r n , fallow, forest}. T h e n Rc — {z € R |

€ L) stands for t h e set o f

grid points that c a n be c o n t r o l l e d with controls c h o s e n from L. L e t H(c,z) yield o f crop c (if a n y ) harvested from cell

and N(z,c)

be t h e

be t h e a m o u n t of nitrogen

that escapes from c e l l z at t i m e t. T h e o t h e r c o n t r o l decision that farmers c a n m a k e is the a m o u n t o f fertilizer to apply: let F ( c , c ) be the a m o u n t o f fertilizer applied for t h e habitat type c at time c. T h e time o f fertilizer application could be a n o t h e r important c o n t r o l parameter, but let us n o t further c o m p l i c a t e t h e problem, and assume that fertilizers are timed according to t h e existing best m a n a g e m e n t practices and t h e only factor we c a n c o n t r o l is t h e total a m o u n t applied. Qualitatively, our goal is t o find t h e o p t i m u m landuse a l l o c a t i o n and fertilizer application to reduce n u t r i e n t outflow o u t o f t h e watershed while increasing total

Optimization

Figure 8.20

341

The study area for the Hunting Creek model

Only the cells that are on the map will take part in the optimization. The cells in the water category will not change and therefore can also he excluded.

yield. S o t h e o b j e c t i v e f u n c t i o n ( p e r f o r m a n c e c r i t e r i o n ) will need t o a c c o u n t for c r o p yield, fertiliser application and nutrient outflow. T h e first t w o factors are easier to c o m p a r e , s i n c e we c a n operate in terms o f prices. T h e revenue from t h e yield o v e r t h e w h o l e study area is A = £ ;Ef?

pH

(c) H (c,

z)

where p H ( c ) is t h e current market price o f crop c. T h e [-rice of fertilizers applied is t h e n B = i>F£ £ ;SK U K T

f(z,O

where pp is t h e unit price o f n i t r o g e n fertilizer. Obviously, A is t o be maximized while 8 is t o be minimized, w h i c h m e a n s that ( A - 8 ) is t o be maximized. A - B is t h e " e c o n o m i c a l " part o f t h e goal f u n c t i o n . T h e r e a r e different ways o f m o d e l i n g t h e " e c o l o g i c a l " part o f t h e p e r f o r m a n c e c r i t e r i o n . O n e possibility is t o take i n t o a c c o u n t t h e total a m o u n t o f n u t r i e n t s generated by all t h e cells in t h e study area,

c = £

E

*6R l
342

Systems Science and Modeling for Ecological Economics w a t e r s h e d ZQ- T h i s takes i n t o a c c o u n t t h e c o m p e n s a t i o n m e c h a n i s m s o f u p t a k e a l o n g t h e p a t h w a y s o f n i t r o g e n w h i l e it travels across t h e w a t e r s h e d a n d e s t i m a t e s t h e a c t u a l water q u a l i t y in t h e river estuary:

c=E

N

o

IcrcT

In b o t h cases, C is to be m i n i m i z e d . T h e crucial p r o b l e m is t o i n t e g r a t e t h e " e c o l o g i c a l " part C a n d che e c o n o m i c parr A -

B into a scalar objective function. For

this purpose, C h a s t o be e x p r e s s e d in u n i t s t h a t c a n b e c o m p a r e d w i t h t h e d o l l a r m e a s u r e t h a t we h a v e in A — 8 . L e t us a s s u m e t h a t t h e r e is a w e i g h t i n g c o e f f i c i e n t /, w h i c h c a n c o n v e r t o u r C m e a s u r e d in gN/m 2 i n t o dollars, w h i c h we use t o m e a s u r e t h e profit A — 8 . T h e n we c a n f o r m u l a t e t h e goal f u n c t i o n as j = A - B - / C T h e o p t i m i z a t i o n task is: Find maps c

(8.4)

a n d F " w h i c h m a x i m i z e J—> m a x . A s

n o t i c e d a b o v e , che real p r o b l e m for t h e o p t i m i z a t i o n a l g o n t h m is t o figure o u t h o w to find every n e x t b e t t e r c o m b i n a t i o n o f p a r a m e t e r s to f u r t h e r i m p r o v e o u r result. W h e n we were d e a l i n g w i t h n u m b e r s t h e r e w e r e s e v e r a l m e t h o d s , c h e m o s t o b v i ous o f w h i c h is t o c o n t i n u e t h e t r e n d . T h a t is, i f we start to c h a n g e a p a r a m e t e r in a c e r t a i n d i r e c t i o n (say d e c r e a s e or i n c r e a s e its v a l u e ) we s h o u l d stay o n t h i s c o u r s e as long as t h e results c o n t i n u e t o i m p r o v e . O r we c o u l d follow t h e g r a d i e n t . T h a r is, c h e c k a p a r a m e t e r c h a n g e in o n e d i r e c t i o n ( i n c r e a s e ) , t h e n t h e o t h e r ( d e c r e a s e ) a n d see w h e r e t h e o b j e c t i v e f u n c t i o n p e r f o r m s b e t t e r (say, h a s t h e m i n i m a l v a l u e ) . T h a t will b e t h e p a r a m e t e r v a l u e t h a t we t a k e as o u r n e x t a p p r o x i m a t i o n . B u t h o w d o we d o t h a t in c a s e o f maps, especially c a t e g o r i c a l maps? If we c h a n g e d from i t o 2 g o i n g from s o y b e a n s t o w m c e r wheac, we c o u l d c o n t i n u e co 3, w h i c h is c o r n . B u t chat would h a v e little sense, s i n c e 3 may h a v e also b e e n forest or fallow. W e just c h o s e t h a t value of 3 t o r e p r e s e n t c o r n . T h e r e is n o real reason t h a t a 3 a n d n o t a 4 should represent c o r n . T h e r e is n o such t h i n g as a n i n c r e a s e or d e c r e a s e o f a c a t e g o r y value: we are s w i t c h i n g to a d i f f e r e n t landuse only, t h e n u m b e r itself h a s n o m e a n i n g . W e may h a v e easily used letters instead o f n u m b e r s o n t h e m a p . W e e n d up with a s o - c a l l e d c o m b i n a t o r i a l optimizaCion p r o b l e m . T o g e t t o t h e s o l u t i o n , we really need t o sort t h r o u g h all t h e possible c o m b i n a t i o n s o f t h e five poss i b l e landuse types o v e r t h e study area. T h e n u m b e r o f possible c o m b i n a t i o n s for t h e task in ( 8 . 4 ) d e p e n d s o n t h e size o f t h e study area. F o r e x a m p l e , foi t h e H u n t i n g C r e e k watershed, w h i c h is r e p r e s e n t e d by

| = 1 6 8 1 c o n t r o l l a b l e cells o f 2 0 0 X

2 0 0 n r with five possible landuse types, we g e t I ( — 5 1 6 8 1 d i f f e r e n t p a t t e r n s of landuse a l l o c a t i o n . R e m e m b e r chat for e a c h o f t h e s e landuse maps we will n e e d t o run o u r m o d e l for a t least 5 5 0 days t o c o v e r t h e v e g e t a t i o n s e a s o n , i n c l u d i n g w i n t e r t o a c c o m m o d a t e for w i n t e r w h e a t , w h i c h is p l a n t e d in t h e fall b u t grows in t h e spring. O n a h i g h - e n d w o r k s t a t i o n , t h e m o d e l cakes a b o u t 3 m i n u t e s t o run. O n t o p o f t h a t we also w a n t t o test for various fertilizer a p p l i c a t i o n rates, b u t e v e n w i t h o u t that ic is c l e a r l y m u c h l o n g e r t h a n t h e t i m e required t o finish reading this b o o k . A c t u a l l y , t h e age of E a r t h is a b o u t 4 . 5 • 1 0 9 years, a n d we are asking for s o m e t h i n g a r o u n d 6 • l O 1 ^ 1 years. E v e n t h e best s u p e r c o m p u t e r will n o t h e l p us. T h e r e should be a b e t t e r way to solve rhe p r o b l e m . Generally, w h e n m a t h e m a t i c i a n s e n d up with a problem that they c a n n o t solve t h e y start s i m p l i f y i n g it by m a k i n g c e r t a i n a d d i t i o n a l a s s u m p t i o n s a b o u t r h e s y s t e m . L e t us do t h e s a m e for o u r system by t a k i n g i n t o a c c o u n t t h e f o l l o w i n g c o n s i d e r a t i o n s .

iiii^niii. •mil Optimization

ii 343

A f t e r all, t h e landscape operates as a c o m b i n a t i o n o f grid cells, a n d perhaps we c a n assume that d i e c o n n e c t i o n s b e t w e e n t h e s e cells are n o t that i m p o r t a n t . This m e a n s t h a t perhaps we c a n get s o m e t h i n g if we solve t h e o p t i m i z a t i o n problem for e a c h individual cell, a n d t h e n p t o d u c e t h e overall landscape by c o m b i n i n g the tnnduses that we tmd o p t i m a l for t h e s e cells In this case we will n e e d t o define a local o b j e c t i v e f u n c t i o n for e a c h grid cell. T h i s is structurally different, liecause it aims t o m a p the regional goal f u n c t i o n o n t o t h e processes in a grid c e l l . T h e basic idea is t o try t o split our global o p t i m i z a t i o n p r o b l e m , w h i c h is spatial, a n d w h i c h has cells spatially c o n n e c t e d , i n t o a c o m b i n a t i o n o f local optimization problems, ignoring rhe spatial c o n n e c t i v i t y between t h e cells f o r every grid cell, A ( ? ) = l>n(c)

H(c,z)

we define t h e o b j e c t i v e f u n c t i o n as a f u n c t i o n :it z. Ler

be t h e local profit from c r o p yield; D ( z ) = l>r£,F(z,i)

local cosr o f fertilizers applied, and C ( ^ ) = l e a c h e d locally. A(z).

be t h e a m o u n t

be the

of nitrogen

B ( j ) a n d C ( r ) a r e n o w c a l c u l a t e d for a specific cell

They do

nut require i n t e g r a t i o n o v e r t h e e n t i r e study area. Based on this, t h e local goal func-

tion

h"" every cell is t h e n : J(?) = A ( z ) m a x , O n c e we find t h e landuse a n d t h e fertilizer application that is o p t i m a l for e a c h individual cell, we c a n t h e n produce che global solution as a m a p m a d e o f t h e s e local o p t i m a l solutions ( a c t u a l l y two maps: o n e for landuse, t h e o t h e r o n e for fertilizer a p p l i c a t i o n ) . T h e problem is n o w reduced t o optimization of landuse a n d iertilirer a p p l i c a t i o n for every grid cell - but n o w this b e c o m e s feasible

Indeed, assuming h o m o g e n e o u s

landuse and several discrete stages of possible total fertilizer input, -say s i x Mages F £ )0, 2 5 , 5 0 , 7 5 , 1 0 0 , 150kg/ha}» our cask

< |F| |L| - .36 c o m b i n a t i o n s . C o n s i d e r i n g

that n o fertilization takes place lor c € {forest, fallow}, we get B = 2 6 c o m b i n a t i o n s Yes, this approach n e g l e c t s a n y neighborhiK>d effects. W e h a v e also implicitly introduced a n o t h e r assumption - that is, char t h e effect of fertilizer is s m o o t h and c o n t i n u ous, with n o significant thresholds. O t h e r w i s e it would be i n c o r r e c t t o use t h e six-step scale of fertilizer application that we described above- But m a k i n g these assumptions we reduced t h e task t o s o m e t h i n g we c a n easily solve. Indeed we need to run t h e model only 2 6 times and t h e n produce t h e global solution by simply c h o u s i n g the o p t i m a l landuse and fertilizer rate for e a c h c e l l . A c t u a l l y , t h e local task gives us a worst-case s c e n a r i o

In terms of n u t r i e n t out-

flow, the global a p p r o a c h c o u l d take i n t o a c c o u n t rhe r e t e n t i o n capability o f the landscape, w h e n che n e x c cell d o w n s t r e a m captures n u t r i e n t s l e a c h e d from o n e c e l l The k>cal approach n o longer allows that, a n d therefore gives us a worst-case upper e s t i m a t e o f che n e t n u t r i e n t outflow. The solution of t h e local task performs a grid search through rhe entire c o n t r o l space, assuming a homogeneous landuse and identical fertilizer a m o u n t s for each cell. S o what we need is to run the H u n t i n g C r e e k model assuming thar t h e whole area is covered by o n e o f rhe agricultural crops, and d o it six times for each c r o p c h a n g i n g che fertilizer application rate. T h a t will be 4 (landuses) X 6 (fertilizer races) = 2 4 model runs. T h e n , in addition, we run t h e model using an all-forested and all-fallow landuse. T h e s e are not fertilized. T h o s e are rhe 2 6 model runs estimated above T h e s e runs are sufficient to give us che A{z), B ( j ) , C(z) o f rhe local o b j e c t i v e function as maps. U s i n g these, we c a n ca leu lace/{>) for all cells and c h o o s e the maximal value for each cell T h i s solution corresponds to a c e r t a i n landuse and ferriluer race in each cell. Putting these

344

Systems Science and Modeling for Ecological Economics 1404

~•—i—•— corn

—

S C V tiwar — . «*icat - frtl ion 1000 -

«aa

ofla -

<:ao -

a 0.001

Figure 8.21

a 3i

a.

I

:00

Distribution of landuse in the optimal pattern as a f u n c t i o n of the environmental

a w a r e n e s s parameter X. W e start w i t h a fully ploughed w a t e r s h e d all c o v e r e d by soybeans, the most profitable c r o p As / g r o w s , there are l e w e r crops a n d mare fores', i n the w a t e r s h e d .

i n t o a n o t h e r m a p , w e get a s o l u t i o n t o t h e g l o b a l task. T h i s pair o f m a p s c a n b e t h e n fed i n t o a spatial s i m u l a t i o n t o c a l c u l a t e t h e value o f t h e global p e r f o r m a n c e criterion. T b e e s t i m a t i o n of l o c a l o p t i m u m l a n d u s e m a p s d o e s n o t r e q u i r e a n y c o m p u t a t i o n a l e f f o r t . O n c e w e h a v e all p o s s i b l e c o m b i n a t i o n s in t h e m a p s A ( ? ) , B ( ? ) a n d C ( ? ) , w e c a n study h o w w e i g h t i n g p a r a m e t e r A a f f e c t s t h e results. A s y o u m a y r e c a l l , t h i s /. p a r a m e t e r r e p r e s e n t s t h e r e l a t i v e i m p o r t a n c e o r w e i g h t o f t h e e n v i r o n m e n t a l c o n c e r n s , in o u r c a s e m e a s u r e d in t e r m s o f n i t r o g e n c o n t e n t in t h e e s t u a r y . F i g u r e 8 . 2 1 s h o w s t h e r e s u l t s o f a p a r a m e t e r study for t h e H u n t i n g C r e e k w a t e r s h e d p l o t t i n g t h e n u m b e r o f d i f f e r e n t l a n d u s e t y p e s as a f u n c t i o n o f /.. T b e c o r r e s p o n d i n g r e s u l t s f o r t h e t o t a l f e r t i l i z e r a p p l i c a t i o n a r e p r e s e n t e d in Figure 8 . 2 2 . A s w e c a n s e e , w h i l e t h e r a t e o f f e r t i l i z e r a p p l i c a t i o n q u i c k l y d r o p s a s e n v i r o n m e n t a l a w a r e n e s s g r o w s , t h e r e is a l s o a s i g n i f i c a n t c h a n g e in t h e c o m p o s i t i o n o f l a n d u s e in t h e w a t e r s h e d

T h e s e graphs d o

n o t tell us m u c h a b o u t t h e s p a t i a l d i s t r i b u t i o n o f l a n d u s e . L e t us t a k e a l o o k at s o m e of t h e spatial o u t p u t . F i g u r e 8 2 3 s h o w s m a p s o f o p t i m u m l a n d for s e v e r a l /. v a l u e s . W e start w i t h a z e r o v a l u e f o r /., w h i c h is t h e " w h y w o u l d I c a r e a b o u t t h e e n v i r o n m e n t ! " s c e n a r i o . In t h i s c a s e , we get t h e m o n o c u l t u r e s o l u t i o n : p l a n t t h e m o s t v a l u a b l e c r o p ( i n t h i s c a s e s o y b e a n s ) in t h e e n t i r e s t u d y a r e a , w h e r e v e r p o s s i b l e . T h e o n l y o t h e r c e l l s t h a t r e m a i n are t h e r e s i d e n t i a l a n d o p e n - w a t e r o n e s , s i n c e t h o s e a r e n o n - c o n t r o l l a b l e c e l l s . A s we s t a r t i n c r e a s i n g /., s o m e f o r e s t a p p e a r s . T h e m o r e w e g e t c o n c e r n e d w i t h n u t r i e n t o u t f l o w ( a n d push /. u p ) , t h e m o r e f o r e s t will a p p e a r in t h e s t u d y a r e a . A t t h e s a m e time, agncultural cells c h a n g e t o crops with a better nutrient-uptake/yield efficiency. T h i s s u c c e s s i o n o f c r o p s a l s o d e p e n d s o n t h e m a r k e t p r i c e s o f t h e c r o p . It w e w e r e t o run these c a l c u l a t i o n s today t h e results would most likely b e different, b e c a u s e o f t h e j u m p iti p r i c e o f c o r n , i n s t i g a t e d b y g r o w i n g d e m a n d f o r c o r n - b a s e d e t h a n o l .

Optimization

a.ei l a i ' i ' n - I

:

g j J

ni

345

i

Application of fertilizer as a function of /..

Application of fertilizers plummets as environmental concerns about water quality start to dominate

W h a t is also r e m a r k a b l e is that w e c a n a c t u a l l y s e e how, w i t h g r o w i n g I , forests first a p p e a r a l o n g r h e s t r e a m n e t w o r k a n d t h e n gradually spread o u t . T h e r e is n o t h ing in t h e o b j e c t i v e f u n c t i o n t h a t would b e d i r e c t l y r e s p o n s i b l e for r h a t , yet t h e r e is a c l e a r p a t t e r n . C o u l d w e i n t e r p r e t this as y e t a n o t h e r e v i d e n c e o f t h e i m p o r t a n t role o f forest buffers? C l e a r l y , we g e t t h e most " b a n g for b u c k s " w h e n forests a r e l o c a t e d along t h e streams. O f c o u r s e , o n c e t h e g l o b a l s o l u t i o n is p r o d u c e d , b a s e d o n t h e l o c a l o n e , it m a k e s s e n s e t o c h e c k w h e t h e r it is really o p t i m a l in t h e g l o b a l s e n s e

T h e most obvious

way t o d o this is t o disturb t h e s o l u t i o n a n d s e e if t h e results w e g e t a r e c o n s i s t e n t l y " w o r s e " t h a n what t h e o p t i m a l result d e l i v e r s . W e c a n use t h e M o n t e C a r l o m e t h o d a n d r a n d o m l y c h o o s e s o m e c e l l s and r a n d o m l y c h a n g e t h e landuse in t h e m . T h e n w e c a i i run rhis n e w d i s t u r b e d m a p through

the model

and check

il che result gees a n y

b e t t e r t h a n t h a t f o u n d a b o v e . In t h e c a s e of t h e o b j e c t i v e f u n c t i o n t h a t uses water q u a l i t y as a n i n d i c a t o r of e n v i r o n m e n t a l quality, we d o n o t find a n y b e t t e r s o l u t i o n s t h a n t h a t identified by t h e l o c a l a l g o r i t h m . A p p a r e n t l y for this task o u r m o d e l s i m p l i f i c a t i o n was n o t d a m a g i n g in a n y way, a n d we a c h i e v e a s o l u t i o n t o t h e g l o b a l o p t i m i z a t i o n task. F o r n u t r i e n t c o n t e n t , t h e n e i g h b o r h o o d c o n n e c t i o n s s e e m t o be negligible. U n f o r t u n a t e l y , t h e l o c a l m e t h o d d o e s n o t work s o well in all cases. F o r e x a m ple, a n o t h e r way t o a c c o u n t for e n v i r o n m e n t a l c o n d i t i o n s is t o l o o k a t w a t e r s h e d hydrology. A s we h a v e s e e n , c h a n g i n g l a n d u s e types also c h a n g e s t h e i n f i l t r a t i o n a n d e v a p o r a t i o n p a t t e r n s , w h i c h in t u r n a f f e c t h o w m u c h water e n d s u p in surface runoff. Very o f t e n as a result o f d e f o r e s t a t i o n we s e e a n i n c r e a s e in p e a k flow ( t h e m a x i m u m flows a f t e r rainfalls a r e e l e v a t e d ) , w h i l e t h e baseflow p l u m m e t s ( t h e flow b e t w e e n rainfalls under dry c o n d i t i o n s ) . I f we try t o i n c o r p o r a t e t h i s c o n c e r n

into

o u r o b j e c t i v e f u n c t i o n by, say, m a x i m i z i n g t h e baseflow, w e g e t a n o t h e r o p t i m i z a t i o n task. If w e try t o apply t h e s a m e set of a s s u m p t i o n s , a n d find a n o p t i m u m using t h e l o c a l a l g o r i t h m , w e m a y be q u i t e d i s a p p o i n t e d to find t h a t t h e c o r r e s p o n d i n g g l o b a l s o l u t i o n d o e s n o t s e e m t o b e o p t i m a l . R u n n i n g s o m e M o n t e C a r l o tests, w e easily

346

Systems Science and Modeling for Ecological Economics

I B • I 1 I I •

Figure 8.23

"Low. Dens_Res" "Converted* "f ALLOW" CORN" "W1NTERWHEA1" SOYBEANS'

(111) (37) (14| (183i (294) (363)

(Hi} (37) (16) (?0fi) (295) (306)

(141) |37| (U) (213) (285) (297)

Change in landuse pattern as a function of /..

find t h a t t h e l o c a l s o l u t i o n c a n b e u n p r o v e d by c h a n g i n g s o m e c e l l c a t e g o r i e s o n t h e m a p . S o t h e m e t h o d is n o t universal a n d does n o t work for all systems a n d o b j e c t i v e f u n c t i o n s . H o w e v e r , w h e n ir d o e s work, it produces a v e r y fast a n d e f f i c i e n t way t o find t h e o p t i m a . F o r e x a m p l e , in t h e s a m e system if we w e r e t o o p t i m i z e for N P P ( n e t p r i m a r y p r o d u c t i o n - a n o t h e r i m p o r t a n t p r o x y used in e c o s y s t e m s e r v i c e s a n a l y s i s ) , we would find t h e m e t h o d w o r k i n g v e r y nicely.

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348

Systems Science and Modeling for Ecological Economics g r o w r o o t s co g e t m o r e w a t e r f r o m c h e g r o u n d . T h e y m a y e v e n s h e d s o m e l e a v e s co c u t o n e v a p o t r a n s p i r a t i o n . W h e n c o n d i t i o n s a r e f a v o r a b l e , p l a n t s will g r o w l e a v e s . If t h e r e is n o t e n o u g h l i g h t , t h e y will try t o g r o w t h e t r u n k a n d t h e b r a n c h e s , t o g e t h i g h e r up towards t h e s u n s h i n e . A l l t h e s e m e c h a n i s m s c a n b e pretty hard t o describe a n d model. R a c s k o a n d S v i r e z h e v c a m e up with a n i n t e r e s t i n g idea to m o d e l based o n o p t i m a l l y principles ( R a c s k o ,

this

1978).

S u p p o s e t h e p l a n t h a s a g o a l , w h i c h is t o g r o w a s m u c h as p o s s i b l e u n d e r t h e e x i s t i n g c o n d i t i o n s . I f t h a t is t h e c a s e , t h e n o n e a c h t i m e - s t e p t h e n e w fuel s h o u l d be d i s t r i b u t e d a m o n g t h e d i f f e r e n t p a r t s o f p l a n t in a w a y t h a t w i l l e n s u r e

maxima)

p r o d u c t i o n o f n e w fuel o v e r t h e n e x t t i m e - s t e p . T h i s m e a n s t h a t f o r e v e r y t i m e - s t e p w e will s o l v e a n o p t i m i z a t i o n p r o b l e m : m a x F ( t , c, cj) t> w h e r e F is t h e n e w l y p r o d u c e d fuel a t t i m e t t h a t is t o b e d i s t r i b u t e d a m o n g branches +

and roots

+ Pj ~ 1•

c

according

to t h e proportions

~ (ci)

leaves,

p — ( p j , [)>, £>3), r e s p e c t i v e l y ,

the vector of ambient conditions

(temperature,

s o i l m o i s t u r e , e t c . ) , a n d q is t h e v e c t o r o f m o d e l p a r a m e t e r s ( p h o t o s y n t h e t i c r a t e , resp i r a t i o n r a t e , e t c . ) . S o a t e a c h t i m e - s t e p w e a s s u m e t h a t t h e a m b i e n t c o n d i t i o n s will b e t h e s a m e as d u r i n g t h e p r e v i o u s t i m e - s t e p , a n d t h e n o p t i m i z e for t h e b e s t d i s t r i b u t i o n o f t h e fuel a v a i l a b l e t o p r o d u c e t h e m o s t fuel d u r i n g t h e n e x t t i m e - s t e p . N o t e that by m a k i n g this assumption about s o m e o p t i m a l l y principle

involved

in r h e p r o c e s s o f p l a n t g r o w t h , w e h a v e e l i m i n a t e d a l o t o f u n k n o w n p a r a m e t e r s t h a t o t h e r w i s e w o u l d h a v e t o be. e i t h e r m e a s u r e d o r c a l i b r a t e d . T h e o n l y p r o b l e m is t h a t in m o s t c a s e s w e d o n o t r e a l l y k n o w w h e t h e r t h e s e o p t i m a l i t y p r i n c i p l e s r e a l l y e x i s t a n d w e a r e r e p r o d u c i n g s o m e real p r o c e s s ( p l a n t s deciding

what to do depending on

e n v i r o n m e n t a l c o n d i t i o n s ! ) , o r w h e t h e r t h e o p t i m a l i t y t h a t s e e m s t o b e in p l a c e is j u s t a n a r t i f a c t o f a c o m b i n a t i o n o f m a n y o t h e r p r o c e s s e s s u c h as t h e o n e s b r o u g h t by t h e e v o l u t i o n a r y process a n d natural s e l e c t i o n m living systems, m a n y of w h i c h we d o n o t r e a l l y k n o w o r u n d e r s t a n d . H e r e is a n o t h e r e x a m p l e . A b o v e , w e h a v e s e e n h o w d i f f e r e n t l e v e l s ot e n v i r o n m e n t a l a w a r e n e s s r e s u l t in different patterns o f landuse distribution. E a c h value of t h e awareness coefficient X c r e a t e s a l a n d u s e d i s t r i b u t i o n t h a t is o p t i m a l in a c e r t a i n s e n s e . T h a t is, d e p e n d i n g u p o n h o w h i g h w e v a l u e e n v i r o n m e n t a l q u a l i t y in c o m p a r i s o n w i t h e c o n o m i c profit, w e g e t d i f f e r e n t p a t t e r n s o f l a n d u s e ( F i g u r e 8 . 2 3 ) . T h i s begs for a r e v e r s e

problem

s t a t e m e n t . W e k n o w w h a t t h e e x i s t i n g l a n d u s e d i s t r i b u t i o n is, Is t h e r e a A t h a t w i l l d e s c r i b e it? O r , in o t h e r w o r d s , c a n w e j u d g e t h e e n v i r o n m e n t a l a w a r e n e s s by t h e landuse pattern observed i n a n area? T o answer this q u e s t i o n , we will need t o c o m p a r e maps from t h e s e t g e n e r a t e d (.luring o u r o p t i m i z a t i o n p r o c e s s w i t h a m a p o f l a n d u s e for, say, 1 9 9 0 , w h i c h is a v a i l a b l e . W e k n o w h o w t o c o m p a r e n u m b e r s . W e c a n figure o u t h o w t o c o m p a r e

many

n u m b e r s at t h e s a m e t i m e . T h a t is w h a t we a r e d o i n g w h e n c a l i b r a t i n g a m o d e l a n d u s i n g a n e r r o r m o d e l . T h i s e r r o r m o d e l is o u r way o f w r a p p i n g u p s e v e r a l

numbers

i n t o o n e t o m a k e t h e c o m p a r i s o n s n e e d e d . H o w e v e r , in t h e c a s e o f m a p c o m p a r i s o n s t h e task b e c o m e s m o r e c o m p l i c a t e d . It is n o t j u s t t h e t o t a l n u m b e r o f c e l l s in differe n t c a t e g o r i e s t h a r w e a r e i n t e r e s t e d i n ; it is a l s o t h e i r s p a t i a l a r r a n g e m e n t . F o r e x a m p l e , in F i g u r e 8 . 2 4 we s e e t h a t t h e m a p o n t h e left h a s t h e s a m e n u m b e r o f b l a c k c e l l s a s t h e m a p o n t h e r i g h t . T h e n u m b e r o f gray c e l l s is a l s o t h e s a m e . H o w e v e r , t h e m a p s o b v i o u s l y l o o k v e r y d i f f e r e n t . O n t h e o t h e r h a n d , m a p s in F i g u r e 8 . 2 5 a l s o h a v e t h e s a m e n u m b e r o f c e l l s in d i f f e r e n t c a t e g o r i e s a n d d o l o o k a l i k e ,

Optimization

Figure 8 . 2 4

349

While the number of cells in different categories can be the same, the maps will look

quite different.

Figure 8 . 2 5

In other cases the number of cells in diHerent categories can be the same, but there will

be no single cell that exactly matches the corresponding cell in the other map - yet the maps will look similar

e v e n t h o u g h n o t a s i n g l e pair o f c o r r e s p o n d i n g c e l l s o n t h e t w o maps h a s m a t c h i n g c o l o r s . S o it is n o t just t h e total n u m b e r o f cells t h a t m a t t e r s , b u t also t h e p a t t e r n , r h e spatial a r r a n g e m e n t , of t h e cells. T h e h u m a n e y e is a pretty powerful t o o l for spatial m a p c o m p a r i s o n s . W e a r e q u i t e g o o d at d i s t i n g u i s h i n g p a t t e r n s a n d finding similar maps, a s l o n g as we h a v e a n agreem e n t o n a c r i t e r i o n for c o m p a r i s o n s . Figure 8 . 2 6 shows s o m e maps that were offered as part ol a survey to c o m p a r e s o m e m a c h i n e a l g o r i t h m s w i t h h u m a n

identification.

M o s r o f t h e a l g o r i t h m s o f m a p c o m p a r i s o n that try t o a c c o u n t for p a t t e r n a r e based o n t h e idea of a m o v i n g w i n d o w wnere, in addition t o a c e l l - b y - c e l l c o m p a r i s o n , w e start l o o k i n g at a n i n c r e a s i n g l y e x p a n d i n g v i c i n i t y o f c e l l s a n d s e a r c h for similarities in t h e s e n e i g h b o r h o o d s , n o t just at t h e c e l l - b y - c e l l c o m p a r i s o n (Figure 8 2 7 ) S o m e of t h e s e m e t h o d s g e t q u i t e c l o s e t o visual c o m p a r i s o n s , a n d c a n b e used for o b j e c t i v e automated map comparisons.

350

S y s t e m s Science and M o d e l i n g for Ecological Economics

I-

I • -

r

v -

- V

i j V

r

1

Z'j-. —r \ .

1

" . v ; -

10

• iFEi gHu^r lei S8i.a2 i6l

Pairs of maps offered for comparison in the survey. Most of the participants said that

pair 4 is the closest

wouid you agree?

Optimization

351

m ra as

M

1

1 11 1

#

I L

11 Figure 8.27

How a moving w i n d o w algorithm works.

We start with a cell-by-cell comparison. Then w e start including the vicinity of a cell and see if there are any matches in the neighborhood. Then w e can gradually expand the w i n d o w to see if there are more matches. The matches in the wider windows get a lower ranking in the overall comparison index. Oo you think you were doing something like this when you where visually comparing the maps in Figure 8.26?

T h i s is e x a c t l y w h a t we need t o s o l v e t h e p r o b l e m : find a n o p t i m u m l a n d s c a p e t h a t will m a t c h a real landuse m a p . U s i n g t h e l o c a l m e t h o d d e s c r i b e d a b o v e , we c a n easily g e n e r a t e t h e w h o l e series o f maps for t h e various values o f /.. B e f o r e w e g e t i n t o t h e c o m p l e x m e t h o d s o f m a p c o m p a r i s o n , let us first find t h e X for w h i c h t h e n u m b e r o f c e l l s in d i f f e r e n t c a t e g o r i e s o f t h e o p t i m a l s o l u t i o n m a t c h t h o s e o f t h e 1 9 9 0 landuse. m a p . T h a t is w h e r e a surprise is w a i t i n g for us. A s w e c h a n g e r h e value o f x , we c a n see h o w t h e n u m b e r o f f o r e s t e d c e l l s gradually increases. A t X = 3 5 0 , we find t h a t t h e n u m b e r s o f c e l l s in all c a t e g o r i e s ( f o r e s t a n d a g g r e g a t e d a g r i c u l t u r e - we did n o t have any information about the actual crop allocations) m both t h e optimal map and t h e landuse d a t a m a t c h . W h a t is really surprising, is t h a t , l o o k i n g at t h e m a p t h a t c o r r e s p o n d s t o this /. - 3 5 0 , we find t h a t ic is n o c just t h e n u m b e r o f c e l l s t h a t we h a v e m a t c h i n g ; it is also t h e p a t t e r n t h a t looks a m a z i n g l y a l i k e . S e e for yourself, l o o k i n g at t h e t w o maps in Figure 8 . 2 8 . E v e n w i t h o u t a n y m a p c o m p a r i s o n algor i t h m s , it is pretty c l e a r t h a t t h e maps h a v e a l o t in c o m m o n . D o e s this m e a n that we h a v e i n a d v e r t e n t l y found a n o t h e r o p t i m a l i t y p r i n c i p l e t h a t , in this c a s e , g o v e r n s landuse c h a n g e ? A r e t h e landuse p a t t e r n s t h a t we c u r r e n t l y h a v e indeed results o f s o m e o p t i m i z a t i o n ? Ic is clearly t o o p r e m a t u r e to j u m p t o this kind o f c o n c l u s i o n . M a n y m o r e case studies should be c o n s i d e r e d a n d a variety o f o b j e c t i v e f u n c t i o n s should be tested t o find o u t if t h e r e is really s o m e t h i n g m e a n i n g ful in this result. H o w e v e r , it does m a k e sense to a s s u m e that indeed h u m a n s apply

352

Systems Science and Modeling for Ecological Economics

1990 landuse

350

H I

"Open_Water"

(113)

(113)

IR^

"Forest"

(11721

(1160)

^ B

"Low . D e n s . R e s "

(14l|

(141)

• I

"Convened"

(37)

(37)

"Cropland"

(481)

(493)

I

I

Figure 8 . 2 8

Comparison of the real 1990 landuse map w i t h the result of optimal landuse allocation

w i t h / = 350 r u e patterns seem to be l e m a i k a b l y close Does this mean that the existing landuse is a result of some optimization process? If w e find this process, w o u l d it help us figure out w h a t can be the future lancuse maps?

s o m e o p t i m a l i t y p r i n c i p l e s in t h e i r l a n d u s e a l l o c a t i o n d e c i s i o n s

A s a matter o f tact,

it s h o u l d n o t b e s u r p r i s i n g a t all t h a t a g r i c u l t u r a l l a n d is a l l o c a t e d in a r e a s w h e r e t h e y i e l d s a n d profits a r e m a x i m i z e d . W h a t is s u r p r i s i n g is t h a c t h e /. f a c t o r a c t u a l l y d o e s play a r o l e . B u t a g a i n , c h a n c e s a r e t h a t it is n o t t h e e n v i r o n m e n t a l c o n c e r n s t h a t s t o p f u r t h e r e x p a n s i o n o f a g r i c u l t u r e b u t s o m e t h i n g e n t i r e l y d i f f e r e n t , a n d it is just t h a t t h e a g r i c u l t u r e s t a y s i n a r e a s w h e r e it is m o s t p r o f i t a b l e w h i l e o t h e r c e l l s g e t t r a n s f e r r e d t o o t h e r landuse. S t i l l , t h i s c o m p a r i s o n g i v e s us a n u m e r i c v a l u e f o r /'. t h a t s u d d e n l y b e c o m e s a meaningful

i n d e x t h a t c a n b e used t o v a l u e c e r t a i n e c o s y s t e m s e r v i c e s . A s n o t e d

a b o v e , t h e u n i t s f o r /. a r e $ / ( g N / m ' ) > s o e v e r y g N / n r t h a c is a l l o w e d t o e s c a p e f r o m t h e land a n d travel t o t h e estuary ol t h e w a t e r s h e d h a s this dollar v a l u e under current landuse conditions

($350}

W e c a n n o w run t h e m o d e l w i t h n o forest, c o m -

p a r e t h e a m o u n t of n i t r o g e n t h a t will b e r e l e a s e d in t h a t c a s e t o w h a t w e h a v e n o w , a n d derive t h e value o f t h e n i t r o g e n - r e t e n t i o n service provided by t h e forest ecosyst e m o n t h i s w a t e r s h e d . S o m e b a c k - o f - t h e - e n v e l o p e c a l c u l a t i o n s c a n tell us t h a t it. a c c o r d i n g t o F i g u r e 8 . 1 9 , t h e d i l f e r e n c e in n i t r o g e n r u n o f f b e t w e e n a n a l l - f o r e s t e d w a t e r s h e d in 1 6 5 0 a n d a n a l l - a g r i c u l t u r a l w a t e r s h e d c a n b e a n o r d e r o t m a g n i t u d e o r m o r e , t h e n e a c h s q u a r e m e t e r o f forest will b e p r o d u c i n g $ 3 , 5 0 0 w o r t h of e c o s y s t e m s e r v i c e . If w e m u l t i p l y t h a t by t h e t o t a l a r e a o f forest in H u n t i n g C r e e k

watershed,

w e will g e t . . . w e l l , a l o t o f d o l l a r s . A n y w a y , o p t i m i z a t i o n is a n e x c i t i n g t o o l t o e x p l o r e

It c a n h e l p us t o u n d e r s t a n d

m a n y o f t h e f e a t u r e s ol t h e s y s t e m s t h a t w e a r e s t u d y i n g . B y r u n n i n g t h e m o d e l s o

Optimization

many

rimes under various conditions,

ir r e v e a l s b e h a v i o r a l

353

patterns that

otherwise

would remain obscure and unexplored.

Further reading A

good

fmmer

on optimization

by D i w c k a r ,

is a book

U. (2003).

to

Introduction

Applied

K l u w e r , 3 3 3 pp.

Optimization.

F o r more details

on the Chernobyl

accident

see the Nuclear

Energy

nea.fr/hanl/rp/chernobyl/allchernobyl.html,

where

Assessment

2 0 0 2 Update

of Radiological

and Health

Impacts.

Agency

at http://www.

website

you car. doundoad

the report

is given at h t t p : / / w w w . u i c . c o m . a u / n i p 2 2 . h t m . M o r e hnlcs can be found

brief account

"Chernobyl:

Ten Years O n " . A good

of Chernobyl.

at http.//www.

chernobyllegacy.com/. is a large body

There

on discounrijio. A brief

of literature

to sicswmabiiiry. can be found

analysis

and how u applies

of discounting

in V y i n o v , A . a n d Farley, |. ( 2 0 0 7 ) . R e c o n c i l i n g

Sustainability,

S y s r c m s T h e o r y a n d D i s c o u n t i n g . Ecological Economics, 6 3 : 1 0 4 - 1 1 3 . A m x h i ' ) interesting is S u m a i t a , U . , W a l t e r s , C . ( 2 0 0 5 ) . In t e r g e n e r a t i o n a l d i s c o u n t i n g : si n e w m i u m v e Ecol

52, 1 35-142.

Econ.

The

IPCC

- The Intergovernmental

respected

bodies

c h / ) . Despite series

on globed

on Climate

it gets from

Change -

the climate

reports available,

T.he "IPCC

in udn'c/t the climate

more efforts Seconil

were

requested

Assessment-

change

ruiy-snyers,

a vast majority the governments

Change

Clmiate

ihey

have

getting

tmxluced

There

grim

increasingly

ai

a

arc cvrorganisa-

and international

1995" is ax-ailabie

most

(hrrp-//wvvv.' ipcc

of scientist.',.

change forecast fuis been from

one of tJie

n probably

on diis planet

of cltrnue

that present the e x i t i n g consensus among

alannmg, while

tions.

Panel

and the future

u arming

all the criticism that

of reports

renrl\ four and

paper

approach.

http://uww.ipcc.

ch/pdf/cl i m a r e - c h a n g e s - 1 9 9 5 / i p c C ' Z n d - a s s e s s t n e t u / 2 n d - a s s e s s m e n t - e n . p d f . A model appears

of a /hhfxmd that can be considered as a mote

complex

version

in S v i r c z h e v . Yii.M., K r y s a n o v a , V P . , V o t n o v . A . A . ( 1 9 8 4 )

a (ish p o n d e c o s y s t e m . Ecoi.Modd/ing, A general description of the Patuxent

of the one presented

here

Mathematical modelling o f

21:315-337.

model

that we used here can be found

in C o s r a n z a , R . , V o i n o v ,

A . , Bonman.s, R . , M a x w e l l , T . , V i l l a , F., V o m o v , H . a n d W a i n g e r , L. ( 2 0 0 2 ) . I n r c g r a r c d e c o logical e c o n o m i c m o d e l i n g o f t h e P a r u x e n c river w a t e r s h e d , M a r y l a n d - Ecological 72 ( 2 ) : 2 0 3 - 2 3 1 . For more

details

on che processes

Bouinans, R., Costanza, R . ( 2 0 0 4 )

and model

structure

Monographs,

see V o i n o v , A . , Fitz, C . ,

M o d u l a r e c o s y s t e m m o d e l i n g . E n v i r o n m e n t s ! .Modelling a n d

Software. 19, 3 : pp. 2 S 5 - 3 0 4 . The

spatial

optimization

Methodology

151/2-3

Modeling,

technique

for Land

was developed

in S e p p e l t R . , V o m o v , A . , 2 0 0 2 . O p t i m i z a t i o n

U s e Patterns Using Spatial

pp. 1 2 5 - 1 4 2 .

Further

Explicit Landscape

are in S e p p e l t

details

Models.

R., Vomov,

Ecological A . . 200.3.

O p t i m i z a t i o n M e t h o d o l o g y for Land U s e P a t t e r n s - E v a l u a t i o n based o n M u l t i s c a l c P a t t e r n C o m p a n i o n . Ecoiogicai Modc'img, v o l 1 6 8 ( 3 ) : Tlie application

of optimality

sertciuo?! of Peter

Racsko,

principles

with

for modeling

Hahirat

217-231.

plant

growth

Yuri S v i r e z h e v as his advisor.

was the topic

The PhD thesis

of the PhD d»s-

is available

only in

Russian Racsko, P ( 1 9 7 9 ) . I m i t a c i o n n a j a model dereva kak elemenra lesnogo biogenocenoza (Simulation Kibern,

m o d e l o f t h e tree g r o w t h , as r h e e l e m e n t

m a r y ) , and was never agricultural after

published

crops a n d modified

a certain

the plant

o f t h e forest b i o g e o c e n o s i s ) .

5 2 , U S S R A c a d e m y o f S c i e n c e s , M o s c o w , pp. 7 3 - 1 1 0 ( I n Russian

This

biological

in any internauonal to include

time become

is described

journals

the reproductive

the mam and only

The principle

organs recipient

(seeds)

with

was further

that plants

of die newly

English

applied

grow

produced

Vop>. sumto

and that

material

m

in R a c s k o , P. a n d S e m e n o v , M . ( 1 9 8 9 ) . A n a l y s i s o f M a t h e m a t i c a l

P r i n c i p l e s in C r o p G r o w t h S i m u l a t i o n M o d e l s , Ecological Mode/lmg, 4 7 : 2 9 1 - 3 0 2 . A good k ' M W of model

comparison

techniques

is offered

by K u h n e r t , M . , V o i n o v . A . , S e p p e l t , R .

( 2 0 0 6 ) . C o m p a r i n g R a s r e r M a p C o m p a r i s o n A l g o r i t h m s foe S p a t i a l M o d e l i n g a n d A n a l y s i s P l i o t o g r a m m e m c Engineering & Remote Sensing, V o l . 7 1 , N o . 8 : 9 7 5 - 9 8 4 -

r

9. T h e Practice of Modeling 9.1

Why models don't work

9.2

Participatory and adaptive modeling

9.3

Open-source, web technologies and decision support

9.4

Conclusions

SUMMARY T h e r e a r e n o f o r m u l a s a n d a l m o s t n o figures m t h i s c h a p i c r , a n d t h i s is b e c a u s e n o w w e a r e figuring o u t w h a t h a p p e n s t o m o d e l s w h e n t h e y a r e b r o u g h t o u t for h u m a n use a n d a c t i o n . T h i s is w h e n w e n e e d t o s p e a k , i n t e r a c t a n d c o m m u n i c a t e m o r e t o e x p l a i n w h y a n d h o w w e b u i l t o u r m o d e l s a n d w h a t use t h e y a r e . W e will l o o k a t a brief history o f global c l i m a t e - c h a n g e m o d e l i n g as a terrible e x a m p l e o f failure r o c o m m u n i c a t e t h e s c i e n t i f i c r e s u l t s in a t i m e l y f a s h i o n , t o m a k e p e o p l e

understand

t h e p o s s i b l e d i s a s t e r s a n d m a k e t h e m a c t a c c o r d i n g l y . P a r t i c i p a t o r y m o d e l i n g will b e t h e n c o n s i d e r e d as o n e p o s s i b l e t o o l o f s t a k e h o l d e r i n t e r a c t i o n t h a t h a s t h e p o t e n tial o f o v e r c o m i n g t h e s e d i s c o n n e c t s b e t w e e n t h e m o d e l e r s a n d t h e s o c i e t y a t large Participatory

m o d e l i n g uses r h e m o d e l i n g process a s a w a y o f j o i n t

u n d e r s t a n d i n g t o build c o n s e n s u s and h e l p m a k e b e t t e r d e c i s i o n s . T h e

learning

and

open-source

paradigm offers a promising framework to support participatory m o d e l i n g a n d o t h e r o p e n and shared decision support tools.

Keywords G l o b a l c l i m a t e c h a n g e , failures o f g o v e r n a n c e , u n c e r t a i n t i e s , l l ' C C , K y o t o P r o t o c o l , Shared

Vision Planning,

scenarios,

model

mediated modeling, C o m p a n i o n

transparency,

modularity,

open-source

modeling,

software,

stakeholders,

Linux,

General

P u b l i c L i c e n s e , gilt e c o n o m y , web tools, i n t e l l e c t u a l property, c o l l a b o r a t i v e r e s e a r c h , c o m m u n i t y modeling, open data, o p e n access publication, watershed m a n a g e m e n t .

9.1

Why models don't work S o n o w we k n o w h o w t o b u i l d a m o d e l , h o w t o m a k e it r u n , h o w t o a n a l y z e it a n d p r o d u c e results. D o e s t h i s m e a n t h a t we a r e ready for s u c c e s s s t o n e s ? U n f o r t u n a t e l y , t h e r e is still o n e e l e m e n t m i s s i n g . H o w d o we m a k e p e o p l e l i s t e n a n d a c t a c c o r d i n g t o t h e f i n d i n g s f r o m o u r m o d e l ? A f t e r a l l , in m o s t c a s e s we w e r e b u i l d i n g t h e m o d e l t o find o u t s o m e t h i n g a n d t o m a k e t h e r i g h t d e c i s i o n s b a s e d o n o u r findings. T h e

355

356

11 W — — P — • • ' " " "HI— I Systems Science and Modeling for Ecological Economics

gist o f m o d e l i n g is t o simplify t h e reality t o i m p r o v e t h e u n d e r s t a n d i n g o f r e a l - w o r l d p r o c e s s e s . But w e d o it for a purpose: we w a n i t o find s o l u t i o n s for t h e real world p r o b l e m s a n d t o m a k e b e t t e r d e c i s i o n s t o i m p r o v e life a n d avoid disaster. O t h e r w i s e , why b o t h e r m o d e l i n g . '

The thoughtless and selfish, indeed, who fear any interference with the enjoyment of the present, will be apt to stigmatize all reasoning about the future as absurd and chimerical. But the opinions of such are closely guided by their wishes. W.S. J e v o n s , 1865

I n m o s t cases, p e o p l e

have preconceived

notions about t h e problem.

They

c o m e t o t h e t a b l e w i t h s o m e ideas a b o u t t h e s o l u t i o n . In m a n y cases, t h e y a r e .so e n t r e n c h e d in t h e i r o p i n i o n s t h a t it b e c o m e s almost i m p o s s i b l e t o find a c o m m o n g r o u n d . W e all build m o d e l s . B u t out m o d e l s a r e d i f f e r e n t . I n e n g i n e e r i n g o r physics, after a l l , w e h a v e t h e " h a r d " s c i e n c e , t h e e x p e r i m e n t a n d , u l t i m a t e l y , t h e system t h a t will e i t h e r w o r k o r n o t . If we w a n t t o m o d e l a bridge, we k n o w t h a t t h e r e will he t h e u l t i m a t e test t h a t will tell us w h e t h e r t h e m o d e l was right o r w r o n g . If t h e bridge c o l l a p s e s , t h e n t h e m o d e l was w r o n g - a n d n o o n e w a n t s t h a t t o h a p p e n . I f we m o d e l w e a t h e r a n d f o r e c a s t a rainfall, we will s o o n k n o w w h e t h e r we w e r e right or wrong; we c a n t h e n adjust o u r m o d e l a n d , most importantly, again t h e s o c i e t y a t large w a n t s o u r m o d e l to b e c o r r e c t . In e c o l o g y or e c o l o g i c a l e c o n o m i c s , t h i n g s g e t messy

First, t h e r e is t h e addi-

t i o n a l u n c e r t a i n t y t h a t c o m e s from h u m a n s b e i n g part o l t h e system. I i u m a n b e h a v ior may b e very c o m p l e x , u n p r e d i c t a b l e a n d h e t e r o g e n e o u s . T h e r e is n o single law ( l i k e N e w t o n ' s law o r t h e law o f gravity in p h y s i c s ) t h a t c a n he d i r e c t l y a p p l i e d t o e v e r y h u m a n a n d will h o l d . T h e s c i e n c e of p s y c h o l o g y is trying t o c o m e u p with s o m e g e n e r a l rules o f h u m a n b e h a v i o r , but we a r e c l e a r l y n o t t h e r e y e t . D i f f e r e n t p e o p l e h a v e d i f f e r e n t p r e f e r e n c e s f o r g o o d s a n d services, h a v e d i f f e r e n t g o a l s a n d a s p i r a t i o n s , a n d d i f f e r e n t levels of e c o l o g i c a l a w a r e n e s s . A l l t h e s e f a c t o r s cause diff e r e n t p a t t e r n s ot h u m a n b e h a v i o r in a n e c o l o g i c a l - e c o n o m i c system, a n d will steer t h e system t o d i f f e r e n t o u t c o m e s . E c o n o m i c s h a s tried t o e x p l a i n e c o n o m i c b e h a v i o r . S o m e 2 0 years ago, t h e a s s u m p t i o n t h a t i n d i v i d u a l s are purely r a t i o n a l , fully i n f o r m e d m a x i m u e r s ( t h e y m a x i m i z e e i t h e r t h e i r u t i l i t y o r p r o f i t ) was a s h a r e d

understand-

ing a m o n g e c o n o m i s t s . H o w e v e r , t h e e v i d e n c e from e x p e n m e n t a l e c o n o m i c s h a s a l t e r e d t h e s e views. A p p a r e n t l y , p e o p l e a r e s o m e t i m e s i r r a t i o n a l , d o n o t possess all t h e i n f o r m a t i o n t o m a k e a d e c i s i o n in a c o m p l e x e c o l o g i c a l - e c o n o m i c e n v i r o n m e n t , h a v e d i f f e r e n t l e v e l s o f risk a w a r e n e s s , a n d so o n . T h i s a c t u a l l y m e a n s t h a t p e o p l e m a y r e s p o n d d i f f e r e n t l y t o t h e s t a t e o f a n e c o l o g i c a l - e c o n o m i c system a n d e x h i b i t d i f f e r e n t s t r a t e g i e s o f r e s o u r c e c o n s u m p t i o n . For e x a m p l e , s o m e A m e r i c a n

Indians

c e r t a i n l y h a d very d i f f e r e n t views o n n a t u t a l resources ( r e m e m b e r i h e s e v e n t h - g e n e r a t i o n p r i n c i p l e w e m e n t i o n e d in C h a p t e r 2 , page 1 4 ) t h a n t h e w h i t e p e o p l e w h o c a m e t o t h e i r land. W h i c h b e h a v i o r m o d e l s h o u l d w e c h o o s e , if we g o b e y o n d o n e i n d i v i d u a l a n d n e e d ro m o d e l r h e e c o n o m i c , b e h a v i o r for a w h o l e r e g i o n o r c o u n try? If t h e r a t i o n a l m a x i m i z i n g individual m o d e l is n o t valid, t h e n w h i c h o n e is? Now, e c o n o m i s t s talk a b o u t h e t e r o g e n e o u s c o n s u m e r m o d e l s , b u t t h o s e a r c far m o r e

The Practice of Modeling

c o m p l i c a t e d , with many m o r e u n c e r t a i n t i e s

357

T h u s , a l t h o u g h in " h a r d " s c i e n c e w e

k n o w for sure that water will flow d o w n h i l l (or t h a t an o b j e c t with a c e r t a i n mass will fall down t o t h e ground with a c e r t a i n speed), we c a n never he 1 0 0 p e r c e n t sure a b o u t h u m a n behavior. It is as if we are m o d e l i n g a budge, n o t k n o w i n g for sure w h e t h e r t h e right a m o u n t of c o n c r e t e will be poured o r w h e t h e r t h e bridge will he a c t u a l l y used for travel or for a r o c k - ' n ' - i o l l c o n c e r t . S e c o n d , h u m a n s are also users o f t h e model

T h e y may h a v e t h e desire a n d

power t o ignore, twist and distort t h e results of t h e model T h e y h a v e rheir o w n priorities a n d vested interests, a n d may " l i k e " some results a n d " d i s l i k e " others. T h e r e are likely t o be parties t h a t d o n o t want o u r model t o produce c e r t a i n results. For e x a m p l e , if you are predicting rainfall and I make a living b e t t i n g o n drought a n d selling sunglasses, I will want your model t o be wrong - o r a t least to m a k e sure t h a t n o b o d y believes in your forecast.

There w a s a w e l l - k n o w n d i s p u t e o e t w e e n Paul Ehrlich a n d Jul",an Simon. Ehrlich, an ecoiogist a n a p r o f e s s o r of population s t u d i e s , f o r e c a s t that s o m e of t h e m a m r e s o u r c e s w o u l d g a m in o n c e over the next several y e a r s . S i m c n . a m a i n s t r e a m e c o n o m i s t , claimed that their price w o u l d drop. His theory w a s that it is not t h e natural r e s o u r c e s that m a y b e a limiting factor, but rather t h e n u m b e r of h u m a n brains that a r e there to s o l v e p r o b l e m s . S o t h e higher t h e world population, the merrier it will be. Eventually, S i m o n b e t that the price of any s e t of r a w materials w o u l d b e lower 1 0 y e a r s f r o m n o w than it is today Ehrlich a n d his s u p p o r t e r s took u p the challenge and, in O c t o o e r 1 9 8 C . c h o s e five m e t a l s : c h r o m e , c o p p e r nickel, tin a n d t u n g s t e n . S i m o n w o n the b e l a s . by O c t o b e r 1 9 9 0 . t h e c o m p o s i t e price index of t h e s e f i v e m e t a l s had fallen by m o r e than 4 0 percent. Note, h o w e v e r , that this w a s not a fair g a m e , s i n c e actually Ehrlich {as weil a s other environmentalists) w a s working hard during t h o s e y e a r s l o try to l o w e r d e m a n d for natural r e s o u r c e s . S o actually h e w a s betting a g a i n s t himself This is just t o illustrate ihat it m a k e s little s e n s e t o predict t h e behavior of o p e n s y s t e m s , w h e r e h u m a n s t h e m s e l v e s a r e likely to c h a n g e t h o s e s y s t e m s . II y o u think that global w a r m ing Is h a p p e n i n g

y o c a r e m o r e likely to d o s o m e t h i n g a b o u t it. S o don't b e t on it. s m c e it is

like'y. chanics to t h e e f f o r t s of yourself a n d p e o p l e w h o think your w a y , that t h e o r o c e s s m a y b e s l o w e d d o w n . You m a y b e betting a g a i n s t yourself. Instead, k e e p o n with the g o o d w o r k

T h e r e c e n t global c l i m a t e c h a n g e saga provides a s p e c t a c u l a r e x a m p l e o f h o w this happens. It dates back almost 2 0 0 years, t o w h e n E d m c M a r i o t t e , H o r a c e B e n e d i c t de Saussure, Fourier and Poulliet did their e x p e r i m e n t s , c o l l e c t e d data and laid t h e f o u n d a t i o n for some t h e o r e t i c a l generalizations. By t h e 1850s, Tyndall was measuring various gases' a b s o r p t i o n - e m i s s i o n behavior. A r i h e n i u s wrapped it all up in his 1 8 9 5 talk to t h e Swedish Royal A c a d e m y and in a subsequent April 1 8 9 6 paper, " O n t h e I n f l u e n c e o f C a r b o n i c A c i d in t h e A i r upon t h e T e m p e r a t u r e o f t h e G r o u n d " (The London, Edinburgh.

and Dublin

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M agazme

arid Journal

of Science).

Thar C O -

c a n c h a n g e t h e absorption right in t h e middle o f Earth's outgoing blackbody spectrum has b e e n understood for a long time. In

1 9 3 8 , G u y Stewart Callendar discovered

t h a t global

w a r m i n g c o u l d be

b r o u g h t a b o u t by increases in t h e c o n c e n t r a t i o n o f a t m o s p h e r i c c a r b o n d i o x i d e d u e to h u m a n a c t i v i t i e s , primarily through b u r n i n g fossil fuels. T h i s is t h e u n t o l d story, just r e c e n t l y d i s c o v e r e d by a h i s t o r i a n , J a m e s F l e m i n g , o f t h e r e m a r k a b l e s c i e n t i s t w h o established t h e c a r b o n - d i o x i d e theory o f c l i m a t e c h a n g e . T h e s e findings were based o n s o m e simplified t h e o r e t i c a l models. T h e n , during t h e t w e n t i e t h century,

358

Systems Science and Modeling for Ecological Economics t h e s e m o d e l s were c o n s i s t e n t l y i m p r o v e d ,

incorporating

m o r e d e t a i l and m e c h a -

nisms, c u l m i n a t i n g in a dozen o r m o r e G l o b a l C i r c u l a t i o n M o d e l s ( G C M s ) d e s i g n e d to u n d e r s t a n d g l o b a l c l i m a t e m as m u c h d e t a i l s as possible a n d t o p r e d i c t its possible future. C u r r e n t l y , t h e r e a r e at least s o m e 2 0 m o d e l s used for c l i m a t e p r e d i c t i o n around t h e world, w i t h t h e s a m e a c r o n y m , they are n o w c a l l e d G l o b a l C l i m a t i c Models. By t h e 1 9 9 0 s , s c i e n t i s t s h a d s t a r t e d t o raise red flags a n d blow all sorts o f w h i s t l e s and horns, trying t o focus t h e a t t e n t i o n o f t h e p u b l i c o n t h e s i m p l e fact t h a t i n c r e a s e d C O ? a n d o t h e r G H G c o n c e n t r a t i o n s leads t o global w a n n i n g , a n d t h a t we are rapidly i n c r e a s i n g t h e a m o u n t o f C C h m t h e a t m o s p h e r e by b u r n i n g huge a m o u n t s o f fossil fuels. It s e e m s s i m p l e - b u t t h e r e is o n e i m p o r t a n t c a v e a t . If this is i n d e e d a p r o b l e m , t h e n t h e most o b v i o u s s o l u t i o n is t h a t we n e e d to burn less fossil fuels; h o w e v e r , thus also m e a n s t h a t w e will n e e d t o c o n s u m e less g a s o l i n e , d r i v e less a n d , c o n s e q u e n t l y , m o s t likely d e l i v e r less profit t o t h e oil c o r p o r a t i o n s . W e h a v e already s e e n in C h a p t e r 7 h o w c o r p o r a t i o n s a n d t h e i r lobbyists c a n rule t h e world. A p p a r e n t l y , this is e x a c t l y w h a t is h a p p e n i n g . T h e B u s h A d m i n i s t r a t i o n , w h i c h is n o t o r i o u s for its links with B i g O i l . h a s d i l i g e n t l y b e e n d o i n g its j o b o f s l a n d e r i n g c l i m a t e c h a n g e r e s e a r c h a n d prev e n t i n g any m e a n i n g f u l m i t i g a t i o n efforts. T h i s m i g h t be c o m p a r e d w i t h a n o s t r i c h s t i c k i n g its h e a d i n t o t h e sand just to ignore t h e fact t h a t there is i m m i n e n t d a n g e r e x c e p t n o w we k n o w t h a t a c t u a l l y n o o s t r i c h would really do t h a t . W h e n o s t r i c h e s feed, they s o m e t i m e s do lay t h e i r h e a d s flat o n t h e ground t o swallow sand a n d p e b bles, w h i c h helps t h e m t o grind t h e food t h a t they e a t ; a l t h o u g h from a d i s t a n c e it may indeed l o o k as t h o u g h t h e bird is burying its h e a d in t h e sand, it is c l e a r l y s m a r t e n o u g h n o t t o d o t h a t w h e n t h e r e is s o m e t h i n g d a n g e r o u s c o m i n g up. W h a t a b o u t the politicians? T h e official U S s t a n c e for t h e past 6 years h a s b e e n that s c i e n t i f i c findings a r e " u n c o n v i n c i n g , " a n d " t o o u n c e r t a i n " t o call for a n y a c t i o n . T h e r e was always s o m e t h i n g missing from t h e m o d e l s . T h a t is n o t surprising. A s we know, m o d e l s are always designed to simplify, t o e x p l a i n . T h e c l i m a t i c s y s t e m is so c o m p l e x t h a t t h e r e will always be c e r t a i n t h i n g s t h a t t h e m o d e l s will n o t c o v e r . Besides, as M a r i k a H o l l a n d , o f t h e N a t i o n a l C e n t e r for A t m o s p h e r i c R e s e a r c h , says, t h e r e are s o m e processes t h a t "are just n o t well u n d e r s t o o d , a n d b e c a u s e o f t h a t h a v e n o t b e e n i n c o r p o r a t e d

into

c l i m a t e m o d e l s " (http://www.cgd.ucar.edu/oce/mholland/). H o w e v e r , it does n o t m e a n t h a t w i t h t h o s e processes i n c l u d e d t h e results will turn a r o u n d . In fact, a c c o r d i n g to D r H o l l a n d , t h e s e a i c e is m e l t i n g foster t h a n m o d e l s h a v e p r e d i c t e d . T h e r e are m a n y reasons for t h e u n d e r e s t i m a t e s . F o r e x a m p l e , m o d e l s d o n o t fully c a p t u r e h e a t transport b e t w e e n o c e a n and a t m o s p h e r e , or faster w a r m i n g as r e f l e c t i v e i c e gives way t o darker, h e a t - a b s o r b i n g waters. A c t u a l l y , it h a s b e e n c o n s i s t e n t l y o b s e r v e d t h a t m o d e l ers t e n d to b e c o n s e r v a t i v e in t h e i r p r e d i c t i o n s , filtering o u t m o d e l s t h a t clearly overe s t i m a t e t h e c h a n g e s s e e n s o far, but a c c e p t i n g t h e results w h e r e e v e r y t h i n g is t o o w e l l - b e h a v e d a n d stable. F o r a n y m o d e l e r , it is o b v i o u s t h a t t h e r e is s o m e t h i n g n o t i n c l u d e d in t h e m o d e l , and t h a t t h e r e is always u n c e r t a i n t y 111 t h e results. D o e s this m e a n t h a t m o d e l s are useless? C e r t a i n l y n o t ! If s e v e r a l m o d e l s , e s p e c i a l l y built i n d e p e n d e n t l y , p o i n t in t h e s a m e d i r e c t i o n , t h e n t h a t is a h u g e r e a s o n for c o n c e r n . If t h e s e m o d e l s are scrupulously tested by third parties, a n d if t h e r e is a s c i e n t i f i c c o n s e n s u s t h a t t h e m o d e l s are c o r r e c t , t h e n we had b e t t e r start to a c t . H o w e v e r , t h a t is w h e r e we find a gap b e t w e e n m o d e l i n g a n d real life. It turns o u t t h a t n o m a t t e r h o w good a m o d e l is, w h e t h e r it will be used f o r t h e b e t t e r m e n t o f h u m a n i t y o r n o t m a y be d e c i d e d by forces t h a t h a v e n o t h i n g t o d o with s c i e n c e or m o d e l i n g .

Th^ Practice oI Modeling

359

Twenty years of denial: a brief history of climate research June 23. fQ&8 J W M Hansen (NASA) testifies to the Senate committee about the greenhouse effect He says he is 99 percent sure that "the greenhouse effect has been detected, sod rt is changing our c*mate r,ow" Componies and industry associations teorcsentiryj petroleum, steel, aytos ana utilities form lobbying groups wfth nfw^cs l.v g the Global Climate Coe'-ition (GCCl and the Information Council on the Environment (ICE) Tne gcal « to 'reposition giobal warming as tneory rathe.then fact." and to sew doubt about climate 'esearch." 'CE ads ask, "if the Earth e atmosphere is not likely to significantly contribute :o the greenhouse effect it's |ust all part of the hoax." In the Newsweek Poll. 42 pe'eent say ihe press "exaggerates the threat of climate change" 1996 William O'Keefe, Vice President of the American Petroleum Institute and Leader of •he GCC. suggests that there i$ too n x c h "scientific uncertainty" to justify CufOS on green house emissions. The "Leipag Declaration on Global Climate Change' a released, where over too Scientists and others including TV weathermen, say they "cannot subsenbe ro the politically inspired world,view that envisages climate catastrophes* Few of the Leipzig s>grors had actually cerned out climate research T997 Kyoto, -apen, over 100 nations negotiate a treaty on making Rio's voluntary and largely ignored greenhov3e curbs mancatory The worried coel ano oil industries rsrnp j p Their message lhaf theie cs too much soentific uncertainty to |ustify any such cuts Tho intergovernmental Panel on Climate Change UPCO - the international body thar periodically assesses climate research - issues its second report Its 2.500 scientists con elude that, although both natural swings and changes in the Sun's output might be contributing to Cj;mate change, "the balance of evidence suggests a discernible human influence on climate" US President Clinton, while a strong suppc-rter o' GHG cuts, does not even try to get the Senate to ratify the Kyoto treaty The Republican Party has a majority n both houses and s in denial. Republicans have also received significantly more campaign cash from the energy and Other mdustnes that dispute climate soencc April 1.998 A dozen people from the Marshall institute. Fred Sanger's group ««3 meet at tne American Fettoleun Institute's Washington headquarters They propose a S5 million

360

Systems Sc wncv « n d M o d o ' i ^ g
campaign to conw.co ir»o puUic wet tho seance ol g t o M w e f ring a ndcted with c M K N V S y and unoenamty January 2V0Q h i e Na'>tr*l Acedomy pt *nr»WX«4 tr*t. contrary 10 i t * c W r thai satellites timing no w»nr*ig qio right and ground stations showing warming are wrong it njin» that tne satellites c o oH f h o wanai n indeed and a: a 'aie. w x e I960 much greorer then in tha pest 2001 Inaugurate* ol Pins-dom George W Bush As acanc*daie ha had pfcdged » cap t a ' b o n dO»ide emi$«ion» h o was ••piKtixJ to l i t e r a t e mm pieck^c- in his spee<^ afte- inew juration The line we» nevoi w d B i n h disavows his campaign plodge, and o March wrttxlrawa from the Cyoio t'eaty Tho IPCC releasee us third assessment ol t h e studies of clWTorV.» r^m, Chapman of 'he Environment Committee »" a 2 n o w soeech. r-e detxrces r e aa-m ot soentj*ic consensus on climate change Despite i f * <*»oc»«rv thai salable data show werrrang, he argues that "satellites, wicefv conanlered tha m o w a e t a t a m a a s ^ e of •joba' temperatures. confirmed - t f e absence o l atmosphere: w * r m n g M g n ' flic6* tm " t h * ? e e * s t '*:>#• w petpetntfed on tha - m e r c a n peoo4e v A r o n c * o e r j a s s parts- underwriter with153,000 hom rhe AW 16 Fatxuaty 7005 T h a Kvo^o Protocol corres te*e atter Russia ' m a f V f i f t y it l h a r a k caretu m a n a j t w r o i i cr i ^ t isoefal y ^ r o s t s and o f f e r s M:» artJ Forma« coal anfl oo ov«4%«e»» maka w e that w r roport and soeec* cast d i r w i e soer tottvj*. w l t r j *o» the Wft.tr N m 4 Council on Environmental Quatit/ edits a 2032 report o r d r r a i a tv "lack of unileistanding" and " c o r t s i d e i i t * u n c e r t a r t y ' o v o u o i k w : m e t e c 200? Tha Democats take tne Congress and r * Serate A| G c « t n t i h e s to D o n c t w r v hors on climate charge, i v o f a tells allies he v»« t-Wwste' any cfr^ale t>« t-at mandate* green house cuts f6l>tv&y 2 0 0 1 The IPCC r e u s e s its Fourtn Assessment ana ' e o o r t l thai .1 n ' "Vry Mealy" |>90 corce^tl heat capping eniis$.or>s i?om human itctnrflies 'iave cau.ied 'moat o* the ohservod increase m gtobai^ r.-et9ged temcwatures w^re the mid /Oth century In the Newsweek Poll. 08 pctceni o< t i o s e su%i £19 million over the y^a-s t o tha CEI ard others who a-a "p«odv-cing very questionable deta" on climate change Senator Jey Pocko'eller say* me company has t o cui back its support Jcr s-jtt groups

Bush announces that he will corwone o gooa- worming 8>jmmit n«»t month, w i f a 2006 goal ot cjRmg greenhouse emissions The Newsweek fbll fmds less than half .>. l a v e o< r e d i n g h.flh-milaage cars oi energy efficient appliances and tu>td.ngs While majorities in fcuiope end Japan ecojnlza a broad

Tho Practice of Modeling

361

consensus omcng cfamote experts that greenhouse gases are altonng donate. *.'•« mtitwiQc •:•' tne denial machine USA remains strong. Arthough l~e figure ts f a n in ear^e- pol» 39 OOf«nt (down Irom 64 percent last yeail say there >s *a lot of disagreement » w . g <*m»'r» scientists" on the b o » question Ol whether the panet is warm ng i2 percent say there ts * tot ol disagreement that human activities are a major cause of gtobi
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WKKKKKKKMWKT ulwww 362 Systems Science and Modeling for Ecological Economics (chough unpopular) decisions quickly and h a s t h e m e a n s to e n f o r c e t h e m . (For e x a m ple, t h e o n e - c h i l d family p l a n n i n g regulation lias managed t o curb population growth; a n o t h e r e x a m p l e is rhe recent decision o l t h e C h i n e s e G o v e r n m e n t to stop all further c o n v e r s i o n o f agricultuial lands for hiofuel production. W h i l e t h e U S A c o n t i n u e s to subsidize hiofuel, unable to o v e r c o m e t h e lobbying power of agricultural corporations, C h i n a has made s o m e very swift a n d timely decisions in this regard.) C e r t a i n l y , this type o l d e c i s i o n - m a k i n g may be efficient - decisions are made and i m p l e m e n t e d quickly

However, t h e downside is that, as m e n t i o n e d a b o v e , we are b e t t i n g o n o n e

W i s e King. Everything may work well as long as he is indeed wise - but if he goes crazy, we have little power t o replace him A l s o n o t e t h a t centralized decisions chat are unpopular are hard to i m p l e m e n t , require m u c h e n f o r c e m e n t , and usually fail. A l t e r n a t i v e l y , we n e e d t o invest heavily in e d u c a t i n g t h e public a n d in creating m e a n s a n d m e t h o d s for public participation

in t h e d e c i s i o n - m a k i n g

process.

R e c o g n i z i n g t h e n e e d t o r e i n f o r c e t h e process with local k n o w l e d g e a n d iterative participatory i n t e r a c t i o n s in order t o derive politically feasible a n d scientifically sound solutions, g o v e r n m e n c s and i n t e r n a t i o n a l organizations h a v e e m b r a c e d c o n cepts of public i n v o l v e m e n t , a n d d e v o l u t i o n of d e c i s i o n m a k i n g t o lower a n d lower levels. For e x a m p l e , t h e S h a r e d Vision P l a n n i n g process that has b e e n d e v e l o p e d in t h e A r m y C o r p s o v e r t h e past 3 0 years is a promising way t o hnd u n d e r s t a n d i n g a n d a c c e p t a n c e a m o n g che various s t a k e h o l d e r s t h a t may be in ceres ced in d i e o u t c o m e s of a p r o j e c t a n d k n o w l e d g e produced by models. T h e n e w web t e c h n o l o g i e s a n d services provide n e w m e a n s o f i n t e r a c t i o n and d i s s e m i n a t i o n o f data and knowledge. A s h u m a n d o m i n a t i o n over t h e e n v i r o n m e n t grows and as t h e complexity of natural systems is lurcher elevated by t h e c o m p l e x h u m a n s o c i o - e c o n o m i c systems builc o n t h e m , d e c i s i o n - m a k i n g processes b e c o m e more constrained by feasible options and time horizons, while t h e c o n s e q u e n c e s o f wrong decisions b e c o m e more dramatic and affect hirgei geographic areas. U n d e r such c i r c u m s t a n c e s , standard scientific activities are inadequate ii we wish co c o n t i n u e o n t h e d e m o c r a t i c path ot d e v e l o p m e n t . T h e y must be reinforced with local knowledge a n d iterative participatory i n t e r a c t i o n s in order t o derive solutions w h i c h are well understood, politically feasible and scientifically sound. W e need new ways t o understand and e m b r a c e the i n c o n v e n i e n t truths of today.

9.2

Participatory and adaptive modeling (This section was written

in collaboration

with Erica Brown

Gaddis.)

A s argued by O r e s k e s et at. ( 1 9 9 4 ) , a n d as we discussed in C h a p t e r 4 , models d o n o r tell us t h e " t r u t h " a b o u t che system. T h e y should be r a t h e r viewed as a process o f striving towards che truth. T h e best model is a process in w h i c h we learn a b o u t t h e system and understand h o w best t o m a n i p u l a t e a n d m a n a g e it A s we stare administering this m a n a g e m e n t , o r as s o m e t h i n g starts co c h a n g e in che e n v i r o n m e n t , che system also c h a n g e s and t h e m o d e l is n o longer valid model

is u'ewed

as a process

W e c a n s u c c e e d o n l y if the

t h a t is designed t o a c c o m m o d a t e t h e s e c h a n g e s and adapt

to t h e m . A g o o d m o d e l should e v o l v e with t h e system; it should be able t o c h a n g e b o t h q u a n t i t a t i v e l y and qualitatively as t h e system c h a n g e s and as our understanding a b o u t t h e system improves. h i r e c e n t years, there has been a shift from top-down prescriptive m a n a g e m e n t o f ecological resources towards p o l i c y - m a k i n g a n d p l a n n i n g processes chac require o n g o i n g a c t i v e e n g a g e m e n t and c o l l a b o r a t i o n b e t w e e n stakeholders, scientists and

The Practice of Modeling

363

d e c i s i o n - m a k e r s . P a r t i c i p a t o r y m o d e l i n g ( P M ) is t h e process o f i n c o r p o r a t i n g stakeh o l d e r s ( o f t e n i n c l u d i n g t h e p u b l i c ) and d e c i s i o n - m a k e r s i n t o an o t h e r w i s e purely a n a lytic m o d e l i n g process to support d e c i s i o n s i n v o l v i n g c o m p l e x e c o l o g i c a l q u e s t i o n s . St is recognized as a n i m p o r t a n t m e a n s by w h i c h n o n - s c i e n t i s t s a t e e n g a g e d in t h e s c i e n t i f i c process, a n d is b e c o m i n g a n i m p o r t a n t part o f e n v i r o n m e n t a l p l a n n i n g , r e s t o r a t i o n and m a n a g e m e n t . Previously s c i e n c e was c o n d u c t e d o u t s i d e o f t h e p o l i c y - m a k i n g process, a l l o w i n g s c i e n t i s t s t o d e v e l o p e c o l o g i c a l m o d e l s derived from analysis a n d o b s e r v a t i o n o f t h e natural world, t h e r e b y c o n t r i b u t i n g an o b j e c t i v e o p i n i o n t o che p o l i c y - m a k i n g process w i t h o u t a c c o u n t i n g for t h e values, k n o w l e d g e 01 priorities o f che h u m a n system t h a t affects a n d is a f f e c t e d by e c o l o g i c a l systems. T h e shift towards m o r e o p e n a n d integrated p l a n n i n g processes has required t h e a d a p t a t i o n of t h e s c i e n tific m o d e l i n g p r o c e s s t o i n c o r p o r a t e c o m m u n i t y k n o w l e d g e , p e r s p e c t i v e a n d values. P a r t i c i p a t o r y m o d e l i n g is particularly c o m p a t i b l e w i t h che rising focus o n ecosyst e m - b a s e d m a n a g e m e n t , i n t e g r a t e d water resources m a n a g e m e n t a n d a d a p t i v e m a n a g e m e n t , all o f w h i c h i n c o r p o r a t e systems t h e o r y a n d a i m t o p r o t e c t a n d i m p r o v e e c o l o g i c a l r e s o u r c e s w h i l e c o n s i d e r i n g e c o n o m i c a n d s o c i a l c o n c e r n s in t h e c o m m u nity T h e s e a p p r o a c h e s h a v e b e e n a d o p t e d by, a m o n g o t h e r s , t h e W a t e r

Framework

D i r e c t i v e ol t h e E u r o p e a n C o m m i s s i o n , anil supported hy t h e N a t i o n a l

Research

C o u n c i l in t h e U n i t e d S t a t e s . T h e l a t t e r r e c o m m e n d s t h a t t h e processes o l analysis and vie l i b e r a t i o n be i n t e g r a t e d in s u c h a way t h a t s y s t e m a t i c analysis is c o m b i n e d w i t h c o m m u n i t y values c r i t i c a l t o d e c i s i o n - m a k i n g . P M provides a p l a t f o r m for integ r a t i n g s c i e n t i f i c k n o w l e d g e w i t h l o c a l k n o w l e d g e a n d , w h e n e x e c u t e d , provides a n o b j e c t i v e , v a l u e - n e u t r a l p l a c e f o r a d i v e r s e group o f s t a k e h o l d e r s t o c o n t r i b u t e inform a t i o n regarding a n e c o s y s t e m o f i n t e r e s t

R e c o g n i t i o n thac effective e n v i r o n m e n t a l

m a n a g e m e n t r e q u i r e s input from b o t h s c i e n t i f i c a n d s o c i a l processes is k e y t o d e v e l o p i n g e f f e c t i v e p a r t n e r s h i p s b e t w e e n s c i e n t i s t s a n d s t a k e h o l d e r s t h a t live a n d work w i t h i n an e c o s y s t e m . P M (of w h i c h c l o n e s a r e also k n o w n as " m e d i a t e d m o d e l i n g " , " c o m p a n i o n m o d eling" or "shared

vision m o d e l i n g " ) draws o n t h e t h e o r y o f p o s t - n o r m a l

science,

w h i c h d i c t a t e s t h a t in p r o b l e m s c h a r a c t e r i s t i c o f h i g h l y c o m p l e x systems, t h e r e is n o o n e c o r r e c t , v a l u e - n e u t r a l s o l u t i o n . S t a k e h o l d e r p a r t i c i p a t i o n in e n v i r o n m e n t a l r e s e a r c h a n d m a n a g e m e n t h a s b e e n justified lor m u l t i p l e reasons. P M supports d e m o c r a t i c p r i n c i p l e s , is e d u c a t i o n a l , i n t e g r a t e s social a n d n a t u r a l processes, c a n legitimize a local d e c i s i o n - m a k i n g process, a n d c a n lead p a r t i c i p a n t s t o be i n s t r u m e n t a l in p u s h i n g forward an agreed a g e n d a . T h e e x t e n t co w h i c h c h e public o r r e p r e s e n t a t i v e s t a k e h o l d e r group c a n e f f e c t i v e l y p a i t i c i p a t e in e c o l o g i c a l r e s e a r c h a n d m a n a g e m e n t is d e t e r m i n e d by r h e m e t h o d s e m p l o y e d in e n g a g i n g s t a k e h o l d e r s

the inclusion of

diverse groups, group size, i n c o r p o r a t i o n o f local k n o w l e d g e a n d e x p e r t i s e , a n d che t i m e a v a i l a b l e for t h e process t o d e v e l o p . T h e d e v e l o p m e n t o f u n i q u e , p r a c t i c a l a n d a f f o r d a b l e s o l u t i o n s co e c o l o g i c a l p r o b l e m s is o f t e n best a c c o m p l i s h e d by e n g a g i n g s t a k e h o l d e r s a n d d e c i s i o n - m a k e r s in t h e r e s e a r c h p r o c e s s . T h e idea s t e m s from t h e f e e l i n g t h a t a n y m o d e l e r d e v e l o p s w h i l e w o r k i n g o n t h e m o d e l . Ynu m a y h a v e e x p e r i e n c e d it yourself w h e n w o r k i n g w i t h s o m e o f che m o d e l s e a r l i e r p r e s e n t e d in chis b o o k . T h e f e e l i n g is i h a t as you g o t h r o u g h all t h e e s s e n t i a l steps of m o d e l b u i l d i n g you get a really g o o d u n d e r s t a n d i n g o f all t h e processes a n d interactions involved and develop a certain intimacy with t h e system, learning what is m o r e i m p o r t a n t a n d w h a t c a n be a p p r o x i m a t e d , g e t t i n g a h a n d l e o n t h e inputs a n d u n d e r s t a n d i n g h o w t h e y m a y a f f e c t t h e outputs. You also learn t o a p p r e c i a t e t h e u n c e r t a i n t i e s e m b e d d e d in che system, a n d realize t h a t e v e n with t h e s e u n c e r t a i n t i e s t h e r e is c e r t a i n l e v e l of c o n f i d e n c e , o r a c o m f o r t zone, t h a t may he large e n o u g h t o

364

Systems Science and Modeling for Ecological Economics

m a k e a d e c i s i o n . A n d t h a r d e c i s i o n will p r o b a b l y b e i h e b e s t - i n f o r m e d o n e for r h e c u r r e n t s t a l e of k n o w l e d g e . Ir s e e m s as i h o u g h you s t a r t t h i n k i n g t h a t if o n l y e v e r y o n e c o u l d share your u n d e r s t a n d i n g of w h a t is g o i n g o n in r h e s y s t e m , t h e n ir s h o u l d n o t he a p r o b l e m ro c o m m u n i c a t e r h e results a n d m a k e t h e right d e c i s i o n s . S o t h a t is e x a c t l y w h a t you d o w h e n you o p e n up t h e m o d e l i n g process a n d i n v i t e e v e r y b o d y p o t e n t i a l l y i n t e r e s t e d in t h e s y s t e m a n d t h e d e c i s i o n s to p a r t i c i p a t e in this c o l l a b o r a t i v e group study. I f it is r e c o g n i z e d t h a t during che m o d e l i n g process t h e m o d e l e r g a i n s m u c h u n d e r s t a n d i n g a b o u t t h e system workings, a b o u t w h a t is m o s t e s s e n t i a l a n d w h a t c o n t r o l s t h e syst e m b e h a v i o r , t h e n this r i c h a n d e x c i t i n g e x p e r i e n c e t h a t c o m e s from che m o d e l i n g process s h o u l d b e s h a r e d , a n d t h e w h o l e d e c i s i o n - m a k i n g process d e s i g n e d a r o u n d i h e m o d e l i n g process. T h e m o d e l i n g process itself b e c o m e s t h e d e c i s i o n - m a k i n g tool, a n d t h e d e c i s i o n - m a k i n g b e c o m e s part o f t h e m o d e l i n g process. M o d e l s are used t o formalize c o n c e p t s o f e c o l o g i c a l a n d s o c i o - e c o n o m i c

proc-

esses a n d , as s u c h , e x p l o r e e x i s t i n g d y n a m i c s a n d c h a r a c t e r i s t i c s . M o d e l s c a n also b e p r e d i c t i v e or used t o c o m p a r e proposed m a n a g e m e n t p l a n s a n d e x p l o r e t h e i r e f f e c t s o n o t h e r processes. M o d e l i n g tools are e s p e c i a l l y useful in c o m m u n i c a t i n g c o m p l e x processes, spatial p a t t e r n s a n d data in a visual f o r m a t rliat is c l e a r a n d c o m p e l l i n g a n d ; w h e n appropriately applied, c a n e m p o w e r s t a k e h o l d e r s t o m o v e forward with concerted

efforts to address a n e n v i r o n m e n t a l

or socio-economic

problem.

Both

m o n i t o r i n g a n d m o d e l i n g a r e s c i e n t i f i c t o o l s t h a t c a n support good d e c i s i o n - m a k i n g in e c o s y s t e m - b a s e d m a n a g e m e n t , a n d are o f t e n most powerful w h e n used t o g e t h e r . M o n i t o r i n g data c o l l e c t e d at varying s c a l e s c a n be used as inputs to models, to c a l i brate a n d v a l i d a i e che a c c u r a c y o f a m o d e l , or to address specific research q u e s t i o n s using statistical models. D e v e l o p m e n t of e c o l o g i c a l m o d e l s o f t e n i n d i c a t e s t h e types o f i n f o r m a t i o n t h a t are i m p o r t a n t in u n d e r s t a n d i n g d y n a m i c s but for w h i c h n o d a t a are a v a i l a b l e . W h e r e a s s e l e c t i v e m o n i r o r i n g c a n give a good d e s c r i p t i o n o f p a t t e r n s and l i n k a g e s w i t h i n a system, it may h e m o r e difficult a n d e x p e n s i v e t o d e t e r m i n e t h e d r i v i n g forces o f t h e s e p a t t e r n s . S i m u l a t i o n m o d e l s h e l p t o d e t e r m i n e t h e m e c h a n i s m s and u n d e r l y i n g d r i v i n g forces o f p a t t e r n s o t h e r w i s e d e s c r i b e d statistically. In m a n y cases, che m o n i t o r i n g efforts thar go a l o n g w i t h m o d e l i n g c a n serve as a good v e h i c l e to e n g a g e t h e l o c a l s t a k e h o l d e r s in che process. W h e n s t a k e h o l d e r s s e e h o w s a m p l e s are t a k e n or, ideally, t a k e part in s o m e o f t h e m o n i t o r i n g programs, t h e y b o n d with che researchers a n d b e c o m e b e t t e r partners in t h e future d e c i s i o n support efforts. T h e m o d e l i n g o f physical, biological a n d s o c i o - e c o n o m i c d y n a m i c s in a system requires a t t e n t i o n t o b o t h t e m p o r a l d y n a m i c s and spatial relationships. T h e r e are m a n y m o d e l i n g tools t h a t focus o n o n e o r t h e other. T o be useful in a participatory framework, models need t o be t r a n s p a r e n t a n d flexible e n o u g h t o c h a n g e in response to t h e needs o f t h e group. S i m u l a t i o n ( p r o c e s s ) m o d e l s may be formalized in software such as S t e l l a , S i m i l e or M a d o n n a , w h i c h we h a v e considered in this b o o k . T h e s e and o t h e r software packages h a v e user-friendly G r a p h i c U s e r I n t e r f a c e s ( G U I ) w h i c h m a k e t h e m especially helpful w h e n m o d e l s are d e m o n s t r a t e d to s t a k e h o l d e r s or w h e n they are form u l a t e d in their p r e s e n c e a n d with t h e i r inpuc. In this c o n t e x t , c o m p l e x s i m u l a t i o n m o d e l s or p r o g r a m m i n g directly in C + + or o t h e r languages may b e less effective, n o m a t t e r h o w powerful t h e resulting m o d e l s are. In s o m e cases, tools as g e n e r i c a n d s i m ple as E x c e l turn o u t co be e v e n more useful in e n g a g i n g che s t a k e h o l d e r s in a m e a n ingful c o l l a b o r a t i v e work t h a n t h e far m o r e powerful a n d a c c u r a t e c o m p l e x models. T o m a k e t h e s e s c a r e - o f - t h e - a r t c o m p l e x m o d e l s useful for che d e c i s i o n - m a k i n g process, a d d i t i o n a l efforts a r e essential t o build i n t e r f a c e s or wrappers t h a t will a l l o w t h e m to b e p r e s e n t e d to t h e s t a k e h o l d e r s , or e m b e d d e d inco o c h e r m o d e l s ( m o d u larity). In g e n e r a l , process m o d e l s m a y be very helpful t o e x p l a i n a n d u n d e r s t a n d

The Practice of M o d e l i n g

365

t h e systems t o he analyzed; h o w e v e r , t h e y a r e n o t p r a c t i c a l for e x p l o r i n g t h e role o f t h e spatial s t r u c t u r e of an e c o s y s t e m . A l t e r n a t i v e l y , G e o g r a p h i c I n f o r m a t i o n .Systems ( C I S ) e x p l i c i t l y m o d e l t h e spatial c o n n e c t i v i t y a n d l a n d s c a p e p a t t e r n s present in a w a t e r s h e d , b u t are weak i n t h e i r a b i l i t y t o s i m u l a t e a system's b e h a v i o r o v e r t i m e . E c o s y s t e m - b a s e d m a n a g e m e n t d e m a n d s t h e c o u p l i n g of t h e s e a p p r o a c h e s s u c h that spatial r e l a t i o n s h i p s , l i n k a g e s a n d t e m p o r a l d y n a m i c s c a n be c a p t u r e d s i m u l t a n e o u s l y . T h e r e are m a n y specific m o d e l s d e v e l o p e d t o a n a l y z e t h e s p a t i o - t e m p o r a l d y n a m i c s o f specific system^ o r processes. S o far, t h e r e a r e n o t m a n y g e n e r i c t o o l s t h a t c o m b i n e t e m p o r a l a n d spatial m o d e l i n g - O n e is t h e S p a t i a l M o d e l i n g E n v i r o n m e n t

(SME),

w h i c h we h a v e s e e n a l i o v e . S i m i l e , t o o , offers s o m e powerful l i n k a g e s t o spatial d a t a a n d p r o c e s s i n g . T h e r e a r e also m o d u l e s p r o g r a m m e d as c o m p o n e n t s of G I S s , say using t h e scripting language o r A v e n u e in A r c l N F O . A g e n t - b a s e d m o d e l s p r o v i d e yet a n o t h e r m o d e l i n g t e c h n i q u e thai

is useful in

p a r t i c i p a t o r y w o r k s h o p s . T h e y offer s o m e powerful t e c h n i q u e s to e n g a g e t h e s t a k e h o l d e r s in a d i a l o g u e , w i i h s o m e r o l e - p l a y i n g g a m e s l e a d i n g t o m o r e clearly defined rules o f b e h a v i o r for a g e n t s . A g a i n , for t h e p a r t i c i p a t o i y c o n t e x t a G U I is e s s e n t i a l . N e t L o g o o r S t at L o g o a r e t w o m o d e l i n g f r a m e w o r k s i h a t offer very user-friendly i n t e i f a c e s a n d h a v e n r e l a t i v e l y s i m p l e t e a m i n g c u r v e . N e t Logo also h a s a module c a l l e d H u h N e t (see, for e x a m p l e , http://ccl. n o r t h w e s t e r n e d u / ' n e d o g o / m o d e Is/Com p H u b N etT r a g e d y o f t h e G o i u m o n s H u h N e t ) , w h i c h allows several p e o p l e t o work o n t h e m o d e l w h i l e sitting b e h i n d d i f f e r e n t c o m p u t e r s at d i f f e r e n t p l a c e s T h i s c a n be an e x c e l l e n t e n v i r o n m e n t t o work o n p a r t i c i p a t o r y m o d e l i n g p r o j e c t s .

Forms of participation S t a k e h o l d e r paiticipant.s e n g a g e in t h e d e c i s i o n - m a k i n g process in t h e l o i m of m o d e l selection a n d d e v e l o p m e n t , data c o l l e c t i o n and integtation, scenario d e v e l o p m e n t , i n t e r p r e t a t i o n o f results, a n d d e v e l o p m e n t of policy a l t e r n a t i v e s . It is g e n e r a l l y l e c o g n i s e d t h a t e n g a g i n g p a r t i c i p a n t s in as m a n y o f t h e s e phases as possible a n d as early as possible, b e g i n n i n g with s e t t i n g t h e goals lor t h e p r o j e c t , drastically i m p r o v e s t h e v a l u e o f t h e r e s u l t i n g m o d e l in t e r m s o f its usefulness t o d e c is i o n - m a k e r s , its e d u c a t i o n a l p o t e n i i a l tor t b e p u h h c , a n d its c r e d i b i l i t y w i t h i n t h e c o m m u n i t y

Model selection

and

development

S e l e c t i n g t h e c o r r e c t m o d e l i n g tool is o n e o f t h e most i m p o r t a n t p h a s e s o f a P M e x e r c i s e , a n d s h o u l d be d e t e r m i n e d based o n t b e g o a l s o f t h e p a r t i c i p a n t s , t h e a v a i l a bility of d a t a , t h e p r o j e c t d e a d l i n e s a n d f u n d i n g l i m i t a t i o n s , r a t h e r t h a n b e i n g d e t e r m i n e d by s c i e n t i s t s ' preferred m o d e l i n g p l a t f o r m a n d m e t h o d o l o g y . In t e r m s o f m o d e l d e v e l o p m e n t , s t a k e h o l d e r s are very helpful

in identifying

w h e t h e r t h e r e are processes o r e c o l o g i c a l p h e n o m e n a thai h a v e b e e n n e g l e c t e d in t h e m o d e l i n g process. S t a k e h o l d e r s c a n also b e called u p o n t o verify basic a s s u m p t i o n s about t h e d y n a m i c s , history a n d p a t t e r n s o f t h e e c o s y s t e m . I n a d d i t i o n , c o m m u n i t y s t a k e h o l d e r s c a n f r e q u e n t l y v a l i d a t e a s s u m p t i o n s about typical h u m a n b e h a v i o r m t h e system. T h i s often a n e c d o t a l e v i d e n c e may be t h e o n l y s o u r c e o f m o d e ! a s s u m p t i o n s a b o u t h u m a n b e h a v i o r in a system. W h e n c o m b i n e d w ith t e c h n i c a l k n o w l e d g e o f e c o logical processes, such e v i d e n c e m a y be key t o i d e n t i f y i n g n e w a n d m o r e appropriate m a n a g e m e n t s o l u t i o n s T h e P M a p p r o a c h is based o n t h e a s s u m p t i o n that t h o s e w h o live a n d work in a system m a y be well i n f o r m e d about its processes a n d perhaps h a v e o b s e r v e d p h e n o m e n a that would n o t be c a p t u r e d by s c i e n t i s t s . T h i s i w o - w a y flow o f i n f o r m a t i o n is a key c h a r a c t e r i s t i c o f successful P M .

mwywmvpmMHPT! 366

S y s t e m s Science a n d Modeling for Ecological Data collection

and

Economics

availability

S t a k e h o l d e r s o f t e n play a k e y role in r e s e a r c h a c t i v i t i e s by c o n t r i b u t i n g e x i s t i n g data to a r e s e a r c h process or by a c t i v e l y p a r t i c i p a t i n g in t h e c o l l e c t i o n o f n e w data. S o m e s t a k e h o l d e r s , particularly from g o v e r n m e n t a l a g e n c i e s , may h a v e a c c e s s to data t h a t are o t h e r w i s e u n a v a i l a b l e due t o privacy r e s t r i c t i o n s o r c o n f i d e n t i a l i t y

agreements.

T h e s e d a t a c a n o f t e n be provided t o r e s e a r c h e r s if aggregated t o p r o t e c c privacy c o i v c e m s , o r if p e r m i s s i o n is g r a n t e d from p r i v a r e citizens. In a d d i t i o n , s o m e s t a k e h o l d e r s are a w a r e o f dara s o u r c e s rhar are m o r e s p e c i f i c t o a p a r t i c u l a r e c o s y s t e m o r l o c a l e , s u c h as c l i m a t i c data a n d b i o l o g i c a l surveys. S t a k e h o l d e r s c a n also e n g a g e in e c o l o g i c a l s a m p l i n g a n d m o n i t o r i n g . T h i s c a n be a particularly effective e n t r y p o i n t t o a c o m m u n i t y t h a t is ready LO " a c t " o n a perc e i v e d p r o b l e m a n d is n o t satisfied with m o r e m e e t i n g s a n d discussions o f a problem. M o n i t o r i n g by citizen s t a k e h o l d e r s , in particular, provides o t h e r benefits t o t h e research process. In many cases, they live c l o s e t o m o n i t o r i n g sites or h a v e a c c e s s 10 privare property s u c h t h a t m o r e f r e q u e n t and/or more c o m p l e t e m o n i t o r i n g c a n t a k e p l a c e at significantly less cost t h a n o n e individual r e s e a r c h e r c o u l d c o m p l e t e i n d e p e n d ently. C i t i z e n s also gain benefits by b e c o m i n g more familiar w i t h t h e i r e c o s y s i e m - a n e d u c a t i o n a l o p p o r t u n i t y t h a t may b e shared with o i l i e r c o m m u n i t y m e m b e r s .

Scenario

development

S t a k e h o l d e r s a r e best placed ro pose s o l u t i o n s c e n a r i o s t o a p r o b l e m . M a n y o f rhern h a v e d e c i s i o n - m a k i n g p o w e r and/or i n f l u e n c e in t h e c o m m u n i t y , a n d u n d e r s t a n d t h e r e l a t i v e feasibility a n d c o s t - e f f e c t i v e n e s s of proposed s o l u t i o n s . In a d d i t i o n , e n g a g i n g local d e c i s i o n - m a k e r s in t h e s c e n a r i o - m o d e l i n g stage o f t h e r e s e a r c h process c a n lead to d e v e l o p m e n t of m o r e i n n o v a t i v e s o l u t i o n s .

Interpreting

results and developing

policy

alternatives

A primary goal of a P M e x e r c i s e is t o resolve che d i f f e r e n c e b e t w e e n p e r c e i v e d a n d actual sources o f a n e c o l o g i c a l p r o b l e m . W h e r e a s s t a k e h o l d e r s m i g h t have proposed s c e n a r i o s based o n their p e r c e p t i o n o f t h e p r o b l e m or system, they may be particularly adept at proposing n e w policy a l t e r n a t i v e s following initial m o d e l results from a s c e n a r i o - m o d e l i n g e x e r c i s e . T h e P M process c a n further f a c i l i t a t e d e v e l o p m e n t of n e w policies t h r o u g h d e v e l o p m e n t o f a c o l l a b o r a t i v e n e t w o r k b e t w e e n s t a k e h o l d e r s a n d t h e i r respective a g e n c i e s o r c o n s t i t u e n t s t h r o u g h o u t t h e research process. S t a k e h o l d e r s are i m p o r r a n t c o m m u n i c a t i o n agenrs to d e l i v e r t h e findings a n d t h e decision a l t e r n a tives t o t h e d e c i s i o n - m a k e r s in t h e federal, state or local g o v e r n m e n t s . T h e y are t h e m o r e likely t o be listened t o than che scientists, w h o m a y b e p e r c e i v e d as foreign t o t h e p r o b l e m or t h e locality. G o v e r n m e n t s c e r t a i n l y h a v e a b e t t e r e a r for t h e e l e c t o r a t e .

Criteria for Successful Participatory

Modeling

P M is a r e l a t i v e l y n e w a c t i v i t y , and as s u c h che field is just b e g i n n i n g t o define itself a n d t h e c r i t e r i a t h a t qualify a p r o j e c t as a good or successful P M e x e r c i s e . B e l o w a r e s o m e o f t h e key c r i t e r i a rhat m a y be useful. 1. Representmiue mvotaemenc, openness. R e g a r d l e s s of t h e m e t h o d used t o s o l i c i t s t a k e h o l d e r i n v o l v e m e n t , every a t t e m p t s h o u l d b e m a d e t o i n v o l v e a d i v e r s e g r o u p o f s t a k e h o l d e r s t h a t represent a variety o f i n t e r e s t s regarding t h e q u e s t i o n a t h a n d .

The Practice of M o d e l i n g

367

W h i l e key s t a k e h o l d e r s should he carefully identified and i n c i t e d co t h e process, there should be also an open i n v i t a t i o n to all interested parties t o j o i n . T h i s will add to t h e public a c c e p t a n c e of and respect for rhe results of che analysis

If a

piocess is perceived to be e x c l u s i v e , key m e m b e r s of che s t a k e h o l d e r and decisionmaking c o m m u n i t y may reiccc model resulcs2 . Scientific credibility.

A l t h o u g h P M i n c o r p o r a t e s values, t h e scientific c o m p o n e n t s

o f che model must adhere to standard scientific p r a c t i c e and o b j e c t i v i t y .

This

c r i t e r i o n is essential in order foi d i e model t o m a i n t a i n credibility a m o n g decis i o n - m a k e r s . scientists, s t a k e h o l d e r s and t h e public. T h u s , while p a r t i c i p a n t s may d e t e r m i n e che q u e s t i o n s chac the m o d e l should answer and may supply key m o d e l parameters, t h e structure or t h e modi-*! must be scientifically sound. It does not m e a n that t h e model should be all e n c o m p a s s i n g and c o m p l e x ; to t h e contrary, tc should be as simple as possible. Ic is c r u c i a l , however, co be e x t r e m e l y c l e a r and h o n e s t about all t h e a s s u m p t i o n s and s i m p l i f i c a t i o n s m a d e 3 . Objectivity, F a c i l i t a t o r s of a P M p r o j e c t muse be trusted by t h e stakeholder c o m munity as being o b j e c t i v e and impartial, and t h e r e f o r e should not t h e m s e l v e s be direct stakeholders. In this regard, f a c i l i t a t i o n by university researchers o r outside c o n s u l t a n t s o f t e n reduces t h e i n c o r p o r a t i o n of s t a k e h o l d e r biases i n t o d i e scientific c o m p o n e n t s of t h e m o d e l . O n che o t h e r h a n d . ;t is essential thac s t a k e h o l d e r s trust t h e facilitators and scientists, and a c e r t a i n track record in t h e local area and perhaps e v e n r e c o g n i t i o n o f researchers by t h e local stakeholders, based o n passed research or i n v o l v e m e n t , c a n be helpful. 4 . Transparency, K e y co e f f e c t i v e s t a k e h o l d e r e n g a g e m e n t in P M is a process t h a t is t r a n s p a r e n t . T r a n s p a r e n c y is not only c r i t i c a l to g a i n i n g trust a m o n g s t a k e h o l d e r s and e s t a b l i s h i n g model credibility with d e c i s i o n - m a k e r s , but also key to t h e educ a t i o n a l goals o f t e n associated with P M 5 . L'ruferswining uncertainty. M a n y e c o l o g i c a l and s o c i o - e c o n o m i c q u e s t i o n s require an.ilvsis of c o m p l e x

systems. As p r o b l e m c o m p l e x i t y

increases,

model

results

b e c o m e less c e r t a i n . U n d e r s t a n d i n g s c i e n t i f i c u n c e r t a i n t y is critically linked to t h e e x p e c t a t i o n s of real-world results associated with decisions made as a result o f the modeling process. T h i s issue is best c o m m u n i c a t e d through direct Participat i o n in che m o d e l i n g process itself. 6 . Flexibility. T h e m o d e l i n g process should be flexible and adjustable co a c c o m m o d a t e t h e new knowledge and understanding chat c o m e s from che s t a k e h o l d e r workshops. S t a k e h o l d e r s might c o m e up with ideas and factors that modelers had not a n t i c i p a t e d , hue modelers should be ready to i n c o r p o r a t e these i n t o the model. 7 . Model adapifibiliry

T h e model d e v e l o p e d should be relatively easy t o use and

update after t h e researchers h a v e m o v e d o n . T h i s requires e x c e l l e n t

documen-

t a t i o n a n d a good user i n t e r l a c e . If n o n - s c i e n t i s t s c a n t i o c underscaod o r use che model, it will not he applied by local d e c i s i o n - m a k e r s to solve real problems. 8 . JncoTjxjrarion

of

stakeholder

knowiedge

Key

co success

with

any

participatory

a p p r o a c h is chat t h e c o m m u n i t y participating in t h e research be c o n s u l t e d from che i n i t i a t i o n o f t h e p r o j e c t , and h e l p to set che goals for t h e p r o j e c t and tin: specific issues to be studied. 9.

Influence

on decision-making. Results from che m o d e l i n g exercise should h a v e an

e f f e c i , t h r o u g h some m e c h a n i s m , o n decisions made a b o u t t h e system under study. Is t h e r e a n y t h i n g special about models thac would be most appropriate for P M ? Indeed, there ate c e r t a i n features chac would make a model becter suited for use with stakeholders.

368

Systems Science and Modeling for Ecological Economics

Choosing a t o o l T h e p r o b l e m o f c h o o s i n g an a p p r o p r i a t e tool is difficult, b e c a u s e l e a r n i n g e a c h o n e requires s o m e t i m e a n d effort, w h i c h c a n b e q u i t e c o n s i d e r a b l e . T h e r e f o r e , it is o f t e n t h e case t h a t o n c e m o d e l e r s h a v e m a s t e r e d a particular m o d e l i n g language or system, t h e y a r c i n c l i n e d t o use t h e s a m e a c q u i r e d skills n e x t t i m e t h e y n e e d t o analyze a diff e r e n t system - e v e n w h e n this o t h e r system is q u i t e u n l i k e t h e first o n e , a n d e v e n w h e n t h e m o d e l i n g goals are d i f f e r e n t . A s B e r n a r d B a r u c h (or, a c c o r d i n g co a l t e r n a t i v e sources, A b r a h a m M a s l o w ) is supposed co h a v e said, " I f all you h a v e is a h a m m e r , e v e r y t h i n g looks like a n a i l . " A n y w a y , it is q u i t e n a t u r a l for p e o p l e t o try t o d o w h a t t h e y already k n o w h o w co d o . A s a result, m o d e l e r s w h o are e q u a l l y p r o f i c i e n t in a variety o f m o d e l i n g t e c h n i q u e s are q u i t e rare, a n d good c o m p a r i s o n s o f m o d e l i n g tools are also hard t o find. I n c h o o s i n g a tool, t h e f o l l o w i n g s h o u l d be c o n s i d e r e d :

1 . Jnciusk'oness. P M c a n n o t rely o n s e v e r a l p a r t i c u l a r m o d e l s . T h e m o d e l i n g e n g i n e s u p p o r t i n g P M s h o u l d b e a b l e t o i n c o r p o r a t e a variety o f m o d e l s , p r e s e n t e d a s m o d u l e s . T h e s e m o d u l e s s h o u l d b e i n t e r c h a n g e a b l e co serve p a r t i c u l a r n e e d s o f a p r o j e c t , a n d co p r e s e n t s t a t e - o f - t h e - a r t m o d e l i n g a n d daca analysis. T h e m o d e l i n g i n t e r l a c e s e r v i n g t h e s e n e e d s s h o u l d o p e r a t e as a m i d d l e w a r e p r o d u c t , or c o u p l e r , chat c a n t a k e various m o d u l e s a n d m a k e t h e m work i n c o n c e r t . M o d u l e s in t h i s c o n t e x t p r e s e n t b o t h s o f t w a r e o b j e c t s for s i m u l a t i o n a n d data o b j e c t s . T h e c h a l l e n g e is t o m a k e t h e s e m o d u l e s talk t o e a c h o t h e r a n d p e r f o r m across a v a r i e t y o f t e m p o r a l a n d spatial s c a l e s a n d r e s o l u t i o n s . 2 . Modukrw)>. I n t h e m o d u l a r a p p r o a c h , we d o n o t i n t e n d t o design a u n i q u e g e n e r a l m o d e l . I n s t e a d , t h e goal is t o offer a f r a m e w o r k t h a t c a n b e easily e x t e n d e d a n d is f l e x i b l e r e g a r d i n g m o d i f i c a t i o n . A m o d u l e t h a t performs besc in o n e case m a y n o t b e a d e q u a t e in a n o t h e r . T h e goals a n d s c a l e o f a p a r e i c u b r study may r e q u i r e a c o m p l e t e l y d i f f e r e n t s e t o f m o d u l e s t h a t will be i n v o k e d a n d further t r a n s l a t e d i n t o a w o r k i n g m o d e l . T h e r e is a c e r t a i n disparity b e t w e e n t h e s o f t w a r e d e v e l o p er's a n d t h e researcher's views u p o n m o d e l s a n d m o d u l e s . F o r a software d e v e l o p e r , a m o d u l e is a n e n t i t y , a b l a c k b o x , w h i c h s h o u l d be a s i n d e p e n d e n t as possible, and as easy as possible co c o m b i n e w i t h o t h e r m o d u l e s . T h i s is e s p e c i a l l y crue for t h e f e d e r a t i o n a p p r o a c h t o m o d u l a r m o d e l i n g , a n d is well d e m o n s t r a t e d by w e b based m o d e l i n g systems. T h e uciltcy o f s u c h a p p l i c a t i o n s m a y b e m a r g i n a l from the research v i e w p o i n t For a r e s e a r c h e r , a m o d e l is p r e d o m i n a n t l y a t o o l for u n d e r s t a n d i n g t h e s y s t e m . By plugging t o g e t h e r a n u m b e r o f b l a c k b o x e s , for w h i c h specifics a n d b e h a v i o r is o b s c u r e a n d hardly u n d e r s t o o d , w e d o n o t s i g n i f i c a n t l y i n c r e a s e o u r k n o w l e d g e a b o u t che syscem. T h e results g e n e r a c e d a r e difficult co incerpret w h e n t h e r e is n o t e n o u g h u n d e r s t a n d i n g o f t h e p i o c e s s e s t h a t are a c t u a l l y m o d e l e d . T h e d e c o m p o s i t i o n o f s u c h systems r e q u i r e s c a r e f u l analysis of spatial a n d t e m p o r a l s c a l e s o f p r o c esses c o n s i d e r e d , a n d is very c l o s e l y related t o specific goals o f t h e m o d e l built. In this c o n t e x t , t h e m o d u l a r a p p r o a c h c a n be useful if t h e focus is shifted from reusability a n d " p l u g - a n d - p l a y " t o transparency, analysis a n d h i e r a r c h i c a l description o f various processes a n d system c o m p o n e n t s . W i t h t h e m o d u l e s being transpare n t and o p e n for e x p e r i m e n t a n d analysis, t h e r e s e a r c h e r c a n b e t t e r understand t h e specifics o f t h e m o d e l f o r m a l i s m t h a t is i n h e r i t e d . It is t h e n easier to d e c i d e w h e t h e r a m o d u l e is suitable, o r w h e t h e r >t should b e modified a n d tuned t o t h e specific goals o f a p a r t i c u l a r study.

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3 . Transparency.

369

In P M , r h e models are used t o e x p l a i n rather t h a n t o predict. It is

i m p o r t a n t t h e r e f o r e t o he a b l e to dive i n t o t h e model structure and be c l e a r a b o u t the processes t h a t are i n c l u d e d a n d t h e a s s u m p t i o n s made. T h i s i m m e d i a t e l y adds value t o simpler models a n d m o d e l i n g tools. In s o m e cases, t h e benefits of valuing s t a k e h o l d e r " b u y - i n " i n t o t h e m o d e l a n d process by working t o g e t h e r o n simple models t h a t they u n d e r s t a n d o u t w e i g h t h e lack of detail a n d lower a c c u r a c y that we get from Mich models in c o m p a r i s o n with t h e more sophisticated but less c o m p r e h e n s i b l e models. A simple model that c a n be well c o m m u n i c a t e d a n d e x p l a i n e d m a y he more useful t h a n a c o m p l e x model that may be raking more features i n t o a c c o u n t but with narrow applicability, high costs o f m o d e l and data, and much uncertainty. It is also important ro m a k e sure t h a t we clearly draw t h e b o u n d a r i e s of t h e system that is researched a n d m o d e l e d , a n d realize t h a t we are n o t supposed t o be m o d e l i n g t h e w h o l e world in all its c o m p l e x i t i e s . For e x a m p l e , if a study is c o n c e r n e d with s c e n a r i o s driven by global warming, it should n o t be o u r goal t o reprod u c e . understand a n d defend t h e e x t r e m e l y c o m p l e x G l o b a l C i r c u l a t i o n

Models

that are used to g e n e r a t e future c l i m a t e s . W e will he m u c h better off clearly describing t h e output from those models, with t h e associated range o f predicted c h a n g e , as a forcing f u n c t i o n that ts o u t o f t h e scope o f our analysis and should lie used as a given for our purposes. O t h e r w i s e , we are at risk o f g e t t i n g ourselves involved in the highly c o n t e n t i o u s d e b a t e about t h e " t r u t h s " o f c l i m a t e c h a n g e instead of a n a lyzing t h e risks and o u t c o m e s t h a t we f a c e within our system. 4- Visualization

M o d e l s should b e impressive o n t h e o u t p u t side; they must present

results ui an appealing a n d e a s y - t o - u n d e r s t a n d form. I n t e r f a c e s must allow multiple levels o f c o m p l e x i t y a n d i n t e r a c t i v i t y ro serve d i f f e r e n t s t a k e h o l d e r groups. 5 . Affmdabtliiy.

T h e models used in t h e P M process should be affordable lor the

s t a k e h o l d e r s in different levels of g o v e r n a n c e . T h i s m e a n s t h a t e i t h e r t h e m o d e l i n g tools should be made a v a i l a b l e o v e r t h e w e b and urn o n t h e server -ade, s o thar users will n o t n e e d t o purchase e x p e n s i v e licenses, o r t h e models t h e m s e l v e s should be freeware o r shareware. 6 . Flexibility,

extetulibility

W h e n s o m e t h i n g is missing in models, t h e r e should be a

way t o add it t o t h e e x i s r m g structure r a t h e r t h a n rebuilding t h e whole model again from s c r a t c h simply because o n e e l e m e n t is missing. T h i s is especially crucial in t h e P M process, when models should b e developed quickly in response ro t h e c o n c e r n s a n d n e w i n f o r m a t i o n c o m i n g from t h e stakeholders.

It is really important t o b e inventive on t h e visualization s i d e For e x a m p ' e , o n e very popular w a y l o p r e s e n t t h e ' e v e l of a certain >moact is to u s e a color c o d e ranging f r o m g r e e n (safe and g o o d ) to y e l l o w ( m o d e r a t e but b e a r a b i e i a n d then to ted (bad. u n s a f e and unhealthy). This coior c o d e is w i d e ' y a d o p t e d m s o m e of t h e E P A reports a n d w e b p a g e s ( s e e . for e x a m p l e , h t t p : / / w w w e p a gov/ieg3artd/airquaiity/airquality.htm) C h n s J o r d a n , a graphic d e s i g n e r a n d photographic artist, u s e s an i n g e n i o u s w a y to s h o w t h e s c a l e of various p r o c e s s a n d s t o c k s . He s t a r t s to picture simply certain i t e m s isay, plastic bottles or a l u m i n u m c a n s ) and then z o o m s out, getting m o r e a n d m o r e i t e m s into t h e picture. S h o w i n g , say. 2 5 million plastic bottles, which is h o w m a n y a r e u s e d in t h e U S e v e r y hour, c r e a t e s a p o w erful m e s s a g e Or t h e 1 1 . 0 0 0 iet trails, equal to the n u m b e r of c o m m e r c i a l flights in the U S e v e r y 8 hours, or t h e 2 3 million f o l d e d prison uniforms, e q u a l to the n u m b e r of A m e r i c a n s incarcera t e d in 2 0 0 5 . S e e http / / w w w c h i r s j o r d a n . c o m / c u r r e n t _ s e t 2 . p h p , or check out t h e P B S w e b s i t e at http://www.pbs.org/nnoyers/journal/09212007/profile4 html

384 Systems Science and Modeling for Ecological Economics S o m e o f t h e s e c o n s i d e r a t i o n s c l e a r l y p o i n t us in t h e d i r e c t i o n o f o p e n - s o u r c e ( O S ) d e v e l o p m e n t . T h e O S paradigm d e l i v e r s u l t i m a t e t r a n s p a r e n c y a n d f l e x i b i l i t y in t h e p r o d u c t s d e v e l o p e d . T h e s e p r o d u c t s a r e also free for che user. I t o n l y m a k e s s e n s e chat t a x p a y e r s ' m o n e y be s p e n t o n p r o d u c t s t h a t will be a v a i l a b l e for che t a x payers, s t a k e h o l d e r s , ac n o a d d i t i o n a l cosc. F e d e r a l a g e n c i e s s h o u l d p r o m o t e a n d support o p e n - s o u r c e s o f t w a r e for a variety o f reasons, s u c h as t r a n s p a r e n c y , e x t e n d i b i l i c y , security, low c o s t , e t c . T h e r e a r e n u m e r o u s i m p l e m e n c a c i o n s o f che m e c h o d t h a t vary in t h e i r level o f s u c c e s s and a c h i e v e m e n t . L e t us m e n t i o n a few.

Solomons Harbor Watershed, Maryland E x c e s s i v e n u t r i e n t loads t o t h e C h e s a p e a k e B a y from s u r r o u n d i n g c i t i e s a n d rural c o u n c i e s h a s led t o e u t r o p h i c a c i o n ,

especially

in small

harbors

a n d inlets. T h e

M a r y l a n d Tribucary S t r a t e g i e s , C h e s a p e a k e B a y 2 0 0 0 A g r e e m e n t a n d C a l v e r t C o u n t y Comprehensive

P l a n call for r e d u c t i o n s i n n u t r i e n t s e n t e r i n g t h e B a y in order t o

r e d u c e i m p a c t s o n a q u a t i c n a t u r a l resources. T h o u g h

t h e goal s e t for p h o s p h o r u s

appears t o b e a c h i e v a b l e , r e d u c t i o n s m n i t r o g e n lag well b e h i n d t h e target. M o s t sewage in rural residential areas o f M a r y l a n d , s u c h as C a l v e r t C o u n t y , is t r e a t e d by o n - s i t e sewage disposal systems ( s e p t i c s y s t e m s ) . A l m o s t all o f t h e n i t r o g e n p o l l u t i o n t h a t e n t e r s local waters from C a l v e r t C o u n t y c o m e s from n o n - p o i n t sources, o f w h i c h t h e M a r y l a n d D e p a r t m e n t o f P l a n n i n g e s t i m a t e s 2 5 p e r c e n t c o m e s f r o m s e p t i c systems. I n this p r o j e c t we i n i t i a t e d a P M effort t o focus o n t h e most densely p o p u l a t e d w a t e r s h e d in C a l v e r t C o u n t y t h a t drains t o S o l o m o n s H a r b o r . D e s p i t e h i g h population d e n s i t i e s , o n l y a small p o r t i o n o f t h e watershed is serviced by sewer. T h e r e are n o m a j o r p o i n t s o u r c e s o f n i t r o g e n in t h e w a t e r s h e d . T w o d i f f e r e n t m o d e l i n g cools were used t o analyze a n d visualize t h e fate o f n i t r o g e n from t h r e e a n t h r o p o g e n i c sources: s e p t i c t a n k s , a t m o s p h e r i c d e p o s i t i o n , a n d fertilizer. T h e first is a s i m p l e d y n a m i c m o d e l o f a s e p t i c t a n k a n d l e a c h - f i e l d system using S t e l l a r M software, w h i c h a l l o w s t h e user t o e v a l u a t e a l t e r n a t i v e s e p t i c t e c h n o l o g i e s . T h e s e c o n d m o d e l i n g tool is t h e spatially e x p l i c i t L a n d s c a p e M o d e l i n g F r a m e w o r k ( L M F ) , d e v e l o p e d by t h e G u n d I n s t i t u t e for E c o l o g i c a l E c o n o m i c s a n d discussed in Chapter 6. Participation

in t h e study

was s o l i c i t e d

from

community

stakeholders w h o

were i n s t r u m e n t a l in u n d e r s t a n d i n g h o w models could b e applied t o local d e c i s i o n making,

in m a k i n g appropriate

model

assumptions

and m developing

politically

feasible s c e n a r i o s . T h e m o d e l results found chat s e p t i c t a n k s may be a less s i g n i f i c a n t c o n t r i b u t o r to surface water n i t r o g e n p o l l u t i o n m t h e s h o r t cerm, w h e r e a s fertilizer used at t h e h o m e s c a l e is a m o r e significant source t h a n previously t h o u g h t . S t a k e h o l d e r s used t h e m o d e l results t o d e v e l o p r e c o m m e n d a t i o n s for t h e C a l v e r t C o u n t y B o a r d of C o m m i s s i o n e r s . R e c o m m e n d a t i o n s i n c l u d e m a n d a t i n g n i t r o g e n r e m o v a l s e p t i c canks for s o m e h o m e s , b u t primarily focus o n i n t e n s i v e c i t i z e n e d u c a t i o n a b o u t fertilizer usage, local regulation o f fertilizer sales, r e d u c t i o n in a u t o m o b i l e traffic, and c o o p e r a tion with regional regulatory a g e n c i e s w o r k i n g to reduce regional N O x emissions.

St Albans Bay Watershed, Vermont L a k e C h a m p l a m h a s r e c e i v e d e x c e s s n u t r i e n t runoff for che pasc 5 0 years due t o c h a n g e s in agricultural p r a c t i c e s and rapid d e v e l o p m e n t o f o p e n s p a c e for residential use. T h e effect o f e x c e s s n u t r i e n t s h a s b e e n most d r a m a t i c a l l y witnessed in bays s u c h

""""""'"'''''''^BBHmiBB The Practice of Modeling 371

as S t A l b a n s Bay, w h i c h e x h i b i t s e u t r o p h i c algal b l o o m s every A u g u s t . T h e w a t e r s h e d feeding S t A l b a n s Bay is d o m i n a t e d by a g r i c u l t u r e at t h e s a m e t i m e t h a t t h e urban area is growing. In t h e 1 9 8 0 s , urban p o i n t sources of p o l l u t i o n were t c d u c c d by upgrading t h e S t A l b a n s sewage t r e a t m e n t p l a n t . A r t h e same t i m e , agricultural n o n - p o i n t s o u r c e s were addressed through t h e i m p l e m e n t a t i o n o f " B e s t M a n a g e m e n t

Practices"

( B M P s ) o n 6 0 p e r c e n t o f t h e farms in t h e w a t e r s h e d , at a cost of $ 2 . 2 m i l l i o n ( U S D A , 1991)

D e s p i t e t h e c o n s i d e r a b l e a m o u n t o f m o n e y a n d a t t e n t i o n paid t o p h o s p h o r u s

loading in S t A l h a n Bay, it r e m a i n s a p r o b l e m today. T h e focus h a s r e m a i n e d primarily o n agricultural landuses in t h e w a t e r s h e d , a n d as a result h a s caused c o n s i d e r a b l e t e n s i o n b e t w e e n farmers, c i t y dwellers, a n d l a n d o w n e r s w i t h l a k e - f r o n t property R e c e n t l y , t h e L a k e C h a m p l a i n T M D L a l l o c a t e d a p h o s p h o r u s load t o t h e S t A l b a n s B a y w a t e r s h e d t h a t would require a 3 3 p e r c e n t r e d u c t i o n oi total p h o s p h o r u s ro r h e bay. W e i n i t i a t e d a P M effoit r o a p p o r t i o n r h e total load o f p h o s p h o r u s from all s o u r c e s , i n c l u d i n g diffuse t r a n s p o r t p a t h w a y s , a n d identify t h e most c o s t - e f f e c t i v e i n t e r v e n t i o n s t o a c h i e v e target r e d u c t i o n s A group of s t a k e h o l d e r s was i n v i t e d r o p a r t i c i p a t e in t h e 2-year r e s e a r c h p r o c e s s a n d m e m b e r s were e n g a g e d in t h e r e s e a r c h a t m u l t i p l e levels, i n c l u d i n g water q u a l ity m o n i t o r i n g , soil p h o s p h o r u s s a m p l i n g , m o d e l d e v e l o p m e n t , s c e n a r i o analysis, a n d future p o l i c y d e v e l o p m e n t

Statistical, mass-balance a n d dynamic landscape simula-

t i o n m o d e l s were used t o assess t h e s t a t e o f t h e w a t e r s h e d a n d t h e l o n g - t e r m a c c u m u l a t i o n o f p h o s p h o r u s in it, a n d t o d e s c r i b e t h e d i s t r i b u t i o n ot i h e a v e r a g e a n n u a l p h o s p h o r u s load t o s t r e a m s in t e r m s o f s p a c e , t i m e a n d t r a n s p o r t process. W a t e r s h e d interventions, matched

t o t h e most s i g n i f i c a n t p h o s p h o r u s s o u r c e s a n d t r a n s p o r t

processes, w e r e d e v e l o p e d w i t h s t a k e h o l d e r s a n d e v a l u a t e d using t h e f r a m e w o r k . M o d e l i n g results suggest t h a t t h e S t A l b a n s B a y w a t e r s h e d h a s a l o n g - t e r m n e t a c c u m u l a t i o n of p h o s p h o r u s , most o f w h i c h a c c u m u l a t e s in agricultural

soils.

D i s s o l v e d p h o s p h o r u s in surface runoff from t h e agricultural l a n d s c a p e , d r i v e n by h i g h soil p h o s p h o r u s c o n c e n t r a t i o n s , a c c o u n t s lor 4 1 p e r c e n t o f t h e total load t o watershed streams

D i r e c t d i s c h a r g e f r o m f a r m s t e a d s a n d s t o r m w a t e r loads, primarily

f r o m road sand washoft, were a l s o f o u n d t o he s i g n i f i c a n t s o u r c e s . T h e P M a p p r o a c h e m p l o y e d in t h i s study led t o i d e n t i f i c a t i o n o f d i f f e r e n t solutions t h a n s t a k e h o l d e r h a d previously assumed would b e required t o r e d u c e t h e p h o s p h o r u s load t o r e c e i v i n g w a t e r s T h e a p p r o a c h led t o g r e a t e r c o m m u n i t y a c c e p t a n c e a n d utility o f m o d e l results, as e v i d e n c e d by l o c a l d e c i s i o n - m a k e r s n o w m o v i n g forward t o i m p l e m e n t t h e s o l u t i o n s i d e n t i f i e d t o b e m o s t c o s t - e f f e c t i v e

Redesigning the American Neighborhood, South Burlington, Vermont U r b a n sprawl a n d its a s s o c i a t e d o f t e n p o o r l y - t r e a t e d s t o r m w a t e r h a v e a big i m p a c t o n water quality and quantity

in V e r m o n t . C o n v e r t i n g agricultural a n d forested

land t o r e s i d e n t i a l a n d c o m m e r c i a l use h a s s i g n i f i c a n t l y c h a n g e d t h e capacity o f r h e w a t e r s h e d s t o r e t a i n w a t e r a n d a s s i m i l a t e n u t r i e n t s a n d o t h e r materials. C u r r e n t l y , as s o m e studies suggest, s t o r m d i s c h a r g e s m a y be 2 0 0 t o 4 0 0 t i m e s g r e a t e r t h a n histori c a l levels ( A p f e l b a u m . 1 9 9 5 ) . A s m e n t i o n e d in C h a p t e r 6 , R e d e s i g n i n g t h e A m e r i c a n N e i g h b o r h o o d ( R A N ) is a p r o j e c t c o n d u c t e d by t h e U n i v e r s i t y o f V e r m o n t t o find c o s t - e f f e c t i v e s o l u t i o n s t o t h e e x i s t i n g r e s i d e n t i a l s t o r m w a t e r p r o b l e m s a t t h e s c a l e o f small, h i g h - d e n s i t y lesid e n t i a l n e i g h b o r h o o d s (http://www.uvm.edu/~ran/ran). T h e p r o j e c t is f o c u s i n g o n a case study of t h e B u t l e r Farms/Oak C r e e k V i l l a g e c o m m u n i t i e s in S o u t h B u r l i n g t o n ,

372

Systems Science and Modeling for Ecological Economics V T , t o address t h e issue o f t a r g e t i n g a n d prioritising best m a n a g e m e n t

practices

( B M P s ) . T h e idea was t o e n g a g e local h o m e o w n e r s in a p a r t i c i p a t o r y study that would s h o w t h e m h o w t h e y c o n t r i b u t e t o che s t o r m water p r o b l e m a n d i n t r o d u c e t h e m t o e x i s t i n g a l t e r n a t i v e m e t h o d s of s t o r m water m i t i g a t i o n t h r o u g h l o w - i m p a c t distributed structural a n d n o n - s t r u c t u r a l t e c h n i q u e s . T h e p r o j e c t started slowly, w i t h o n l y a few h o m e o w n e r s willing t o p a r t i c i p a t e in t h e process. I lowever, s o o n t h e n e i g h b o r h o o d l e a r n e d t h a t t h e i r h o m e s were s u b j e c t to l o n g - e x p i r e d S t a t e 5 c o n n water d i s c h a r g e p e r m i t s , a n d that t h e i r

neighborhood's

s t o r m water system d i d n o t m e e t s t r i n g e n t n e w standards. A s is o f t e n t h e case, problems w i t h h o m e sales, frustration with localised f l o o d i n g , a n d c o n f u s i o n a b o u t rhe r e l a t i o n s h i p b e t w e e n t h e C i t y ' s s t o r m w a t e r utility a n d che S t a t e p e r m i t impasse led t o frustration a n d e v e n o u t r i g h t a n g e r o n t h e part o f residents T h e t e n s i o n i n c r e a s e d a l t e r t h e h o m e o w n e r s realized that in order for t h e C i t y t o t a k e o v e r t h e e x i s t i n g d e t e n t i o n p o n d s a n d o t h e r s t o r m w a t e r structures, t h e y h a d t o he upgraded t o c u r rently active 2 0 0 2 standards

S i n c e t h e n , t h e i n t e r e s t a n d i n v o l v e m e n t o f residents

in S t o r m w a t e r S t u d y Group; h a s b e e n h e i g h t e n e d , hut s t a k e h o l d e r m e e t i n g s h a v e b e c o m e forums for c o n f l i c t b e t w e e n h o m e o w n e r s and local m u n i c i p a l i t i e s , T h e m o d e l i n g c o m p o n e n c w a s moscly based o n spatial analysis using t h e E S R I A r c G I S 9 . 2 c a p a b i l i t i e s for h y d r o l o g i c m o d e l i n g . A s h i g h - r e s o l u t i o n b e c a m e available,

LIDAR

data

it b e c a m e possible ro g e n e r a t e c l e a r visualization a n d s u b s t a n -

tial u n d e r s t a n d i n g about t h e m o v e m e n t o f w a t e r t h r o u g h t h e n e i g h b o r h o o d s , a n d t o d e v e l o p n e w a p p r o a c h e s cu resolve t h e s t o r m w a t e r m a n a g e m e n t c o n u n d r u m . The M i c r o S t o r m w a t e r N e t w o r k h a s h e l p e d t o visualise rain flowpachs ac a scale w h e r e residents h a v e b e e n able t o m a k e t h e c o n n e c t i o n w i t h processes in t h e i r b a c k y a r d . T h e M i c r o S t o r m w a t e r D r a i n a g e D e n s i t y ( M S D D ) i n d e x was i n s t r u m e n t a l in opti nuzing t h e l o c a t i o n of B M P s of small a n d M i d - s c a l e m a n a g e m e n t p r a c t i c e s , a n d h a d a n i m p o r t a n t e d u c a t i o n a l a n d trusc-building value. A r p r e s e n t , t h e h o m e o w n e r s seem co prefer d e c e n t r a l i z e d m e d i u m a n d small s c a l e i n t e r v e n t i o n s { s u c h as t a u i g a r d e n s ) t o c e n t r a l i ; e d a l t e r n a t i v e s such as large detention ponds

Cutler Reservoir TMDL process, Utah C u t l e r R e s e r v o i r , in t h e C a c h e V a l l e y o f N o r t h e r n U t a h , h a s i m p o u n d e d t h e Bear, L o g a n a n d L i t t l e B e a r R i v e r s s i n c e 1 9 2 7 . C u t l e r D a m is o p e r a t e d by P a c i f i C o r p - U t a b Powct a n d L i g h t to provide water for agricultural use a n d [sower ^ e n e r a i i o n . C u r l e r R e s e r v o i r supports r e c r e a t i o n a l uses a n d a w a r m water fishery w h i l e p r o v i d i n g a habitat for waterfowl a n d a water supply for agricultural uses. C u t l e r R e s e r v o i r h a s b e e n identified as w a t e r - q u a l i t y l i m i t e d due t o low dissolved o x y g e n a n d e x c e s s p h o s p h o r u s l o a d i n g . T h e U t a h D i v i s i o n o f W a t e r Q u a l i t y i n i t i a t e d che process o f d e v e l o p i n g a T o t a l M a x i m u m Daily L o a d { T M D L ) f o r d i e C u r l e r R e s e r v o i r in 2 0 0 4 ! with t h e goal o f restoring a n d m a i n t a i n i n g water q u a l i t y t o a l e v e l thac p r o t e c t s t h e b e n e f i c i a l uses described above. P a r t i c i p a t i o n from local s t a k e h o l d e r s is e n c o u r a g e d t h r o u g h o u t t h e T M D L process, a n d h a s b e e n f o r m a l i z e d in t h e d e v e l o p m e n t o f t h e B e a r R i v e r / C u t l e r R e s e r v o i r A d v i s o r y C o m m i t t e e , w h i c h has r e p r e s e n t a t i o n f r o m ail t h e m a j o r seccors a n d intere s t o f t h e local c o m m u n i t y . T h e advisory c o m m i t t e e h a s b e e n m e e t i n g

monthly

s i n c e A u g u s t 2 0 0 5 , a n d h a s i n f o r m e d t h e T M D L process by c o n t r i b u t i n g d a t a a n d k n o w l e d g e o f p h y s i c a l a n d stxiial processes i n t h e w a t e r s h e d , a n d i d e n t i f y i n g solutions t o help reduce pollution sources.

The Practice of Modeling

373

W a t e r s h e d - l o a d i n g m o d e l s a n d a r e s e r v o i r - r e s p o n s e m o d e l ( B a t h t u b ) a r e in p r e l i m i n a r y d e v e l o p m e n t s t a g e s a t t h e r u n e o f w r i t i n g , a n d will b e n e f i t f r o m f e e d b a c k f r o m t h e a d v i s o r y c o m m i t t e e . Ic is e x p e c t e d t h a t c o m m i t t e e m e m b e r s will c o n t i n u e to p r o v i d e f e e d b a c k t o t h e T M D L p r o c e s s w h i l e w o i k i n g w i t h t h e i r r e s p e c t i v e c o n s t i t u e n t s t o p i o v t d e d i r e c t i o n t o U D E Q in d e v e l o p i n g a n d i m p l e m e n t i n g a w a t e r s h e d m a n a g e m e n t p l a n . T h e y will also b e h e l p f u l in i d e n t i f y i n g f u n d i n g n e e d s a n d s o u r c e s o f support for specific p r o j e c t s t h a t m a y be i m p l e m e n t e d .

James River Shared Vision Planning, Virginia T h e l a m e s R i v e r in V i r g i n i a will p o t e n t i a l l y f a c e s i g n i f i c a n t w a t e r supply d e v e l o p m e n t pressures o v e r t h e n e x c s e v e r a l years d u e t o g r o w i n g p o p u l a t i o n a n d d e v e l o p m e n t pressure. T h e C o r p s ' N o r f o l k D i s t r i c t h a s a l r e a d y r e c e i v e d o n e a p p l i c a t i o n f o r a C l e a n W a t e r A c c S e c t i o n 4 0 4 p e r m i t for C o b b C r e e k R e s e r v o i r , a n d i n i t i a l i n q u i r ies by t h e V i r g i n i a D e p a r t m e n t o f E n v i r o n m e n t a l Q u a l i t y i n d i c a t e t h e p o t e n t i a l for m o r e a p p l i c a t i o n s in t h e n e a r f u t u r e . U S E P A R e g i o n 111 h a s f o r m a l l y r e q u e s t e d t h a t Norfolk

District prepare a basin-wide

assessment

that considers all the proposed

w a t e r supply p r o j e c t s o n t h e J a m e s R i v e r a n d m a k e p e r m i t t i n g d e c i s i o n s based o n a c u m u l a t i v e impacts analysis. T h e s e f a c t o r s p o i n t t o t h e n e e d for a c o m p r e h e n s i v e p l a n n i n g process, ing a l l t h e k e y a g e n c i e s a n d s t a k e h o l d e r s ,

in order t o identify broadly

a n d s u s t a i n a b l e s o l u t i o n s for w a t e r m a n a g e m e n t

within

involv-

acceptable

t h e basin. D u e t o h i s t o r i c

water c o n f l i c t s i n t h e state, t h e S h a r e d V i s i o n P l a n n i n g ( S V P ) p r o c e s s ( h t t p : / / w w w . s h a r e d v i s i o n p l a n n i n g . u s ) h a s b e e n p r o p o s e d as t h e m e t h o d lor c o n d u c t i n g t h i s c o m ' prehensive process. T h e A r m y C o r p s o f Engineers has pioneered participatory decis i o n m a k i n g since the 1970s (Wagner and Ortolando, 1 9 7 5 , 1976). T h e Shared Vision P l a n n i n g process is a P M a p p r o a c h in w h i c h s t a k e h o l d e r s are i n v o l v e d i n c r e a t i n g a m o d e l o f t h e system t h a t c a n be used t o run s c e n a r i o s a n d find o p t i m a l s o l u t i o n s t o a problem

S h a r e d V i s i o n P l a n n i n g relies o n a s t r u c t u r e d p l a n n i n g process firmly r o o t e d

in t h e federal P r i n c i p l e s a n d G u i d e l i n e s , a n d in t h e c i r c l e s o f i n f l u e n c e a p p r o a c h t o s t r u c t u r i n g p a r t i c i p a t i o n ( P a l m e r ecai,

2007).

T b e J a m e s R i v e r S t u d y ( J R S ) b e g a n w i t h a g e n e r a l w o r k s h o p in t h e w i n t e r o f 2 0 0 6 , entitled " F i n d i n g a n d C r e a t i n g C o m m o n G r o u n d in W a t e r M a n a g e m e n t . " T h e p u r p o s e o f this o p e n m e e t i n g was t o scart a c o n t i n u i n g d i a l o g u e a m o n g t h e v a r i o u s stakeholders involved, including those with divergent interests. A major o b j e c t i v e o f che w o r k s h o p was t o d e s c r i b e a n d i n t r o d u c e t h e use o f c o l l a b o r a t i v e m o d e l i n g t o facilit a t e l e a r n i n g a n d d e c i s i o n - m a k i n g across v a r i o u s g o v e r n m e n t a l a n d n o n - g o v e r n m e n t a l groups. W h i l e t h e r e was g o o d p a r t i c i p a t i o n i n t h e w o r k s h o p , t h e process s t a l l e d w h e n w o r k i n g groups were t o b e f o r m e d . O n l y a few s t a k e h o l d e r s signed up t o c o n t i n u e w i t h t h e P M effort, a n d during t h e f o l l o w i n g m o n t h s t h e p r o c e s s a l m o s t s t o p p e d . I t t o o k s o m e t i m e t o realize t h a t i n f a c t t h e p r o j e c t g o t s t u c k a m i d s t s o m e m a j o r c o n t r o v e r s y b e t w e e n t w o k e y s t a k e h o l d e r s . I n a d d i t i o n , t h e r e was s o m e i n t e r n a l o p p o s i t i o n t o t h e p r o j e c t w i t h i n t h e A r m y C o r p s . U n d e r t h e s e c o n d i t i o n s , n o t surprisingly, s t a k e h o l d ers w h o k n e w a b o u t t h e s e c o n f l i c t s w e r e s k e p t i c a l a b o u t t h e p r o j e c t a n d r e l u c t a n t t o p a r t i c i p a t e . A s o f today, a c o n s e n s u s s e e m s t o be e m e r g i n g b e t w e e n t h e s t a k e h o l d e r s regarding t h e goals o f t h e p r o j e c t , a n d a fresh start is p l a n n e d in t h e n e a r future. T o a c e r t a i n e x t e n t , t h e s e a n d o t h e r P M p r o j e c t s t e n d t o f o l l o w c h e flow c h a r t f o r a g e n e r i c P M p r o c e s s p r e s e n t e d tn F i g u r e 9 . 1 . N o t e t h a t t h e r e m a y b e a l o t o f varia t i o n s o f a n d d e v i a t i o n s I r o m t h i s r a t h e r idealized s e q u e n c e .

W h e n dealing

with

374

Systems Science and Modeling for Ecological Economics

<

Process initiation

•

Contact key stakeholders Identify m a i n i s s u e a n d p r o b l e m s Plan p r o c e s s

V

<

Process startup

•

First o p e n g e n e r a l w o r k s h o p D e f i n e i s s u e s a n d priorities

r\ $

Workgroup self-selection

It

First round of modeling

4

Workgroup defines - communication m e a n s - schedule

+

- roles a n d responsibilities - data s o u r c e s Modeling work - conceptual model - ballpark calibration

V Process progress report

4

G e n e r a l w o r k s h o p or publication

•

- reality c h e c k - s o l i c i t input a n d d e s c r i b e i n t e r a c t i v e f e a t u r e s

A ^

Second round of modeling

•

Workgroup defines - scenarios - objectives - policies M o d e l i n g work - model relinement - m o d e l testing

V Process report

4

•

R e p o r t to t h e w i d e r s t a k e h o l d e r c o m m u n i t y R e p o r t to d e c i s i o n - m a k e r s Influencing t h e d e c i s i o n s a n d controling e n a c t m e i

V Process continuation

4

^

D e s i g n future c o l l a b o r a t i o n - decide on model "home" - w e b interfaces - adaptive management

V B Fa ifgfumr eT 9 ^ .M1B

A flow chart for a generic PM process.

Note that each particular project will most likely develop in its own way, driven by the stakeholders involved. That is perfectly fine; however, it is good to keep some of the keystones in mind.

The Practice of M o d e l i n g

375

p e o p l e , we h a v e ro he ready f o r surprises a n d need t o adapt t h e w h o l e process t o spei ific n e e d s of p a r t i c u l a r p r o j e c t s a n d s t a k e h o l d e r groups. H o w e v e r , this diagram m a y b e s o m e t h i n g t o k e e p in m i n d w h e n p l a n n i n g a P M process.

Some lessons learned, or a guide to success 1 Identify a clear problem

and lead

stakeholders

A l t h o u g h m o s t w a t e r s h e d m a n a g e m e n t d e c i s i o n s b e n e f i t Irom s t a k e h o l d e r I n p u t a n d i n v o l v e m e n t , s o m e issues m i g h t n o t raise t h e i n t e r e s t o f a wide group o f s t a k e h o l d e r s . If t h e p r o b l e m is n o t u n d e r s t o o d o r c o n s i d e r e d t o bit i m p o r t a n t by s t a k e h o l d e r s , t h e n it will be v e r y difficult t o s o l i c i t i n v o l v e m e n t in a p a r t i c i p a t o r y e x e r c i s e . F o r e x a m ple, t h e V i r g i n i a p r o j e c t h a d a very difficult Startup b e c a u s e t h e r e was c l e a r disagreem e n t b e t w e e n s t a k e h o l d e r s regarding t h e i m p o r t a n c e o f t h e study. W h i l e it was q u i t e c l e a r t o all t h a t t h e r e w o u l d be g r o w i n g p r o b l e m s with water supply in t h e a r e a , t h e s i t u a t i o n did n o t look b a d e n o u g h t o get local p e o p l e really i n v o l v e d , whiie a g e n c i e s h a d t h e i r o w n agendas a n d were n o t e x a c t l y c l e a r o n t h e purposes ot t h e studyE d u c a t i o n of t h e c o m m u n i t y about water resource issues and t h e impact of d e c i s i o n s on t h e c o m m u n i t y is often a g o o d first step. This c a n o f t e n he a c c o m p l i s h e d through t h e media, town hall meetings, o r v o l u n t e e r a n d c o m m u n i t y - o r i e n t e d programs. In s o m e cases, it is helpful w h e n t h e r e is a strong g o v e r n m e n t a l lead in t h e p r o c ess, ' T h e C a l v e r t group s p r o u t e d from a n o p e n m e e t i n g w h e r e all c i t i s e n s

residing

in t h e w a t e r s h e d w e r e i n v i t e d t o c o m m e n t o n proposed r e g u l a t i o n o f s e p t i c s y s t e m s by t h e C o u n t y

Planning and Zoning commission

T h e possibility o f n e w regula-

t i o n c a u g h t t h e a t t e n t i o n a t t h e p u b l i c , a n d i n t e r e s t e d parties were willing t o part i c i p a t e in t h e study, h i o t h e r cases, i n t e r e s t from s o m e s t a k e h o l d e r s m a y o n l y arise a f t e r a policy c h a n g e thai d i r e c t l y i m p a c t s t h e m . T h e R A N p r o j e c t s t a r t e d with s e v eral s t a k e h o l d e r w o r k s h o p s , w h e r e h o m e o w n e r s were addressed a b o u t t h e l o o m i n g p r o b l e m s a s s o c i a t e d with u n t r e a t e d stormwater-

I h e r e c e p t i o n was l u k e w a r m , w i t h

very l o w a t t e n d a n c e . T h i n g s c h a n g e d q u i t e d r a m a t i c a l l y w h e n t h e c i t y o f S o u t h P u i i i n g t o n a p p r o v e d l e g i s l a t i o n that c r e a t e d a s t o r m w a t e r utility, w h i c h would t a k e o v e r s t o r m w a t e r t r e a t m e n t from t h e h o m e o w n e r s , h u t o n l y after t h e y brought t h e i r r u n o f f up t o c e r t a i n standards. It turned out that t h e i r titles were n o l o n g e r valid, s i n c e all t h e i r p e r m i t s r e l a t e d t o s t o r m w a t e r h a d e x p i r e d a w h i l e a g o . T h e i n t e r e s t m r h e R A N p r o j e c t i m m e d i a t e l y j u m p e d , bur e v e n t h e n for s o m e h o m e o w n e r s r h e i n v o l v e m e n t ot university

researchers

was s e e n as a n i m p e d i m e n t .

N e v e r u n d e r e s t i m a t e t h e " l u c k f a c t o r . " W o r k i n g w i t h people, it takes just o n e o r t w o s t a k e h o l d e r s w h o c h o o s e co t a k e an o b s t r u c t i o n i s t p o s i t i o n t o d a m a g e che p r o c ess. S i m i l a r l y , o n e s t a k e h o l d e r that "gets ic" a n d :s i n t e r e s t e d a n d a c t i v e l y p a r t i c i p a t ing c a n s i g n i f i c a n t l y e n h a n c e t h e effort.

2 Engage stakeholders

as early and often as possible

E s t a b l i s h m e n t of a c o m m u n i t y - b a s e d m o n i t o r i n g effort c a n b e a particularly e f f e c t i v e e n t r y p o i n t t o a c o m m u n i t y that is ready to " a c t " o n a p e r c e i v e d p r o b l e m a n d is n o t satisfied with m o r e m e e t i n g s a n d d i s c u s s i o n . M o n i t o r i n g by c i c n e n s , m particular, provides o t h e r b e n e f i t s t o t h e r e s e a r c h process. In m a n y cases, t h e y live c l o s e t o m o n i t o r i n g sites o r h a v e a c c e s s t o p r i v a t e property su
376

Systems Science and M o d e l i n g for Ecological Economics

with o t h e r c o m m u n i t y m e m b e r s . W h e n s t a k e h o l d e r s see h o w s a m p l e s are t a k e n or, ideally, t a k e part ill s o m e of t h e m o n i t o r i n g programs, t h e y b o n d with t h e r e s e a r c h ers a n d b e c o m e b e t t e r p a r t n e r s i n future r e s e a r c h a n d d e c is i o n - s u p p o r t efforts. ln t h e S t A l b a n s B a y w a t e r s h e d , t h e r e -was a lack o f r e c e n t d a t a regarding t h e g e n e r a l stare of t h e w a t e r s h e d , i n c l u d i n g w a t e r quality, discharge, arid soil p h o s p h o rus c o n c e n t r a t i o n s . A t r h e s a m e time, t h e r e was a highly m o t i v a t e d group o f c i t i z e n s o r g a n n e d t h r o u g h t h e S t A l b a n s A r e a W a t e r s h e d A s s o c i a t i o n e a g e r t o begin " d o i n g " something

in che watershed

immediately,

di p a r t n e r s h i p w i t h this group a n d t h e

V e r m o n t . A g e n c y of N a t u r a l R e s o u r c e s , a c i t i z e n s ' v o l u n t e e r m o n i t o r i n g program was e s t a b l i s h e d with 2 5 m o n i t o r i n g sites a r o u n d t h e S r A l b a n s Bay w a t e r s h e d . Most o f t h e 500-+- w a t e r - q u a l i t y s a m p l e s a n d s t a g e - h e i g h t d a t a were c o l l e c t e d by a group o f 1 5 v o l u n t e e r s d r a w n from t h e c o m m u n i t y o v e r 2 years. T h e resulting d a t a woiud n o t h a v e b e e n a v a i l a b l e o t h e r w i s e , a n d t h e process e n g a g e d a group o f local f i l l e r s in t h e research process. T h i s early e n g a g e m e n t proved v a l u a b l e during t h e latter stages o f che p r o j e c t , w h e n a s t a k e h o l d e r group was a s s e m b l e d for t h e P M e x e r c i s e T h e partn e r s h i p rhac grew from t h e m o n i t o r i n g M o r e also built trust b e t w e e n t h e r e s e a r c h e r s a n d w a t e r s h e d a c t i v i s t s w o r k i n g in t h e c o m m u n i t y . A key t o s u c c e s s with a n y p a r t i c i p a t o r y a p p r o a c h is t h a t t h e c o m m u n i t y part i c i p a t i n g m t h e r e s e a i c h be c o n s u l t e d from t h e i n i t i a t i o n of t h e p r o j e c t a n d h e l p to set t h e g o a l s f o r t h e p r o j e c t a n d specific issues t o b e studied ( B e i r c l e a n d C a y f o r d , 2 0 0 2 ) . S t a k e h o l d e r p a r t i c i p a n t s e n g a g e in t h e d e c i s i o n - m a k i n g process in t h e form of m o d e ! s e l e c t i o n a n d d e v e l o p m e n t , d a t a c o l l e c t i o n a n d i n t e g r a t i o n , s c e n a r i o d e v e l o p m e n t , i n t e r p r e t a t i o n of results, a n d d e v e l o p m e n t o f p o l i c y a l t e r n a t i v e s . It is g e n erally r c c o g m z e d t h a t e n g a g i n g p a r t i c i p a n t s in as m a n y o f these p h a s e s as possible a n d as early as possible, b e g i n n i n g with s e t t i n g che goals for t h e p r o j e c t , drastically i m p r o v e s t h e v a l u e of t h e resulting m o d e l

in terms o f ics usefulness t o d e c i s i o n -

m a k e r s , its e d u c a t i o n a l p o t e n t i a l for t h e p u b l i c , a n d its c r e d i b i l i t y w i t h i n t h e c o m munity (Korfmachcr, 2 0 0 1 ) .

3. Create an appropriately

representative

working

group

P a r t i c i p a t o r y m o d e l i n g may be i n i t i a t e d by local d e c i s i o n - m a k e r s , g o v e r n m e n t a l b o d ies, citizen a c t i v i s t s o r scientific, researchers. In rhe U n i t e d S t a r e s , m o s t P M a c t i v i t i e s are i n i t i a t e d by g o v e r n m e n t a l b o d i e s ( D u r a m a n d B r o w n , 1 9 9 9 ) . D e p e n d i n g u p o n t h e type o f p a r t i c i p a t i o n a n d t h e goals a n d l i m e r e s t r i c t i o n s of t h e p r o j e c t , s t a k e h o l d ers m a y be e n l i s t e d co p a r t i c i p a t e in a vai iety o f ways. I n s o m e p r o j e c t s s t a k e h o l d e r s are s o u g h t o u t f o r t h e i r k n o w n " s l a k e " in a p r o b l e m o r d e c i s i o n , a n d a t e i n v i t e d t o j o i n a working group. In o t h e r cases, i n v o l v e m e n t in t h e w o r k i n g group may be o p e n to a n y m e m b e r o f t h e p u b l i c . R e g a r d l e s s o f che n i e c h o d used t o solicit s c a k e h o l d e r i n v o l v e m e n t , every a t t e m p t s h o u l d b e m a d e co i n v o l v e a d i v e r s e group o f s t a k e h o l d e r s t h a t r e p r e s e n t a v a r i e t y of i n t e r e s t s r e g a r d i n g t h e q u e s t i o n at h a n d . W h e n less w e l l - o r g a n i z e d

stakeholder

groups d o noc a c t i v e l y p a r t i c i p a t e , w a t e r s h e d m a n a g e r s c a n o b t a i n i n f o r m a t i o n a b o u t cheir o p i n i o n s t h r o u g h o t h e r m e a n s s u c h as puhlic m e e t i n g s , e d u c a t i o n , o r surveys (Korfmacher, 2001). In this s e n s e , t h e S t A l b a n s B a y watershed m o d e l i n g process m a y h a v e failed s o m e w h a t in t h a t t h e s c a k e h o l d e r group f o r m e d r a t h e r o r g a n i c a l l y from t h o s e t h a r c u r r e n t l y work o n issues o r are directly a f f e c t e d by w a t e r s h e d m a n a g e m e n t , i n c l u d i n g local, s t a l e a n d federal n a t u r a l resources, p l a n n i n g a n d agricultural a g e n c i e s , as well as farmers a n d w a t e r s h e d activists, A d e l i b e r a t e a t t e m p t was made co i n v o l v e m e m b e r s

The Practice of M o d e l i n g

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o f t h e business a n d r e s i d e n t i a l c o m m u n i t y w i t h o u t success, d u e t o a lack o f i n t e r e s t in t h e process - p e r h a p s b e c a u s e t h e y p e r c e i v e d t h e m s e l v e s t o h a v e n o s t a k e in t h e outcome.

4. Gain trust and establish neutrality

as a scientist

T h i s c a n he a c h i e v e d by a d h e r i n g t o t h e s e c o n d a n d t h u d c r i t e r i a o n v . i e i u . t i c c i c d i htliry a n d o b j e c t i v i t y , as p r e s e n t e d a b o v e

It is helpful w h e n we c a n refer t o past e x a m -

pies a n d success stories, o r refer t o e x i s t i n g m o d e l s that are k n o w n t o s t a k e h o l d e r s a n d p e r h a p s published in peer-re\ iewed l i t e r a t u r e . H o w e v e r , it is e v e n m o r e i m p o r t a n t ro k e e p t h e m o d e l c l e a r t o all p a r t i c i p a n t s , t o h a v e a good h a n d o n all a s s u m p t i o n s a n d f o r m a l i z a t i o n s used in t h e model- For e x a m p l e , r h e m o d e l s d e v e l o p e d for use in t h e S t A l b a n s B a y a n d S o l o m o n s H a r b o r w a t e r s h e d s h a v e b e e n peer-rev iewed a n d a c c e p t e d by t h e s c i e n t i f i c c o m m u n i t y ( C a d d i s , 2 0 0 7 ; G a d d i s

2007}- Model development

is still underway f o r i h e l a m e s R i v e r a n d C u t l e r R e s e r v o i r .

5. Know the stakeholders

and acknowledge

conflict

In s o m e cases, s t a k e h o l d e r s may h a v e h i s t o r i c a l d i s a g r e e m e n t s with o n e a n o t h e r . O n e purpose of t h e P M m e t h o d is t o p r o v i d e a n e u t r a l p l a t f o r m u p o n w h i c h disputing parties c a n c o n t r i b u t e a n d g a i n i n f o r m a t i o n . H o w e v e r , it is i m p o r t a n t t o w a t c h for s u c h h i s t o r i c c o n f l i c t s a n d e x t e r n a l issues that may o v e r s h a d o w t h e w h o l e process. In a d d i t i o n , we h a v e round t h a i w h e n t h e o u t c o m e of a m o d e l i n g e x e r c i s e is b i n d i n g , s u c h as in r h e d e v e l o p m e n t o f a T M D L , parties m a y be b o t h m o r e e n g a g e d but a l s o d e f e n s i v e if t h e y p e r c e i v e t h a t t h e process wilt result in a n e g a t i v e i m p a c t o n t h e m o r t h e i r c o n s t i t u e n t s . F o r e x a m p l e , p o i n t - s o u r c e polluters m a y look f o r ways t o h o l d up a T M D L process in o r d e r t o p r o l o n g a l o a d - r e d u c t i o n d e c i s i o n . T h e s e sources o( c o n t e n t i o n m a y he m a s k e d as s c i e n t i f i c d i s s e n t w h e n t h e y a r e a c t u a l l y p o l i t i c a l .

When

c o n f l i c t w i t h i n t h e group b e c o m e s u n m a n a g e a b l e , it is i m p o r t a n t t o set o u t rules for discussion a n d , in s o m e cases, ro hire a professional f a c i l i t a t o r . In t h e J a m e s R i v e r p r o j e c t , t h e r e h a d b e e n a long history o f t e n s i o n

between

s o m e s t a k e h o l d e r s o n issues o f water p l a n n i n g . T h e S h a r e d V i s i o n P l a n n i n g process got c a u g h t in this c o n t r o v e r s y , a n d c o u l d m o v e n o w h e r e further until s o m e c o n s e n s u s was r e a c h e d b e t w e e n s t a k e h o l d e r s . In t h e o r y , t h e m o d e l i n g process was supposed co be o p e n t o all s t a k e h o l d e r s , s h o u l d b e truly d e m o c r a t i c a n d t r a n s p a r e n t , a n d s h o u l d not d e p e n d u p o n local m i s u n d e r s t a n d i n £ b e t w e e n s o m e s t a k e h o l d e r s . I n p r a c t i c e , t h e h i s t o r i c n e t w o r k ol c o n n e c t i o n s ( b o t h professional a n d p e r s o n a l ) b e t w e e n s t a k e h o l d ers is e v i d e n t a n d c a n c o m e t o d o m i n a t e t h e p a r t i c i p a t o r y process.

6. Select appropriate modeling that are clearly identified

tools to answer

questions

A c r i t i c a l step early in t h e P M process is t h e d e v e l o p m e n t o f r e s e a r c h q u e s t i o n s a n d goals of t h e process.

I h e q u e s t i o n s identified s h o u l d be a n s w e r a b l e , given t h e t i m e

and f u n d i n g a v a i l a b l e ro t h e process

In a d d i t i o n , it is i m p o r t a n t t h a t all s t a k e h o l d e r s

agree o n t h e goals o f t h e process such t h a t a c l e a r r e s e a r c h d i r e c t i o n is e m b r a c e d by the. e n t i r e >;roup before d e t a i l e d m o d e l i n g begins. S e l e c t i n g t h e c o r r e c t m o d e l i n g tool is o n e ot t h e most i m p o r t a n t phases u f a P M e x e r c i s e , unci should be d e t e r m i n e d based o n t h e goals o f t h e p a r t i c i p a n t s , che a v a i l a b i l i t y o f d a t a , t h e p r o j e c t d e a d l i n e s a n d funding l i m i t a t i o n s , r a t h e r t h a n b e i n g d e t e r m i n e d by s c i e n t i s t s ' preferred i m x l e l m g p l a t f o r m a n d m e t h o d o l o g y . M o d e l s are

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used co formalize c o n c e p r s o f w a t e r s h e d , s t r e a m a n d r e c e i v i n g - w a t e r processes, a n d as s u c h e x p l o r e e x i s t i n g d y n a m i c s a n d c h a r a c t e r i s t i c s . M o d e l s c a n also be p r e d i c t i v e , o r used t o c o m p a r e proposed m a n a g e m e n t p l a n s and e x p l o r e rheir e f f e c t s o n o t h e r p r o c esses. M o d e l i n g t o o l s c a n be e s p e c i a l l y useful in c o m m u n i c a t i n g c o m p l e x processes, spatial p a t t e r n s a n d data in a visual f o r m a t t h a t is c l e a r a n d c o m p e l l i n g a n d , w h e n a p p r o p r i a t e l y applied, c a n e m p o w e r s t a k e h o l d e r s t o m o v e forward with

concerted

efforts to address an e c o l o g i c a l p r o b l e m . It is i m p o r t a n t t o m a i n t a i n " m o d e l neutrality.'" It is c o m m o n for m o d e l e r s t o t u r n to t h e m o d e l s a n d m o d e l i n g p l a t f o r m s t h a t are most f a m i l i a r ro t h e m . It is i m p o r t a n t , h o w e v e r , always t o survey t h e a v a i l a b l e tools a n d s e l e c t o n e t h a t is m o s t a p p r o p r i a t e to t h e p o i n t s o f i n t e r e s t o f t h e s t a k e h o l d e r s . T h e S o l o m o n s H a r b o r w a t e r s h e d p r o j e c t was i n i t i a l l y geared towards a fairly s o p h i s t i c a t e d spatial m o d e l i n g effort based o n our e x p e r i e n c e in i n t e g r a t i n g d y n a m i c spatial m o d e l s . W h i l e this m o d e l i n g was still b e i n g p e r f o r m e d , che p r o j e c t focus t u r n e d ro s o m e fairly s i m p l e b a l a n c e c a l c u l a t i o n s thac h e l p e d m o v e t h e d e c i s i o n - m a k i n g process in t h e righc d i l e c t i o n . T o b e useful in a p a r t i c i p a t o r y f r a m e w o r k , m o d e l s need t o b e t r a n s p a r e n t a n d flexible e n o u g h t o c h a n g e in r e s p o n s e to r h e n e e d s o f che group. A s we noced a b o v e , m s o m e cases cools as s i m p l e as E x c e l c a n be che right c h o i c e . M a j o r b e n e f i t s of E x c e l are t h a t it is readily a v a i l a b l e in m o s t cases, a n d m a n y s t a k e h o l d e r s a r e already i n t i m a t e l y f a m i l i a r w i t h it. S i m u l a t i o n ( p r o c e s s ) m o d e l s h e l p d e t e r m i n e t h e m e c h a n i s m s a n d u n d e r l y i n g d r i v i n g forces o f p a t t e r n s o t h e r w i s e d e s c r i b e d s t a t i s t i c a l l y ;

however,

t h e y a r e n o r p r a c t i c a l for e x p l o r i n g t h e role o f t h e spatial strucrure o f an e c o s y s t e m . A l t e r n a t i v e l y , G e o g r a p h i c I n f o r m a t i o n S y s t e m s ( G L S ) e x p l i c i t l y m o d e l t h e spatial c o n n e c t i v i t y a n d l a n d s c a p e p a t t e r n s p r e s e n t in a w a t e r s h e d , but are weak in t h e i r a b i l i t y t o s i m u l a t e a syscem's b e h a v i o r o v e r t i m e . M o d e l c o m p l e x i t y muse b e d i c t a t e d by t h e q u e s t i o n s posed hy t h e s t a k e h o l d e r group. M o d e l s thac are coo s i m p l e are less p r e c i s e and e x p l a n a t o r y ; h o w e v e r , a m o d e l t h a t is t o o c o m p l e x c a n lose transpare n c y a m o n g t h e s t a k e h o l d e r group. I n m a n y cases, a s i m p l e m o d e l t h a r c a n be well c o m m u n i c a c e c l a n d e x p l a i n e d is m o r e useful t h a n a c o m p l e x m o d e l t h a t h a s n a r r o w a p p l i c a b i l i t y , high c o s t s o f d a t a , a n d m u c h u n c e r t a i n t y . In addition to a Stella i m p l e m e n t a t i o n o f the simple T R - 5 5 rounng model, the R A N p r o j e c t h a s h e e n using G I S analysis. T h e spatial v i s u a l i z a t i o n o f s t r e a m f l o w s at t h e tine s c a l e t h a r was allowed by t h e L I D A R data was a t u r n i n g p o i n t in r h e discuss i o n s , w h e n s t a k e h o l d e r s c o u l d a c t u a l l y see h o w t h e i r local d e c i s i o n s c o u l d m a k e a difference.

7 Incorporate

all forms of stakeholder

knowledge

T h e k n o w l e d g e , d a t a a n d p r i o r i t i e s o f s t a k e h o l d e r s s h o u l d h a v e a real, n o t just c u r sory, i m p a c t o n m o d e l d e v e l o p m e n t , b o t h in terms o f s e l e c t i n g a m o d e l i n g p l a t f o r m and

in s e t t i n g m o d e l a s s u m p t i o n s a n d p a r a m e t e r s . S t a k e h o l d e r s o f t e n

contribute

e x i s t i n g d a t a t o a r e s e a r c h process or a c t i v e l y p a r t i c i p a t e in che c o l l e c t i o n o f newdata. S o m e s t a k e h o l d e r s , particularly from g o v e r n m e n t a l a g e n c i e s , may h a v e a c c e s s co daca t h a t a r e o t h e r w i s e u n a v a i l a b l e ro r h e p u b l i c d u e ro privacy r e s t r i c t i o n s o r c o n f i d e n t i a l i t y a g r e e m e n t s . T h e s e data c a n o f t e n b e p r o v i d e d to r e s e a r c h e r s if aggreg a t e d to p r o t e c t privacy c o n c e r n s , o r if p e r m i s s i o n is g r a n t e d from p r i v a t e c i t i z e n s . In a d d i t i o n , s o m e s t a k e h o l d e r s are a w a r e o f data s o u r c e s t h a t are m o r e specific t o r h e w a t e r s h e d , s u c h as locally c o l l e c t e d c l i m a t i c data. T h e P M a p p r o a c h is based o n t h e a s s u m p t i o n t h a t t h o s e w h o live a n d w o r k in a system may be well i n f o r m e d a b o u t its processes a n d m a y h a v e o b s e r v e d p h e n o m e n a

mnpnapMamniMMft^Ba" The Practice of M o d e l i n g

379

t h a t would n o t lie c a p t u r e d hy s c i e n t i s t s . S t a k e h o l d e r s c a n also he very helpful in identifying whether there are hydrologic, ecological or h u m a n - d o m i n a t e d

processes

chat h a v e b e e n n e g l e c t e d in t h e m o d e l s t r u c t u r e . S t a k e h o l d e r s c a n also verify basic a s s u m p t i o n s a b o u t t h e d y n a m i c s , history a n d p a t t e r n s of b o t h t h e n a t u r a l a n d s o c i o e c o n o m i c systems. Farmers a n d h o m e o w n e r s possess i m p o r t a n t local a n d lay k n o w l edge a b o u t t h e b i o p h y s i c a l a n d s o c i o - e c o n o m i c system b e i n g r e s e a r c h e d . A n e c d o t a l e v i d e n c e m a y be t h e only s o u r c e ol a s s u m p t i o n s a b o u t h u m a n b e h a v i o r in a watershed, m a n y o f w h i c h are i m p o r t a n t inputs t o a s i m u l a t i o n m o d e l ( i . e . f r e q u e n c y of fertilizer a p p l i c a t i o n ) . T h i s type of k n o w l e d g e , w h e n c o m b i n e d w i t h t e c h n i c a l knowle d g e o f w a t e r s h e d processes, is key t o i d e n t i f y i n g n e w a n d more a p p r o p r i a t e s o l u t i o n s to environmental problems. T h e m o d e l i n g process s h o u l d be f l e x i b l e a n d a d j u s t a b l e t o a c c o m m o d a t e t h e n e w k n o w l e d g e a n d u n d e r s t a n d i n g thac c o m e s from s c a k e h o l d e r w o r k s h o p s . T h i s requires chat m o d e l s be m o d u l a r , robust a n d h i e r a r c h i c a l t o m a k e sure t h a t c h a n g e s in c o m p o n e n t s d o n o t crash t h e w h o l e system.

8. Gain acceptance of modeling presenting model results

methodology

before

G i v i n g s t a k e h o l d e r s che o p p o r t u n i t y t o c o n t r i b u t e to a n d c h a l l e n g e m o d e l SSsumpt i o n s before results a r e reported also c r e a t e s a sense o f o w n e r s h i p of t h e process t h a t m a k e s it more difficult t o r e j e c t results in t h e future. T h i s c a n o n l y occur, h o w e v e r , if c h e m o d e l s d e v e l o p e d a r e t r a n s p a r e n t a n d well u n d e r s t o o d hy t h e p u b l i c o r stakeh o l d e r group ( K o r t m a c h e r , 2 0 0 1 ) . In s o m e cases, it c a n reduce c o n f l i c t b e t w e e n s t a k e h o l d e r s m c h e w a t e r s h e d , s i n c e m o d e l a s s u m p t i o n s are o f t e n less c o n t r o v e r s i a l chan m o d e l results. T h e d e v e l o p m e n t o f t h e m o d e l i n g t o o l s used in t h e S t A l b a n s B a y w a t e r s h e d was very t r a n s p a r e n t . S t a k e h o l d e r s were repeatedly g i v e n t h e o p p o r t u n i t y to c o m ment on model

assumptions and parameters selecced,

a n d were e v e n

consulted

o n a l t e r n a t i v e m o d e l i n g f r a m e w o r k s w h e n a p p r o p r i a t e . H o w e v e r , t h e m o d e l is noc " u s e r - f r i e n d l y " d u e co t h e a r c h i t e c t u r e o f m o d e l i n g f r a m e w o r k selecced.

9. Engage stakeholders

in conversations

regarding

uncertainty

S t a k e h o l d e r s t h a t h a v e p a r t i c i p a t e d in all t h e stages o f che m o d e l building a c t i v i ties d e v e l o p crust in che m o d e l a n d are less likely co q u e s t i o n r h e reliability o f che result-. Primarily, this is b e c a u s e t h e y know- all t h e m o d e l a s s u m p t i o n s , t h e e x t e n t o f m o d e l reliability, a n d chat che m o d e l i n c o r p o r a t e d t h e besc a v a i l a b l e k n o w l e d g e a n d daca; t h e y also u n d e r s t a n d t h a t t h e r e will always be s o m e u n c e r t a i n t y in t h e m o d e l results.

10 Develop scenarios that are both feasible and ideal S t a k e h o l d e r s a r e in a b e t t e r p o s i t i o n t o j u d g e w h a t rhe m o r e realistic a n d effect i v e i n t e r v e n t i o n s a r e , a n d w h a t t h e m o s t feasible d e c i s i o n s m i g h t be ( C a n a n d Kalvorsen, 2 0 0 1 ) . In t h e S o l o m o n s H a r b o r w a t e r s h e d , M a r y l a n d , a n i n t e r e s t i n g q u e s t i o n e m e r g e d from t h e discussion o f s c e n a r i o s that could reduce n i t r o g e n r o S o l o m o n s

Harbor

G i v e n limited resources for m o d e l i n g , is it beccer t o focus o n che s c e n a r i o s chat che r e s e a r c h team suspect will h a v e t h e greatest i m p a c t o n water quality, o r ( h o s e t h a t a r e easiest a n d t h e r e f o r e likely t o be i m p l e m e n t e d politically? S c e n a r i o s a r e very d i f f e r e n t

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Systems Science and M o d e l i n g for Ecological Economics

for e a c h p e r s p e c t i v e . A c o n s e n s u s was r e a c h e d h e r e t h r o u g h discussion t o test b o t h sets o f s c e n a r i o s . By t e s t i n g feasible s c e n a r i o s , we g e t a s e n s e o f w h a t c a n r e a s o n a b l y be a c h i e v e d in t h e s h o r t t e r m , g i v e n c u r r e n t funding and p o l i t i c a l realities. Ideal s c e narios push s t a k e h o l d e r s to t h i n k b e y o n d c o n v e n t i o n a l s o l u t i o n s a n d t o recognize t h e b o u n d a r i e s a n d t i m e lag i n v o l v e d with w h a t t h e y a i m t o a c c o m p l i s h . Besides, a n o t h e r most c o s t - e f f i c i e n t

and productive scenario

emerged

from t h e p a r t i c i p a t o r y

fact-

finding e x e r c i s e : to focus o n r e d u c t i o n o f r e s i d e n t i a l fertilizer a p p l i c a t i o n a n d o t h e r a i r b o r n e sources o f n i t r o g e n in t h e area.

77. Interpret results in conjunction with stakeholder group; facilitate development of new policy and management ideas that arise from modeling results S t a k e h o l d e r s c a n h e l p to i n t e r p r e t t h e results a n d p r e s e n t t h e m in t h e way t h a t will be b e t t e r u n d e r s t o o d by d e c i s i o n - m a k e r s at various levels o f g o v e r n a n c e . T h e y c a n advise o n b o w best to visualize t h e results m order to d e l i v e r a c o m p e l l i n g a n d c l e a r message. In t h e S t A l b a n s B a y w a t e r s h e d , m a n y o f t h e m o d e l i n g results were n o t e x p e c t e d by t h e s t a k e h o l d e r group. S o m e o f t h e m o s t i m p o r t a n t s o u r c e s and p a t h w a y s o f phosphorus m o v e m e n t t o r e c e i v i n g waters (dissolved p h o s p h o r u s f r o m agricultural fields, road-sand w a s h o f f a n d tile d r a i n a g e ) w e r e n o t addressed by most of t h e proposed s c e narios. S o m e processes had previously b e e n c o n s i d e r e d s i g n i f i c a n t by t h e s t a k e h o l d e r group. H o w e v e r , several s t a k e h o l d e r s h a v e i n d i c a t e d t h a t they i n t e n d t o use t b e inform a t i o n g l e a n e d f r o m t h e p r o j e c t t o d i r e c t e x i s t i n g f u n d i n g s o u r c e s a n d adapt p o l i c i e s to t h e e x t e n t possible t o address t h e most s i g n i f i c a n t p h o s p h o r u s transport processes and sources in t h e w a t e r s h e d . T h e m u n i c i p a l i t i e s in t h e watershed h a v e agreed t o i n v e s t i g a t e a l t e r n a t i v e s t o road sand for w i n t e r d e i c i n g o f roads The T M D L

process c u r r e n t l y u n d e r w a y for C u t l e r R e s e r v o i r , U t a h ,

requires

t h a t t h e results o f t h e P M study be i n c l u d e d in t h e prescribed m a n a g e m e n t c h a n g e s i n c l u d e d in r h e T M D L d o c u m e n t s u b m i t t e d for a p p r o v a l to t h e U S E P A . T h e s e d e c i s i o n s i n c l u d e required n u t r i e n t - l o a d

reductions according

t o load a l l o c a t i o n s for

various p o i n t a n d n o n - p o i n t s o u r c e s t h r o u g h o u t t h e w a t e r s h e d , as well as a P r o j e c t I m p l e m e n t a t i o n P l a n designed to a c h i e v e t h e s e r e d u c t i o n s . i n t h e S o l o m o n s H a r b o r w a t e r s h e d , M a r y l a n d , u n e x p e c t e d results led t h e working group to adapt m a n a g e m e n t goals a n d policies for C a l v e r t C o u n t y . Fertilizer a n d a t m o s p h e u c d e p o s i t i o n were found t o h a v e a s i g n i f i c a n t l y larger e f f e c t t h a n t h e c o m m u n i t y had t h o u g h t o n n i t r o g e n loads in S o l o m o n s H a r b o r , a l t h o u g h n o n e o f t h e p r o p o s e d s e p t i c m a n a g e m e n t s c e n a r i o s are likely t o h a v e a real e f f e c t o n t h e t r o p h i c status of r h e h a r b o r in t h e s h o r t - t e r m . N o n e t h e l e s s , upgrading septic, t a n k s is still a g o o d e n v i r o n m e n t a l d e c i s i o n , s i n c e it will i m p r o v e g r o u n d w a t e r q u a l i t y a n d , in t h e long t e r m , affect s u r f a c e - w a t e r quality. F u r t h e r m o r e , it is t h e o n l y r e g u l a t i o n t h a t c a n be easily a n d i m m e d i a t e l y i m p l e m e n t e d at t h e l o c a l level. T h e m o d e l results w e r e first p r e s e n t e d t o t h e s m a l l e r w o r k i n g g r o u p o v e r t w o m e e t i n g s a n d were a s e v e r e test o f p a r t i c i p a n t c o n f i d e n c e , s i n c e n e w results were s o m e w h a t c o n t r a r y t o previous e s t i m a t e s . T h e w o r k i n g g r o u p t o o k a very p o s i t i v e a n d c o n s t r u c t i v e a p p r o a c h . W h i l e a c k n o w l e d g i n g t h e i n h e i e n t u n c e r t a i n t i e s in t h e m o d e l i n g process, they b e g a n to explore new solutions and policy r e c o m m e n d a t i o n s . R a t h e r than abandoning t h e p r o p o s e d p o l i c i e s t o i e d u c e n i t r o g e n from s e p t i c tanks, t h e w o r k i n g group c h o s e to e x p a n d its p o l i c y r e c o m m e n d a t i o n s t o i n c l u d e all s o u r c e s of n i t r o g e n t o t h e waters h e d . T h e r e s e a r c h t e a m found this t o b e a d i s t i n c t l y p o s i t i v e o u t c o m e o f t h e P M

The Practice of M o d e l i n g

381

e x e r c i s e . T h e w o r k i n g g r o u p c a m e u p w i t h t h e f o l l o w i n g c o n c l u s i o n s a b o u t t h e types o f p o l i c y o p t i o n s that are r e a l i s t i c a n d a v a i l a b l e to t h e S o l o m o n s H a r b o r c o m m u n i t y : •

A t m o s p h e r i c d e p o s i t i o n c a n n o t be d i r e c t l y i n f l u e n c e d by local c i n : e n s ,

except

t h r o u g h r e d u c t i o n o f local traffic a n d l o h b y i n g r e g i o n a l officials t o i e d u c e N O x e m i s s i o n s from c o a l - f i r e d p o w e r p l a n t s •

F e i t i l i r e r usage c a n be m o s t easily i n f l u e n c e d t h r o u g h e d u c a t i o n a l i n i t i a t i v e s , s i n c e p o l i c y c h a n g e s will require i n v o l v e m e n t o f o t h e r g o v e r n m e n t a l a n d c i t i z e n groups b e y o n d t h e D e p a r t m e n t o f P l a n n i n g a n d Z o n i n g , w h i c h is c u r r e n t l y l e a d i n g t h e i n i t i a t i v e to r e d u c e n i t r o g e n i o t h e harhor.

12. Involve members of the stakeholder group in presenting results to decision-makers, the public and the press A n i m p o r t a n t hnal s t e p m rhe P M m e t h o d is t h e d i s s e m i n a t i o n o f results a n d c o n c l u s i o n s t o t h e wider c o m m u n i t y . P r e s e n t a t i o n s t o larger s t a k e h o l d e r groups, d e c i s i o n - m a k e r s , a n d t h e press s h o u l d be m a d e by a m e m b e r of t h e s t a k e h o l d e r w o r k i n g group. T h i s solidities t h e a c c e p t a n c e o f t h e m o d e l results a n d c o o p e r a t i o n S t a k e h o l d e r s i h a t w e r e e s i a h i t s h e d during l h e P M e x e r c i s e

between

In a d d i t i o n , m e m b e r s o f

t h e c o m m u n i t y are o f t e n m o r e r e s p e c t e d a n d h a v e a b e t t e r h a n d l e o n t h e i m p a c t o f p o l i c y d e c i s i o n s o n che local c o m m u n i t y ' s issues. ! n t h e S o l o m o n s H a r b o r w a t e r s h e d , t w o m e m b e r s o f t h e w o r k i n g group p i e sented their

recommendations

t o t h e larger s t a k e h o l d e r group f o l l o w i n g a pres-

e n t a t i o n o f t h e m o d e l i n g results by o n e m e m b e r o f o u r r e s e a r c h t e a m . D u r i n g this meeting, the D i r e c t o r of P l a n n i n g and Z o n i n g for Calvert C o u n t y solicited feedback o n proposed policy r e c o m m e n d a t i o n s a n d later refined i h e i n f o r a p r e s e n t a t i o n t o i h e C a l v e r t C o u n t y B o a r d o f C o m m i s s i o n e r s . W c e m p h a s i z e h e r e t h a t t h e role of t h e r e s e a r c h t e a m in this process w-as ro support t h e discussion r a t h e r t h a n t o r e c o m m e n d o u r o w n p o l i c y ideasIn t h e S t A l b a n s B a y w a t e r s h e d , s e v e r a l s t a k e h o l d e r s p a r t i c i p a t e d in t h e prese n t a t i o n of m o d e l results t o t h e local press a n d g e n e r a l p u b l i c in M a y 2 0 0 6 . S e v e r a l i n t e r a g e n c y p a r t n e r s h i p s a p p e a r t o h a v e b e e n s t r e n g t h e n e d a n d trust d e v e l o p e d in previously o p p o s i n g groups a s a result ot r h e P M e x e r c i s e s .

13. Treat the model as a process T h e r e are always c o n c e r n s about t h e f u i u r e of p a r t i c i p a t o r y efforts. W h a t

happens

w h e n t h e r e s e a r c h e r s g o away? If we look at h o w c o l l a b o r a t i v e m o d e l p r o j e c t s are d e v e l o p e d , t h e r e is a c l e a r s i m i l a r i t y with t h e o p e n - s o u r c e paradigm, w h e r e software is a p r o d u c t ol j o i n t efforts ot a d i s t r i b u t e d g r o u p o f players. Ideally, t h e process s h o u l d live o n t h e w e b a n d c o n t i n u e b e y o n d a p a r t i c u l a r p r o j e c t . It is a v a l u a b l e asset for luture d e c i s i o n - m a k i n g a n d c o n f l i c t r e s o l u i i o n . It c a n b e kept a l i v e with

incremen-

tal f u n d i n g o r e v e n d o n a t i o n s , w i t h s t a k e h o l d e r s able t o c h i p in t h e i r e x p e r r i s e a n d k n o w l e d g e t o k e e p it going b e t w e e n p e a k s o f a c t i v i t y w h e n bigger p r o j e c t s surface T h ere are e x a m p l e s of w e b a n d m o d e l i n g t o o l s that c a n p r o v i d e this kind o f f u n c t i o n ality a n d i n t e r o p e t a b i l i t y , so t h e r e is real p r o m i s e thai t h i s m i g h t a c t u a l l y h a p p e n . T h i s last lesson b r i n g s up a w h o l e n e w issue o f h o w i o use a n d reuse m o d e l s . W h e r e a n d h o w d o m o d e l s " l i v e , " a n d h o w c a n we m a k e che most o f r h e m ? It a p p e a r s t h a t t h e n e w w e b t e c h n o l o g i e s a n d t h e n e w dispersed w a y o f c o l l e c t i v e

thinking,

r e s e a r c h a n d d e v e l o p m e n t h a v e t h e p o t e n t i a l ro b e c o m e r h e n e w s t a n d a r d o f m o d eling and decision-making.

382

Systems Science and Modeling for Ecological Economics

9.3

Open-source, web technologies and decision support (Parts of this section stein from and Software

Society

workshop

with Raleigh

Hood,

John

Danes,

Computer

programming

discussions

at the International

on Collaborative Hamed

Assaf

in t h e 1 9 6 0 s

Modeling, and Robert and 1 9 7 0 s

Environmental Modeling-

and the resulting

position

paper

Stuart.) was d o m i n a t e d

by t h e free

e x c h a n g e o f software (Levy, 1 9 3 4 ) . T h i s started t o c h a n g e in t h e 1930s, w h e n rhe M a s s a c h u s e t t s Institute o f T e c h n o l o g y

( M I T ) licensed some o f t h e c o d e

created

by its employees t o a c o m m e r c i a l firm a n d also w h e n software c o m p a n i e s began to impose copyrights ( a n d later software p a t e n t s ) t o p r o t e c t their software from being c o p i e d ( D r a h o s and B r a i t b w a i t e . 2 0 0 2 ) . Probably in protest against these d e v e l o p m e n t s , t h e o p e n - s o u r c e c o n c e p t started to gain ground in t h e 1980s. T h e growing d o m i n a n c e o f W i n d o w s and t h e a n n o y i n g l y s e c r e t i v e policies o f Microsoft c e r t a i n l y added fuel to t h e fire. T h e o p e n - s o u r c e c o n c e p t stems from t h e so-called h a c k e r culture. Hackers are n o t w h a t we usually t h i n k they are - software pirates, vicious producers o f viruses, worms and o t h e r nuisances for o u r c o m p u t e r s . H a c k e r s will insist that those people should be called " c r a c k e r s . " H a c k e r s are t h e real c o m p u t e r gurus, who are addicted to p r o b l e m - s o l v i n g and building things T h e y believe in freedom a n d voluntary mutual help, it is a l m o s t a moral duty for them to share i n f o r m a t i o n , solve problems, and then give t h e solutions awav just so o t h e r hackers c a n solve n e w problems instead o f h a v i n g to re-address old ones. B o r e d o m and drudgery are n o t just u n p l e a s a n t but actually evil. H a c k e r s have a n i n s t i n c t i v e hostility to censorship, secrecy, a n d the use o f force or d e c e p t i o n . T h e idea of software source c o d e shared for free is probably best k n o w n tn c o n n e c t i o n with t h e L i n u x o p e r a t i n g system. A f t e r L m n s Torvalds developed its core and released it t o softwaie developers worldwide, L i n u x b e c a m e a product o f j o i n t efforts of many people, w h o c o n t r i b u t e d c o d e , bug reports, fixes,

enhancements

and plug-ins. T h e idea really t o o k o f f when N e t s c a p e released t h e source c o d e of its Navigator, t h e popular I n t e r n e t browser program, in 1 9 9 8 . T h a t is when t h e term " o p e n source"' was c o i n e d a n d when t h e o p e n - s o u r c e definition was derived

Both

L i n u x and N a v i g a t o r ( t h e latter now d e v e l o p e d as t h e Firefox browser under mozi lla. o r g ) h a v e s i n c e developed i n t o m a j o r software products with worldwide distribution, a p p l i c a t i o n s and input from software developers.

The basic idea behind open source is very simple: when a programmer can read, redistribute, and modify the source code for a piece of software, the software evolves. People improve it, people adapt it, people fix bugs. And this can happen at a speed that, if one is used to the slow pace of conventional software development, seems astonishing. R a y m o n d , 2000a

M o t i v a t e d by t h e spirit o f traditional scientific c o l l a b o r a t i o n , R i c h a r d S t a l l m a n , t h e n a programmer ar M I T ' s Artificial

Intelligence

Laboratory,

founded t h e Free

S o f t w a r e Foundation ( F S F ) in 1 9 8 5 (hrtp-.//www.fsf.org/). T h e F S F is dedicated t o

The Practice of Modeling

383

p r o m o t i n g c o m p u t e r users' rights t o use, study, copy, modify a n d redistribute c o m p u t e r programs B r u c e Perens a n d Eric R a y m o n d c r e a t e d t h e O p e n S o u r c e D e f i n i t i o n in 1 9 9 8 ( P e r e n s , 1 9 9 8 ) . T h e G e n e r a l P u b l i c L i c e n s e ( G P L ) , R i c h a r d S t a l l m a n ' s i n n o v a t i o n , is s o m e t i m e s k n o w n as " c o p y l e f t " - a form of c o p y r i g h t p r o t e c t i o n a c h i e v e d t h r o u g h c o n tract law. A s S t a l l i n a n describes it:

To copyleft a program, first we copyright it; then we add distribution terms, which are a legal instrument that gives everyone the rights to use, modify, and redistribute the program's code or any program derived from it, but only if the distribution terms are unchanged.

T h e G P L c r e a t e s a c o m m o n s in s o f t w a r e d e v e l o p m e n t " t o w h i c h a n y o n e m a y add, but from w h i c h n o o n e may s u b t r a c t . " O n e o f t h e c r u c i a l parts o f t h e o p e n - s o u r c e l i c e n s e is t h a c it a l l o w s m o d i f i c a t i o n s a n d d e r i v a t i v e works, but all of t h e m must b e t h e n d i s t r i b u t e d under t h e s a m e t e r m s as t h e l i c e n s e o f t h e o r i g i n a l software. T h e r e f o r e , u n l i k e simply free c o d e , thar c a n be b o r r o w e d a n d t h e n used in c o p y r i g h t e d , c o m m e r c i a l d i s t r i b u t i o n s , t h e o p e n - s o u r c e d e f i n i t i o n a n d l i c e n s i n g e f f e c t i v e l y m a k e s sure t h a t t h e d e r i v a t i v e s stay in t h e o p c n s o u r c e d o m a i n , e x t e n d i n g a n d e n h a n c i n g it. T h e G P L p r e v e n t s e n c l o s u r e ot t h e tree s o f t w a r e c o m m o n s , a n d c r e a t e s a legally p r o t e c t e d s p a c e for it co flourish.

Because

n o o n e c a n seize t h e surplus v a l u e c r e a t e d w i t h i n c h e c o m m o n s , s o f t w a r e d e v e l o p e r s are willing co c o n t r i b u t e t h e i r t i m e a n d e n e r g y t o i m p r o v i n g u . T h e c o m m o n s is prot e c t e d a n d stays p r o t e c t e d . T h e G P L is t h e c h i e f r e a s o n t h a t L i n u x a n d dozens ot o t h e r programs h a v e b e e n a b l e t o flourish w i t h o u t b e i n g privatized. T h e O p e n S o u r c e S o f t w a r e ( O S S ) paradigm c a n p r o d u c e i n n o v a t i v e , h i g h - q u a l i t y s o f t w a r e that m e e t s t h e n e e d s o f r e s e a r c h scie n t i s t s with respect t o p e r f o r m a n c e , s c a l a b i l i t y , security, a n d t o t a l cosi of o w n e r s h i p ( T C O ) . O S S d o m i n a c e s che I n t e r n e t , with s o f t w a r e s u c h as S e n d m a i l , B I N D ( D N S ) , PHP, O p e n S S L , T C P / I P , a n d H T T P / H T M L . M a n y e x c e l l e n t a p p l i c a t i o n s also e x i s t , i n c l u d i n g A p a c h e w e b server, M o z i l l a F i r e f o x w e b browser a n d T h u n d e r h i r d e m a i l c l i e n t , t h e O p e n O f f i c e suite, a n d m a n y o t h e r s . O S S users h a v e f u n d a m e n t a l c o n t r o l a n d flexibility a d v a n t a g e s

F o r e x a m p l e , if

we were co w r i t e a m o d e l using A N S I s t a n d a r d C T + ( a s o p p o s e d M i c r o s o f t C T + ), we c o u l d easily m o v e t h e c o d e from o n e p l a t f o r m t o a n o t h e r . T h i s m a y b e c o n v e n i e n t for a n u m b e r o f reasons - f r o m simply a p r e f e r e n c e for o n e d e v e l o p e r to a n o t h e r , to m o v i n g f r o m a d e s k t o p P C e n v i r o n m e n t t o a h i g h - p e r f o r m a n c e c o m p u t i n g e n v i r o n m e n t . O p e n S t a n d a r d s , w h i c h a i e p u b l i c l y a v a i l a b l e s p e c i f i c a t i o n s , offer c o n t r o l a n d f l e x i b i l i t y as well. E x a m p l e s in s c i e n c e i n c l u d e E n v i r o n m e n t a l M a r k u p L a n g u a g e ( E M L ) a n d V i r t u a l R e a l i t y M a r k u p L a n g u a g e ( V R M L ) . If t h e s e were proprietary, use would b e likely l i m i t e d t o o n e p r o p r i e t y a p p l i c a t i o n t o i n t e r f a c e wich o n e proprietary format o r n u m e r o u s a p p l i c a t i o n s , e a c h wich its o w n f o r m a t . W e n e e d o n l y imagi n e che l i m i t a t i o n s o n i n n o v a t i o n if c o m m o n l y used p r o t o c o l s like A S C I I ,

HTTP

o r H T M L were proprietary. T o o r g a n i z e t h i s g r o w i n g c o m m u n i t y , c h e O p e n

Source

D e v e l o p m e n t N e t w o r k ( O S D N ) (http://www.osdn.com) was c r c a c e d . L i k e m a n y previous o p e n - s o u r c e spin-offs, n is based o n t h e I n c e r n e c a n d p i o v i d e s c h e t e a m s o f software d e v e l o p e r s d i s t r i b u t e d a r o u n d t h e world w i t h a virtual w o r k s p a c e w h e r e t h e y

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c a n discuss t h e i r ideas a n d progress, any bugs, s h a r e updates a n d n e w releases. T h e o p e n - s o u r c e p a r a d i g m has b e c o m e t h e o n l y v i a b l e a l t e r n a t i v e t o t h e c o p y r i g h t e d , closed and restricted corporate software W h a t u n d e r l i e s t h e O S S a p p r o a c h is t h e s o - c a l l e d "gift c u l t u r e " a n d "gift e c o n o m y " that is based o n t h i s c u l t u r e . U n d e r gift c u l t u r e , you g a i n s t a t u s a n d r e p u t a t i o n w i t h i n II n o t by d o m i n a t i n g o t h e r p e o p l e o r by b e i n g special o r by possessing t h i n g s o t h e r p e o p l e w a n t , but r a t h e r by g i v i n g t h i n g s away - specifically, by g i v i n g away your t i m e , c r e a t i v i t y , a n d t h e results of your skill. W e c a n find this in s o m e o f t h e p r i m i tive h u n t e r - g a t h e r e r s o c i e t i e s , w h e r e a h u n t e r ' s status was n o t d e t e r m i n e d by howm u c h of t h e kill h e a r e , but by w h a t h e b r o u g h t b a c k for o t h e r s . O n e e x a m p l e o f a gift e c o n o m y is r h e p o r l a t c h , w h i c h is part o f t h e p r e - E u r o p e a n c u l t u r e o f t h e Pacific Northwest of North America, In the porlatch ceremony, the host demonstrates his w e a l t h a n d p r o m i n e n c e by giving away possessions, w h i c h p r o m p t s p a m c i p a n r s t o r e c i p r o c a t e w h e n t h e y hold t h e i r o w n pot l a t c h . T h e r e a r e m a n y o t h e r e x a m p l e s o f rhis p h e n o m e n o n . W h a t is c h a r a c t e r i s t i c o f most is that t h e y a r e based o n a b u n d a n c e e c o n o m i e s . T h e r e is usually a surplus o f s o m e t h i n g t h a r it is e a s i e r ro s h a r e t h a n ro k e e p f o r yourself. T h e r e is a l s o t h e u n d e r s t a n d i n g o f r e c i p n t c i t y - that by d o i n g ihis, p e o p l e c a n lower t h e i r i n d i v i d u a l risks a n d i n c r e a s e t h e i r survival ( R a y m o n d , 2 0 0 0 ) . In h u n t e r - g a t h e r e r s o c i e t i e s , freshly k i l l e d g a m e c a l l e d for a gift e c o n o m y b e c a u s e it was p e r i s h a b l e a n d t h e r e was t o o m u c h for a n y o n e person t o ear. I n f o r m a t i o n also loses value o v e r r i m e a n d h a s r h e c a p a c i t y to satisfy m o r e t h a n o n e . I n m a n y cases, i n f o r m a t i o n g a i n s r a t h e r t h a n loses value t h r o u g h sharing. U n l i k e m a t e r i a l o r energy, t h e r e a r e n o c o n s e r v a t i o n laws for i n f o r m a t i o n . O n t h e c o n t r a r y , w h e n divided a n d s h a r e d , t h e v a l u e o ! i n f o r m a t i o n o n l y grows - t h e t e a c h e r does n o r k n o w less w h e n h e shares (us k n o w l e d g e with h i s s t u d e n t s . W h i l e rhe e x c h a n g e e c o n o m y may h a v e b e e n a p p r o p r i a t e tor r h e industrial a g e , t h e gift e c o n o m y is c o m i n g b a c k as w e e n t e r t h e i n f o r m a t i o n aije. Ir s h o u l d be n o t e d t h a t r h e c o m m u n i t y o f s c i e n t i s t s , in a way, follows t h e rules o f a gift e c o n o m y . T h e s c i e n t i s t s with h i g h e s t status are n o t t h o s e w h o possess t h e most k n o w l e d g e , t h e y are t h e o n e s w h o h a v e c o n t r i b u t e d t h e most t o i h e i r fields. A s c i e n tist o f great k n o w l e d g e b u t with n o s t u d e n t s a n d followers is a l m o s t a loser - his o r h e r c a r e e r is s e e n a s a waste ot t a l e n t

H o w e v e r , in s c i e n c e t h e gth c u l t u r e h a s n o t yet

fully p e n e t r a t e d t o t h e level o f d a t a a n d s o u r c e - c o d e s h a r i n g

T h i s culture has been

i n h i b i t e d by a n a n t i q u a t e d a c a d e m i c m o d e l for p r o m o t i o n a n d t e n u r e t h a t is still p r e v a l e n t today T h i s c u l t u r e e n c o u r a g e s d e l a y i n g rhe release of d a t a a n d s o u r c e c o d e to e n s u r e t h a t c r e d i t a n d r e c o g n i t i o n are b e s t o w e d u p o n t h e s c i e n t i s t w h o c o l l e c t e d t h e data and/or d e v e l o p e d t h e c o d e . T h i s m o d e l ( w h i c h was d e v e l o p e d w h e n d a t a w e r e m u c h m o r e difficult t o c o l l e c t a n d analyze, a n d long before c o m p u t e r s a n d p r o g r a m m i n g e x i s t e d ) n o longer applies in t h e m o d e r n s c i e n t i f i c w o r l d , w h e r e n e w s e n sot t e c h n o l o g i e s a n d o b s e r v i n g systems g e n e r a t e m a s s i v e v o l u m e s o f d a t a , a n d w h e r e c o m p u t e r programs a n d n u m e r i c a l m o d e l s h a v e b e c o m e so c o m p l e x that t h e y c a n n o t he fully a n a l y s e d o r c o m p r e h e n d e d by o n e s c i e n t i s t o r evert small t e a m s .

Knowledge-sharing and intellectual property rights For c e n t u r i e s , n o b o d y cared a b o u t " o w n i n g k n o w l e d g e " E i t h e r p e o p l e freely s h a r e d ideas, o r t h e y were kept s e c r e t . T h e idea o f giving k n o w l e d g e out yet r e t a i n i n g s o m e sort o f c o n n e c t i o n t o it, rights for it, was hard t o c o m p r e h e n d , A c t u a l l y , it is still a p r e t t y fluid c o n c e p t , regardless o f t h e n u m e r o u s laws a n d t h e o r i e s t h a t h a v e b e e n

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c r e a t e d s i n c e che British S t a t u r e o f A n n e , from 1 7 1 0 , w h i c h was t h e first copyright act in t h e world. V i c t o r Hugo scruggled wich che c o n c e p t back in 1 8 7 0 :

Before the publication, the author has an undeniable and unlimited right. Think of a man like Dante, Moliere, Shakespeare. Imagine him at the time when he has just finished a great work. His manuscript is there, in front of him; suppose that he gets the idea to throw it into the fire; nobody can stop him. Shakespeare can destroy Hamlet, Moliere Tartu fe, Dante the Hell. But as soon as the work is published, the author is not any more the master. It is then that other persons seize it: call them what you will: human spirit public domain, society. It is such persons who say: I am here; I take this work, I do with it what I believe I have to do, [...] I possess it, it is with me from now on....

An Act for the Encouragement the Authors

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W h e r e a s Printers, B o o k s e l l e r s , and other P e r s o n s , h a v e of late f r e q u e n t l y taken the Liberty of Printing, Reprinting, and Publishing without t h e C o n s e n t of the A u t h o r s or Proprietors of such B o o k s and Writings, to their very g r e a t Detriment, a n d t o o often to t h e Ruin of t h e m and their Families

For Preventing t h e r e f o r e such P r a c t i c e s for t h e future, a n d for t h e E n c o u r a g e m e n t of

L e a r n e d M e n t o C o m p o s e and Write u s e f u l B o o k s ; M a y it p l e a s e Your M a j e s t y , that it m a y b e Enacted,

.. That f r o m and after t h e Tenth Day of April, O n e t h o u s a n d s e v e n hundred a n d t e n ,

the Author of any Book or B o o k s already Printed, . . or . o t h e r P e r s o n or P e r s o n s , w h o hath or h a v e P u r c h a s e d or Acquired t h e C o p y or C o p i e s of a n y Book or B o o k s , in order to Print or Reprint the s a m e , shall h a v e t h e s o l e Right a n d Liberty of Printing such Book and B o o k s for t h e Term of O n e and t w e n t y Years, to C o m m e n c e f r o m t h e saidTenth Day of April, a n d no longer. {http.//www. c o p y r i g h t h i s t o r y . c o m / a n n e . html)

Formally, an i n t e l l e c t u a l property ( I P ) is a knowledge product, w h i c h might be an idea, a c o n c e p t , a m e t h o d , an insight o r a fact, t h a t is manifested e x p l i c i t l y in a p a t e n t , copyrighted material or some o t h e r form, where o w n e r s h i p c a n be defined, d o c u m e n t e d , a n d assigned to a n individual or c o r p o r a t e e n t i t y ( H o w a r d , 2 0 0 5 ) . Ic turned o u t that in most cases it was t h e c o r p o r a t i o n s , c o m p a n i e s , producers and publishers w h o ended up o w n i n g t h e i n t e l l e c t u a l property rights and b e i n g way more c o n c e r n e d about t h e m t h a n authors, even though originally t h e idea was for the " E n c o u r a g e m e n t o f L e a r n e d M e n to C o m p o s e and W r i t e useful B o o k s . " A l t h o u g h t h e c o n c e p t o f public d o m a i n was implicitly considered by t h e S t a t u t e of A n n e , it was clearly arciculated by D e n i s Diderot, w h o was retained by che Parts B o o k G u i l d co draft a treatise o n literary rights. In his Encyclopedic,

Diderot advo-

cated the systemic p r e s e n t a t i o n a n d p u b l i c a t i o n o f k n o w l e d g e o f all t h e m e c h a n i c a l arts and m a n u f a c t u r i n g secrets for t h e purpose o f r e a c h i n g t h e public at large, promot i o n o f research, and w e a k e n i n g t h e grip o f craft guild o n knowledge ( T u o m i , 2 0 0 4 ) . W i t h these p i o n e e r i n g ideas, Diderot set t h e stage for the e v o l v e m e n t o f public d o m a i n , w h i c h includes n o n - e x c l u s i v e I P t h a t is freely, o p e n l y a v a i l a b l e a n d accessible t o any m e m b e r ot t h e society.

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Systems Science and Modeling for Ecological Economics

A c c h e s a m e r i m e , Diderot: w a s p a r t o f a d e b a t e w i t h a n o t h e r F r e n c h

Enlighr-

e n m e n r prominent m a t h e m a t i c i a n , philosopher, and political thinker, t h e Marquis de C o n d o r c e t

(174>-1794),

w h o w a s v o i c i n g e v e n m o r e r a d i c a l ideas a b o u t

intel-

l e c t u a l p r o p e r t y rights. D i d e r o t a r g u e d t h a t ideas s p r a n g d i r e c t l y f r o m t h e i n d i v i d u a l m i n d , a n d t h u s w e r e a u n i q u e c r e a t i o n i n a n d o f t h e m s e l v e s . I n d e e d , t h e y w e r e , in his w o r d s , " t h e v e r y s u b s t a n c e o f a m a n " a n d " t h e m o s t p r e c i o u s part o f h i m s e l f . " I d e a s had n o t h i n g t o d o with t h e physical, natural world; they were s u b j e c t i v e ,

individual

and uniquely c o n s t i t u t e d , a n d thus were t h e most i n v i o l a b l e form o f property. F o r D i d e r o t , p u t t i n g ideas in p u b l i c d o m a i n d i d n o t e n c r o a c h o n t h e p r o p e r t y r i g h t s for t h e s e ideas. F o r h i m , c o p y r i g h t s h o u l d b e r e c o g n i z e d as a p e r p e t u a l p r o p e r t y r i g h t , bestowed upon an a u t h o r a n d inherited by his or h e r offspring. C o n d o r c e t w e n t m u c h f u r t h e r . In s h a r p c o n t r a s t t o D i d e r o t , h e a r g u e d t h a t i d e a s did n o t s p r i n g d i r e c t l y f r o m t h e m i n d b u t o r i g i n a t e d i n n a t u r e , a n d w e r e t h u s o p e n to a l l . C o n d o r c e t s a w l i t e r a r y w o r k s a s t h e e x p r e s s i o n o f i d e a s t h a t a l r e a d y e x i s t e d . T h e form o f a work m i g h t be u n i q u e , but t h e ideas were o b j e c t i v e a n d particular, and could n o t be c l a i m e d as t h e property o f a n y o n e . U n l i k e land, w h i c h could only b e s e t t l e d b y a n i n d i v i d u a l o r a family, a n d passed d o w n by l i n e a g e t o o f f s p r i n g , ideas c o u l d b e d i s c o v e r e d , used a n d c u l t i v a t e d

by a n i n f i n i t e n u m b e r o f p e o p l e a t t h e

same time. F o r C o n d o r c e t , i n d i v i d u a l s c o u l d n o t c l a i m a n y s p e c i a l right o r p r i v i l e g e t o ideas. I n fact, h i s ideal w o r l d w o u l d c o n t a i n n o a u t h o r s a t a l l . I n s t e a d , p e o p l e w o u l d m a n i p u late a n d d i s s e m i n a t e ideas freely for t h e c o m m o n g o o d a n d t h e b e n e f i r o f all. C o p y r i g h t would n o t exist, since n o individual or institution could c l a i m to h a v e a m o n o p o l y o n a n idea. T h e r e g o o u r p a t e n t s ! P u b l i c d o m a i n a n d e x c l u s i v e I P rights r e p r e s e n t t h e t w o e x t r e m e s in I P r e g i m e s , w i t h t h e f o r m e r p r o v i d i n g a free s h a r i n g o f k n o w l e d g e a n d t h e l a t t e r e m p h a s i z i n g r h e r i g h t s o f o w n e r s in l i m i t i n g a c c e s s t o t h e i r k n o w l e d g e p r o d u c t s . S i n c e t h e i n c e p t i o n o f t h e c o n c e p t o f i n t e l l e c t u a l p r o p e r t y r i g h t s , it h a s b e e n argued t h a t p r o t e c t i n g t h e s e r i g h t s p r o v i d e s a d e q u a t e c o m p e n s a t i o n s for o w n e r s a n d e n c o u r a g e s i n n o v a t i o n s a n d technological development. However, historical evidence a n d published research d o n o t s u p p o r t t h e s e c l a i m s , a n d p o i n t t o lack o f c o n c r e t e e v i d e n c e t h a t c o n f i r m s t h e m ( N a t i o n a l A c a d e m y of Engineering, 2 0 0 3 ) . Also, a n d increasingly, m a n y technologic a l i n n o v a t i o n s a r e t h e result o f c o l l a b o r a t i v e e f f o r t s in a n e n v i r o n m e n t t h a t p r o m o t e s n o n - e x c l u s i v e i n t e l l e c t u a l rights. A l t h o u g h most o f t h e s e efforts are in t h e software d e v e l o p m e n t d o m a i n ( e . g . d e v e l o p m e n t o f L i n u x ) , ic is i n t e r e s t i n g t o n o t e t h a t t h e tremendous

growth

and development

in t h e s e m i - c o n d u c t o r

industry

are mainly

attributed t o t h e highly d y n a m i c and c o n n e c t e d social n e t w o r k s o f t h e S i l i c o n Valley in t h e 1 9 6 0 s , w h i c h w a s r e g a r d e d a s a p u b l i c d o m a i n r e g i o n , s i n c e i n f o r m a t i o n a n d k n o w - h o w were freely s h a r e d a m o n g its m e m b e r s . In t h e w o r l d o f b u s i n e s s , p r e s e r v a t i o n o f e x c l u s i v e I P r i g h t s is s e e n as a n e c e s s i t y to maintain c o m p e t i t i v e edge and protect expensively obtained technology.

Patents

that were designed to stimulate innovation are n o w having the opposite effect, espec i a l l y i n t h e s o f t w a r e industry. A s P e r e n s d e s c r i b e s : ' ' P l a g u e d by a n e x p o n e n t i a l g r o w t h in s o f t w a r e p a t e n t s , m a n y o f w h i c h a r e n o t v a l i d , s o f t w a r e v e n d o r s a n d d e v e l o p e r s m u s t n a v i g a t e a p o t e n t i a l m i n e f i e l d t o a v o i d p a t e n t i n f r i n g e m e n t a n d future lawsuits' 1 ( P e r e n s , 2 0 0 6 a ) . T h e b i g c o r p o r a t i o n s s e e m t o s o l v e t h e p r o b l e m by o p e r a t i n g in a detente

m o d e : by a c c u m u l a t i n g

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numbers o f patents themselves,

they

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i n v u l n e r a b l e t o c l a i m s f r o m rivals - c o m p e t i t o r s d o n ' t sue o u t o f f e a r o f r e c i p r o c i t y . H o w e v e r , n o w we see that w h o l e c o m p a n i e s are creaced with t h e sole purpose o f gene r a t i n g profit from patents. T h e s e " p a t e n t parasices" m a k e n o products, and derive

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nil o f cheir i n c o m e from p a t e n t l i t i g a t i o n . S i n c e t h e y m a k e n o products, t h e parasites t h e m s e l v e s are i n v u l n e r a b l e t o p a t e n t i n f r i n g e m e n t lawsuits, and c a n a t t a c k e v e n very large c o m p a n i e s w i t h o u t a n y fear t h a t t h o s e c o m p a n i e s will retaliate. O n e of t h e m o s t e x t r e m e and ugly m e t h o d s is k n o w n as p a t e n t f a r m i n g - i n f l u e n c i n g a standards o r g a n ization co use a p a r t i c u l a r p r i n c i p l e c o v e r e d by a p a t e n t . I n t h e worst a n d most d e c e p tive form o f p a t e n t f a r m i n g , che p a t e n t h o l d e r e n c o u r a g e * t h e standards o r g a n i z a t i o n t o m a k e use o f a p r i n c i p l e w i t h o u t r e v e a l i n g t h e e x i s t e n c e of a p a t e n t c o v e r i n g t h a t p r i n c i p l e . T h e n , later o n , t h e p a t e n t h o l d e r d e m a n d s royalties from all i m p l e m e n t e d o f the standard (Perens, 2 0 0 6 b ) . C e r t a i n l y , t h e s e p a t e n t g a m e s a r e d e t r i m e n t a l for small businesses. A c c o r d i n g t o t h e A m e r i c a n I n t e l l e c t u a l P r o p e r t y L a w A s s o c i a t i o n , s o f t w a r e p a r e n t lawsuits c o m e w i t h a d e f e n s e c o s t of a b o u t $ ) m i l l i o n per a n n u m . A single p a t e n t suit c o u l d b a n k rupt a typical small or m e d i u m - s i z e d a p p l i c a t i o n s d e v e l o p e r ( l e t a l o n e a n o p e n - s o u r c e d e v e l o p e r ) e v e n before che c a s e were fully h e a r d ( N e w s C o m , 2 0 0 5 ) . T h e s m a l l e r p a t e m h o l d e r simply c a n n o t sustain r h e e x p e n s e o f d e f e n s e , e v e n w h e n justified, a n d is f o r c e d t o setcle a n d l i c e n s e p a t e n t s c o t h e larger c o m p a n y

T h e open-source com-

m u n i t y is also c o n s t a n t l y u n d e i t h e t h r e a t o f m a j o r a t t a c k s from large c o r p o r a t i o n s . T h e r e is g o o d r e a s o n t o e x p e c t chat M i c r o s o f t will s o o n b e l a u n c h i n g a p a t e n t - b a s e d legal o f f e n s i v e against L i n u x a n d o t h e r f r e e s o f t w a r e p r o j e c t s ( N e w s F o r g e , 2 0 0 4 >. Unfortunately, universities are increasingly seeking t o capitalize o n knowledge in t h e f o r m o f IP rights. H o w e v e r , o n l y a few of i h e s e u n i v e r s i t i e s a r e g e n e r a t i n g sign i f i c a n t r e v e n u e s f r o m l i c e n s i n g such r i g h t s ( H o w a r d . 2 0 0 5 ) . T h i s a p p l i e s e q u a l l y t o i n d i v i d u a l r e s e a r c h e r s w h o m a y seek to p r o t e c t a n d profit from t h e i r findings.

Software development and collaborative research Just as p u b l i c d o m a i n and e x c l u s i v e IP rights represent t h e t w o e x t r e m e s in IP regimes, t h e software d e v e l o p m e n t process c a n o c c u r in o n e o f t w o ways - e i t h e r t h e ' c a t h e d r a l " o r t h e " b a z a a r " ( R a y m o n d , 2 0 0 0 a ) . T h e a p p r o a c h of most producers o f c o m m e r c i a l , proprietary software is t h a t o f t h e c a t h e d r a l , carefully c r a f t c d by a small n u m b e r ot p e o p l e w o r k i n g in isolation tific research

T h i s is t h e t r a d i t i o n a l a p p r o a c h w e also find in s c i e n -

D i a m e t r i c a l l y o p p o s e d t o t h i s is che bazaar, che a p p r o a c h t a k e n by o p e n -

s o u r c e p r o j e c t s . O p e n source e n c o u r a g e s p e o p l e t o t i n k e r freely wich t h e c o d e , thus p e r m i t t i n g n e w ideas t o b e easily i n t r o d u c e d a n d e x c h a n g e d . A s t h e best o f t h o s e n e w ideas g a i n a c c e p t a n c e , it e s s e n t i a l l y e s t a b l i s h e s a c y c l e o f building u p o n a n d i m p r o v i n g t h e work o f t h e o r i g i n a l c o d e r s ( f r e q u e n t l y in ways t h e y didn't a n c i c i p a c e ) T h e release p r o c e s s c a n be d e s c r i b e d as release early a n d o f t e n , d e l e g a t e e v e r y t h i n g you c a n , b e o p e n . L e a d e r s h i p is essential m t h e O S S world - i . e . , most p r o j e c t s h a v e a lead thac has t h e final word o n w h a t goes in a n d w h a t d o e s n o t F o r e x a m p l e , L i n u s T o t v a l d s has t h e final say o n w h a t is i n c l u d e d in c h e k e r n e l o f L i n u x . I n t h e c a t h e d r a l - b u i l d e r view o f p r o g r a m m i n g , bugs a n d d e v e l o p m e n t p r o b l e m s are tricky, insidious, d e e p p h e n o m e n a . It cakes m o n t h s co weed t h e m all out - thus t h e long release i n t e r v a l s , a n d t h e d i s a p p o i n t m e n t w h e n l o n g - a w a i t e d releases are n o t perfect. I n t h e bazaar view, most bugs b e c o m e s h a l l o w w h e n e x p o s e d t o a t h o u s a n d c o - d e v e l o p e r s . A c c o r d i n g l y , I r e q u e n t release leads t o m o r e c o r r e c t i o n s , a n d , a s a b e n e f i c i a l s i d e - e f f e c t , you h a v e less t o lose if a bug gets through t h e door. It is c l e a r that t h e bazaai a p p r o a c h c a n work i n g e n e r a l s c i e n t i f i c p r o j e c t s , a n d in m o d e l i n g a p p l i c a t i o n s in particular. N u m e r o u s successful e x a m p l e s , e s p e c i a l l y in Earth s y s t e m m o d e l i n g , a t t e s t t o this fact H o w e v e r , w e must also r e c o g n i z e t h a t t h e r e

388

Systems Science and Modeling for Ecological Economics is a difference b e t w e e n software d e v e l o p m e n t a n d s c i e n c e , a n d that software e n g i neers and s c i e n t i s t s have different attitudes regarding software d e v e l o p m e n t . F o r a software e n g i n e e r , t h e e x p o n e n t i a l growth o f c o m p u t e r p e r f o r m a n c e offers u n l i m i t e d resources for t h e d e v e l o p m e n t ol new m o d e l i n g systems. M o d e l s are t h e r e f o r e viewed by engineers as just pieces of software t h a t c a n be built from b l o c k s or o b j e c t s , a l m o s t a u t o m a t i c a l l y , a n d t h e n c o n n e c t e d o v e r t h e w e b a n d distributed over a n e t w o r k o f c o m p u t e r s . It is simply a m a t t e r of c h o o s i n g t h e right a r c h i t e c t u r e a n d writing rhe appropriate c o d e . T h e c o d e is either c o r r e c t o r n o t . e i t h e r it works o r it crashes. N o t so w i t h a scienrtfic model. R a t h e r , most s c i e n t i s t s c o n s i d e r that a model is useful only as an e l o q u e n t simplification o f reality t h a t needs profound u n d e r s t a n d i n g o f t h e system to be built. A model should tell us m o r e a b o u t t h e system t h a n simply t h e data a v a i l a b l e E v e n t h e best model c a n be w r o n g and yet still quite useful i f it e n h a n c e s our u n d e r s t a n d i n g of t h e system. M o r e o v e r , it often rakes a long t i m e t o d e v e l o p a n d test a s c i e n t i f i c m o d e l . A s a result of this difference in point o f view and a p p r o a c h , we t e n d to see m u c h m o r e rapid d e v e l o p m e n t o f n e w languages, software d e v e l o p m e n t tools a n d o p e n c o d e a n d i n f o r m a t i o n - s h a r i n g a p p r o a c h e s a m o n g software e n g i n e e r s . In c o n t r a s t , we see relatively slow adoption ot these tools and a p p r o a c h e s by t h e research m o d e l i n g c o m m u n i t y . T h i s is in spite o f t h e fact t h a t they will undoubtedly catalyze more rapid scientific advancements

As w e b Services e m p o w e r researchers, it ts b e c o m i n g c l e a r

t h a t t h e biggest o b s t a c l e t o fulfilling this vision o f free and o p e n e x c h a n g e a m o n g scientists is cultural. C o m p e t i t i v e n e s s a n d c o n s e r v a t i v e a p p r o a c h e s will always be with us, but d e v e l o p i n g ways t o give meaningful credit t o those w h o share t h e i r data and t h e i r c o d e will be essential in order to c h a n g e attitudes a n d e n c o u r a g e the diversity of m e a n s by w h i c h researchers c a n c o n t r i b u t e to t h e global a c a d e m y {Nature,

2005)

It is c l e a r t h a t a new a c a d e m i c m o d e l thar promotes o p e n e x c h a n g e o f data, software a n d i n f o r m a t i o n is needed

Fortunately, t h e success o f t h e o p e n - s o u r c e a p p r o a c h in

software d e v e l o p m e n t has e n c o u r a g e d researchers t o start c o n s i d e r i n g similar shared o p e n a p p r o a c h e s in scientific research N u m e r o u s c o l l a b o r a t i v e research projects are n o w based o n I n t e r n e t c o m m u n i c a t i o n s , a n d a r e led simultaneously at several instit u t i o n s working o n parts o f a larger e n d e a v o r ( S c h w e t k el al., 2 0 0 5 ) . S o m e t i m e s , such p r o j e c t s are o p e n a n d allow n e w researchers t o participate in t h e work. R e s u l t s and c r e d i t a r e usually shared a m o n g all t h e p a r t i c i p a n t s . T h i s trend is b e i n g fueled by t h e general trend o f increasing funding for large c o l l a b o r a t i v e research p r o j e c t s , particularly in the E a r t h s c i e n c e s .

Open-source software vs community modeling T h e r e c e n t e m e r g e n c e o f o p e n - s o u r c e model d e v e l o p m e n t a p p r o a c h e s m a variety o f different Earth s c i e n c e m o d e l i n g efforts ( w h i c h we refer co here as c o m m u nity m o d e l i n g ) is an e n c o u r a g i n g d e v e l o p m e n t

A l t h o u g h t h e basic a p p r o a c h is t h e

s a m e , we c a n also identify several aspects o f r e s e a r c h - o r i e n t e d c o m m u n i t y m o d e l i n g t h a t distinguish it from a n o p e n - s o u r c e software d e v e l o p m e n t . For e x a m p l e , t h e r e h a v e b e e n a n u m b e r o f successful c o m m u n i t y m o d e l i n g efforts { T a b l e 9 . 1 ) . However, u n l i k e most o p e n - s o u r c e software d e v e l o p m e n t p r o j e c t s , these h a v e b e e n blessed by substantial grant a n d c o n t r a c t support {usually from federal sources), and exist laigely as umbrella p r o j e c t s for e x i s t i n g o n g o i n g r e s e a t c h . It is probably a l s o fair t o say t h a t

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most of t h e e x i s t i n g Earth s c i e n c e c o m m u n i t y models a r e n o t truly " o p e n s o u r c e " - i.e. access to t h e c o d e s a n d rules g o v e r n i n g modification and redistribution are usually more restrictive t h a n , for e x a m p l e , t h o s e under G P L . In general, in c o m m u n i t y m o d e l i n g t h e r e is usually a m u c h smaller n u m b e r o f participants because t h e research c o m m u n i t y is m u c h smaller a n d m o r e specialized t h a n che broad field o f software d e v e l o p m e n t . B e c a u s e t h e pool is smaller, it maybe harder to find t h e right people, b o t h in terms o f their skills and t h e i r willingness to c o l l a b o r a t e w i t h i n an o p e n m o d e l i n g paradigm

Similarly, t h e r e is generally

a m u c h smaller n u m b e r o f users o f o p e n - s o u r c e r e s e a r c h - o r i e n t e d models, w h i c h may be very specialized and usually require specific skills co use. T h i s is mostly because s c i e n t i f i c models are very o f t e n focused o n s i m u l a t i n g a specific p h e n o m e n o n or addressing a specific scientific q u e s t i o n or hypothesis, and also because t h e scientific c o m m u n i t y is very small c o m p a r e d with t h e public at large. A l o n g these same lines, r e s e a r c h - o r i e n t e d models are generally m o r e s o p h i s t i c a t e d and difficult to use t h a n software products that a r e d e v e l o p e d for t h e public. It is c e r t a i n l y m u c h harder to run a m e a n i n g f u l s c e n a r i o with a h y d r o d y n a m i c s i m u l a t i o n m o d e l t h a n to aim a virtual gun at a virtual v i c t i m a n d press t h e " s h o o t " b u t t o n in a c o m p u t e r g a m e ( t h o u g h it might be argued t h a t to a large e x t e n t this difference in difficulty o f use has m o r e to d o with t h e p r i m i t i v e state o f the user interface of most scientific c o d e s ) . It is also generally true t h a t scientific c o d e s require more sophisticated d o c u m e n t a t i o n a n d a s t e e p e r l e a r n i n g curve. D o c u m e n t i n g scientific models is a real problem - it is n o t what researchers n o r m a l l y e n j o y doing, a n d t h e need for doing tt is rarely appreciated a n d funded. O n t h e o t h e r h a n d , d o c u m e n t a t i o n is a crucial part o f t h e process if we a n t i c i p a t e o t h e r s using and taking part m t h e d e v e l o p m e n t o f our models. O p e n research m o d e l i n g is also m u c h more t h a n o p e n programming. A s m e n tioned above, software d e v e l o p m e n t has a clear goal, a n o u t c o m e . T h e product specifications c a n be well established and designed. In contrast, research m o d e l i n g is iterative and i n t e r a c t i v e . T h e goal o f t e n gets modified while t h e p r o j e c t evolves. It is much more a process than a product. It is usually harder to agree o n t h e desired o u t c o m e s and t h e features o f t h e product. In some respects, m o d e l i n g is more like an art t h a n a s c i e n c e . Following this analogy, how d o you get several artists t o g e t h e r to p a i n t o n e picture? T h i s is particularly true in e c o l o g i c a l modeling, where there is n o overa r c h i n g theory to guide model structure and where a variety o f different formulations c a n be used to represent a particular process. T h e s e aspects o f scientific m o d e l i n g actually m a k e it highly a m e n a b l e to o p e n programming approaches, w h i c h naturally allow a high degree o f flexibility.

The Practice of Modeling

A significant infrastructure -

impediment

391

ro d e v e l o p i n g o p e n r e s e a r c h m o d e l s is t h e lack of

t h e r e are still very few g o o d software t o o l s t o support

community

r e s e a r c h a n d m o d e l i n g p r o j e c t s . O n c e a g a i n , t h e r e IS an o b v i o u s gap b e t w e e n software a n d a p p l i c a t i o n . T h e r e is software t h a t p o t e n t i a l l y offers s o m e e x c i t i n g a p p r o a c h e s and

n e w paradigms

ro support

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o r g a n i z a t i o n - all t h e m a j o r c o m p o n e n t s required f o r successful m o d e l and development,

integration

i h e most r e c e n t trends in software d e s i g n are c o m p a r e d w i t h t h e

L e g o c o n s t r u c t o r o v e r t h e well ( M a r k o f t , 2 0 0 6 ) - e x a c t l y w h a t we n e e d for m o d u l a r models. H o w e v e r , this is yet t o he d e v e l o p e d a n d applied t o t h e m o d e l i n g process, a n d e m b e d d e d i n t o t h e m o d e l i n g l e x i c o n a n d p r a c t i c e . Yet a n o t h e r d i f f e r e n c e is t h a t most research m o d e l i n g p r o j e c t s takes years t o d e v e l o p . T h i s is in c o n t r a s t t o s o m e o f t h e software h a c k s t h a t c a n b e i n v e n t e d a n d i m p l e m e n t e d tn a m a t t e r of hours, q u i c k l y g a i n i n g r e c o g n i t i o n a n d respect in t h e solrware d e v e l o p m e n t

community.

R e s e a r c h is a m u c h slower a n d tedious process, w h e r e small i n c r e m e n t a l ideas a n d s u c c e s s e s may be very i m p o r t a n t , but are m u c h h a r d e r t o d o c u m e n t , d i s s e m i n a t e a n d appreciate. Finally, r e t u r n i n g t o t h e c e n t r a l

problem,

we really n e e d t o c h a n g e t h e tra-

d i t i o n a l c u l t u r e a n d a t t i t u d e s of r e s e a r c h s c i e n t i s t s - t h a t is, p r o m o t e a shift in t h e m i n d s e t anil p s y c h o l o g y that drives s c i e n t i f i c r e s e a r c h . Historically, m o s t s c i e n c e has b e e n d r i v e n by individual efforts a n d t a l e n t . T h e t a l e n t a n d i n g e n u i t y of i n d i v i d u als will always lie c r i t i c a l in s c i e n t i f i c e x p l o r a t i o n , but with t h e g r o w i n g a m o u n t of d a t a , k n o w l e d g e a n d i n f o r m a t i o n , m o s t o f t h e b r e a k t h r o u g h a c h i e v e m e n t s are n o w p r o d u c e d by ream efforts, w h e r e t e a m s a n d t e a m w o r k r a t h e r t h a n

individuals are

key. T h i s rrend is beiiyg d r i v e n t o a large e x t e n t by t h e i n c r e a s i n g e m p h a s i s in s c i e n tific research o n large p r o j e c t s a i m e d at s o l v i n g c o m p l e x i n t e r d i s c i p l i n a r y p r o b l e m s , s u c h as s i m u l a t i n g a n d p r e d i c t i n g t h e E a r t h system response t o global warming. It is b e c o m i n g i n c r e a s i n g l y difficult t o identify t h e sole i n d i v i d u a l w h o cries " E u r e k a ! " and solves t h e p r o b l e m . E v e n w h e n this d o e s u c c u r , very o f t e n r h e r e c o g n i t i o n is biased by past success, hierarchy, a n d p e r s o n a l i t i e s T h e r e is a n o b v i o u s need for n e w award and credit systems that will s t i m u l a t e s h a r i n g a n d t e a m w o r k r a t h e r t h a n d i r e c t personal g a m , c r e d i t a n d fame By sharing r h e d a t a a n d c o n c e p t s over t h e web, p o t e n t i a l users are i n v i t e d t o j o i n in c o l l a b o r a t i v e r e s e a r c h a n d analysis of t h e future trends o f w a t e r s h e d d e v e l o p m e n t . T h e i r f e e d b a c k is s o l i c i t e d for f u r t h e r d i s s e m i n a t i o n a n d i m p r o v e m e n t of k n o w l e d g e a b o u t t h e w a t e r s h e d system. T h e m a n a g e m e n t a n d d e c i s i o n - m a k i n g are d i s c l o s e d t o t h e p u b l i c , offering a b r o a d s p e c t r u m ol v i e w s a n d values, a n d i n v i t i n g s t a k e h o l d e r s t o b e c o m e p a r t i c i p a n t s in a truly d e m o c r a t i c process o f d e c i s i o n - m a k i n g B e y o n d s e p a r a t e p r o j e c t s i n v o l v i n g P M . we c a n e n v i s i o n t h e m c o m i n g t o g e t h e r in an i n t e g r a t e d effort t o support w h o l e e c o s y s t e m a n d w a t e r s h e d m a n a g e m e n t , w h i c h is a h o l i s t i c a n d integral w a y o f r e s e a r c h , analysis a n d d e c i s i o n - m a k i n g at a watershed scale

In t h e 1 9 9 0 s a n d e v e n earlier, t h e r e was m u c h h o p e for this a p p r o a c h

II

c e r t a i n l y implies m o r e t h a n just t h e r e g i o n a l scale ol analysis. T h e m e t h o d stresses i h e n e e d to i n t e g r a t e n o t o n l y physical a n d b i o l o g i c a l factors, b u t also p o l i t i c a l a n d socio-economic ones

T h e m a j o r i m p e t u s foi w a t e r s h e d m a n a g e m e n t s t e m m e d from

t h e u n d e r s t a n d i n g t h a t s c i e n c e n e e d s to b e linked co p l a n n i n g , a n d t h a r d e c i s i o n m a k i n g should lie based o n broad c i t i z e n i n v o l v e m e n t

T h u s it is i m p o r t a n t t h a t t h e

i n f o r m a t i o n is s h a r e d h e t w e e n t h e s t a k e h o l d e r s a n d t h a t ic is processed i n t o a f o r m a t readily p e r c e i v e d by wide a n d diverse groups, i n s t i t u t i o n s a n d individuals. M o r e o v e r , t h e w a t e r s h e d d e l i n e a t e s a physical b o u n d a r y a n d n o t a p o l i t i c a l o n e , c r e a t i n g r h e n e e d for m e t h o d s chat will allow m a n a g e m e n t a n d c o m m u n i c a t i o n

between

many

392

Systems Science and Modeling for Ecological Economics a d m i n i s t r a t i v e e n t i t i e s such as towns, c o u n t i e s and states. O n e o f t h e problems rhat watershed m a n a g e m e n t i m m e d i a t e l y e n c o u n t e r e d was t h e m i s m a t c h b e t w e e n t h e e x i s t i n g a d m i n i s t r a t i v e h i e r a r c h i e s and t h e physical a n d societal boundaries and groupings thar represented

t h e watershed d y n a m i c s . A p p r o p r i a t e

institutions are

required that c a n o p e r a t e in a flexible m a n n e r o v e r a l t e r n a t i v e regional divisions. T h e tact thac ecosystem m a n a g e m e n t seeks a l t e r n a t i v e m e c h a n i s m s t o purely market forces based o n t h e existing policy equilibrium seems to be very b o t h e r s o m e to traditional e c o n o m i s t s (Fitzsimmons, 1 9 9 4 ) T h e y argue t h a t t h e ecosystem c o n c e p t is inappropriate for use as a geographic guide for public policies. Mostly, chough, they are c o n c e r n e d t h a t t h e ecosystem approach will significantly expand federal and o t h e r n o n - m a r k e t c o n t r o l of t h e use o f privately-owned land, and lead to increased restrictions o n rhe use o f public lands for e c o n o m i c purposes. Lackey ( 1 9 9 8 ) identified five general c h a r a c t e r i s t i c s for ecosystem m a n a g e m e n t problems. 1. Public and private values and priorities are in dispute, resulting in mutually e x c l u sive d e c i s i o n a l t e r n a t i v e s 2 . T h e r e is political pressure to make rapid and significant c h a n g e s in public policy 3 . Private and public stakes are high, with substantia! costs and risks ( s o m e irreversible) to s o m e groups 4 . T h e t e c h n i c a l , e c o l o g i c a l and s o c i o l o g i c a l facts are highly u n c e r t a i n 5 . Policy decisions will h a v e effects outside t h e scope o f t h e p r o b l e m . T h e s e seem like e x a c t l y t h e type o f difficulties t h a t c a n be resolved with P M . H e c o n c l u d e s t h a t "solving these kinds o f problems in a d e m o c r a c y has been likened to asking a pack o f four hungry wolves and a s h e e p to apply d e m o c r a t i c principles to deciding what to e a t for l u n c h " (Lackey, 1 9 9 8 : 2 2 ) . T h e o u t c o m e may seem quite obvious, e x c e p t that with people t h e r e is always less c e r t a i n t y a b o u t h o w problems are resolved, and in t h e long run there ts still a c h a n c e for t h e s h e e p to persuade t h e wolves to b e c o m e vegetarians. T h e success o f this e n d e a v o r b e c o m e s very m u c h d e p e n d e n t o n how efficiently t h e n e w t e c h n o l o g y is d e v e l o p e d a n d used, s i n c e it is o u r scientific, cultural a n d social d e v e l o p m e n t t h a t makes

H C M O

Sapiens

spe-

cial and leaves c e r t a i n space f o r o p t i m i s m . In this c o n t e x t we d o n o t view t e c h n o l ogy as a p a n a c e a t h a t c a n cure all t h e p r o b l e m s o f e n v i r o n m e n t a l degradation a n d resource d e p l e t i o n , but rather as a m e a n s o f understanding, educating, and resolving conflict. Regional

management

implies a close

interaction

and linkage b e t w e e n t h e

numerous agents a c t i n g in t h e region. T h e efficacy o f this i n t e r a c t i o n is a f u n c t i o n of t h e i n f o r m a t i o n that is shared a m o n g a n d used by all t h e stakeholders. In many cases, it d e p e n d s n o t so m u c h o n t h e quality and a m o u n t o f t h e i n f o r m a t i o n availa b l e ( w h a t s c i e n c e h a s been mostly c o n c e r n e d with all this t i m e ) bur rather on howwell t h e i n f o r m a t i o n is disseminated, s h a r e d a n d used. A n d that is e x a c t l y t h e funct i o n that t h e P M t e c h n i q u e s c a n offer, especially if they are e n h a n c e d by the W e b technologies. A s with t h e a d v e n t o f any n e w t e c h n o l o g y , it has t a k e n s o m e time to realize all t h e benefits and advantages that t h e I n t e r n e t c a n deliver. U n t i l 1 9 9 2 it was t b e realm of a relatively small c o n t i n g e n t o f s c i e n t i s t s a n d engineers, w h o were using it to c o m m u n i c a t e data a m o n g t h e m s e l v e s , and b o t h t h e sender and the r e c i p i e n t o f i n f o r m a t i o n were usually personally defined. T h e W e b o p e n e d a n e w page in t h e

HMNSNMPPHMIMHMMBBOTi The Practice of Modeling

393

use a n d d e v e l o p m e n t o f t h e I n t e r n e t . I n f o r m a t i o n was n o longer personally targeted; o n c e posted ro t h e n e t it b e c a m e open to any user w h o h a d t h e interest a n d time to view it. Basically, t h e W e b to t h e I n t e r n e t is t h e same as t h e radio is to postal services. Instead of mailing a letter t o a definite addressee, i n f o r m a t i o n could be n o w aired as if b e i n g broadcast o v e r a radio o r t e l e v i s i o n n e t w o r k , with t h e sender n o longer k n o w i n g w h o t h e r e c i p i e n t is t o be. In this way, rhe a u d i e n c e s e x p a n d e d dramatically a n d a r c still growing rapidly. A m a j o r a d v a n t a g e of t h e W e h , c o m p a r e d with o t h e r mass media, is t h a t it is relatively c h e a p . A s a result, 111 a d d i t i o n to t h e businesses t h a t are eager t o e m p l o y a n o t h e r o p p o r t u n i t y for a d v e r t i s e m e n t a n d sales, the W e b offers a whole n e w way o f o u t r e a c h a n d c o m m u n i c a t i o n to g o v e r n m e n t a l , a c a d e m i c a n d non-profit organisations. E v e n individuals c a n afford to establish t h e i r p r e s e n c e in this mass mediaA n o t h e r advantage o f t h e W e b is t h a t it provides for direct feedback from t h e r e c i p i e n t , w h o c a n n o w interact

with t h e i n f o r m a t i o n displayed. Instead of just

passively viewing i n f o r m a t i o n , website visitors c a n c h a n g e and modify it r e m o t e l y Users are offered search e n g i n e s t h a t c a n direct t h e m t o t h e most relevant

infor-

m a t i o n a v a i l a b l e ; t h e y c a n revisit sites a n d refer o t h e r s t o t h e m . U n l i k e o t h e r mass media, t h e W e b is more stable a n d persistent in t h e sense t h a t , unlike o t h e r mass media such as radio, where o n c e i n f o r m a t i o n has been aired it is n o longer retractable, o n t h e W e b che i n f o r m a t i o n stays w h e r e it was a n d c a n he easily r e f e r e n c e d a n d downloaded. In spite o f these n o v e l features, most o f t h e use o f t h e I n t e r n e t does n o t seem t o be m u c h different from t h a t o f t h e traditional mass media o r a r c h i v e d informat i o n (libraries, data sets, e t c . ) - Business is driving a vast m a j o r i t y o f web a p p l i c a t i o n s towards a d v e r t i s e m e n t a n d sales in a way very similar t o t h a t w h i c h m a y b e observed o n radio a n d T V , a n d in u n s o l i c i t e d mail a n d catalogs- T h e r e are just a few e x a m ples w h e n t h e W e h is used in an i n n o v a t i v e way t h a t employs s o m e o f its u n i q u e features. The

c o n s e n s u s building power of t h e " i n f o r m a t i o n a l

superhighway"

created

o n t h e W e b h a s n o t b e e n used to "full s p e e d . " W e argue that t h e r e are a n u m b e r o f features t h a t m a k e t h e W e b a n e x c e p t i o n a l l y i m p o r t a n t tool for watershed m a n a g e m e n t in particular, a n d for d e c i s i o n support a n d m a n a g e m e n t in general. T h e W e b is: • Open. T h e I n t e r n e t is o n e of t h e most readily a v a i l a b l e a n d reliable media, providing

information

across

geographical,

administrative,

social

and e c o n o m i c

boundaries. It is relatively c h e a p , and c a n be accessed by all t h e s t a k e h o l d e r s m a watershed a n d outside o f it T h e fact that it requires a c o m p u t e r ( o r a d v a n c e d T V set - " W e b - T V " ) a n d an I n t e r n e t c o n n e c t t o n is b e c o m i n g less a n d less restrictive as m o r e I n t e r n e t S e r v i c e Providers ( I S P ) e n t e r t h e market

For those who do not

h a v e W e b a c c e s s at h o m e o r at work, there are public providers (libraries, " w e b c a f e , " e t c . ) that also h a v e b e c o m e more a v a i l a b l e . T h i s direct a c c e s s t o all t h e necessary i n f o r m a t i o n a n d , reciprocally, t h e ability t o d i s s e m i n a t e t h e facts that are ol c o n c e r n t o particular s t a k e h o l d e r s is an i m p o r t a n t prerequisite of watershed management. •

Interactive.

It is most i m p o r t a n t for m a n a g e m e n t purposes that t h e user has t h e

o p t i o n o f i n t e r a c t i n g with t h e provider o f i n f o r m a t i o n 3 n d with o t h e r s t a k e h o l d ers. W i t h the I n t e r n e t , this c a n he a c c o m p l i s h e d either v i a e - m a i l or directly t h r o u g h forms, wikis or hlogs t h a t c a n be part o f w e b pages a n d transmitted to

394

Systems Science and Modeling for Ecological Economics t h e server. T h e s e c o n t r i b u t i o n s c a n be further m a n u a l l y or a u t o m a t i c a l l y

proc-

essed and posted back o n t h e W e b . I n this case, i n f o r m a t i o n is n o t only passively perceived, as in case of t h e traditional

media

(radio, press, newsletter,

etc.);

it also s t i m u l a t e s direct feedback. M o r e o v e r , users c a n modify t h e c o n t e n t and f o r m a t o f t h e existing pages by ordering e x c e r p t s from data bases o r providing scenarios for m o d e l runs, a n d thus c r e a t i n g their own output to be i m m e d i a t e l y viewed o n t h e W e b . T h e y may also provide additional i n f o r m a t i o n to the W e b in response to t h e published requests o r as a r e p r e s e n t a t i o n o f their own findings and concerns. •

Fust. C o m m u n i c a t i o n s via t h e I n t e r n e t are probably t h e fastesc a n d t h e most e c o n o m i c , s i n c e they d o n o t require a n y i n t e r m e d i a t e carriers (as tn ordinary m a i l ) and materials (paper). O n c e t h e i n f o r m a t i o n is updated o n t h e server, ir b e c o m e s immediately available for further use a n d processing. T h e f e e d b a c k in many cases c a n be h a n d l e d a u t o m a t i c a l l y and directly c h a n n e l e d t o t h e appropriate w e b link or interest group.

• Spatially

distributed.

I n t e r n e t access is offered over t e l e p h o n e lines and t h e r e f o r e

c o v e r s almost t h e e n t i r e p l a n e t . T h e various nodes o n t h e I n t e r n e t c a n correspond and represent t h e spatially distributed data o f different stakeholders in t h e watershed and outside it. T h e w e b tools allow i n f o r m a t i o n to be linked together,- search e n g i n e s are c r e a t e d to find t h e necessary i n f o r m a t i o n and data. In this way, c o n c e r n s and awareness c a n be shared across different g e o g r a p h i c localities. T h i s gives a broader picture o f t h e system at stake w i t h i n t h e f r a m e w o r k o f e x t e r n a l systems and c o n c e r n s . •

Hierarchical.

T h e h i e r a r c h i c a l structure supported by che W e b design allows organ-

ization o f t h e data in logical and efficient ways w h e n various b r a n c h e s o n t h e W e b may present specific fields, d o m a i n s a n d interest groups. T h e links o n w e b pages c a n s t i t c h t h e w h o l e structure together, offering c r o s s - r e f e r e n c e s and a l t e r n a t i v e views w h e n e v e r necessary. F o r e x a m p l e , t h e watershed hierarchy of subwatersheds and sub-subwatersheds c a n he easily mirrored o n t h e W e b , with specific groups o f pages representing e a c h particular level. T h e h i e r a r c h i c a l structure also offers levels o f p r o t e c t i o n for t h e i n f o r m a t i o n , a l l o w i n g c e r t a i n d o m a i n s to be c o m p l e t e l y o p e n to all users, o t h e r s only r e a d - p e r m i t t e d , and still o t h e r s accessible o n l y to limited users and interest groups, providing t h e necessary e x t e n t o f privacy and discretion. •

Flexible.

A d d i t i o n a l benefits t h a t are offered by t h e i n t e r a c t i v e features allow t h e

data to be processed by users a c c o r d i n g to cheir o w n goals and interests. 1 his is especially i m p o r t a n t for m o d e l i n g tools, because by e m p l o y i n g t h e W e b they c a n be made directly accessible to t h e user, and sufficiently flexible and user-friendly to be used m e a n i n g f u l l y a n d efficiently. C u r r e n t l y , web a p p l i c a t i o n s are b e i n g used at t h e h i g h - s c h o o l level t o t e a c h s c i e n c e and ecology. T h e s c o p e o f p o t e n t i a l uses ranges from r u n n i n g particular scenarios, which s t a k e h o l d e r s c a n formulate based on t h e i r c o n c e r n s , to a d j u s t m e n t s in scale and structural detail o f che m o d e l in response to special needs and projects. A l l t h e i m p o r t a n t features and tools to a u g m e n t a n d improve d e c i s i o n support and m a n a g e m e n t seem to b e present, and it t h e n b e c o m e s a m a t t e r o f using t h e m efficiently. T h i s is really handy for supporting t h e P M process and m a k i n g it evolutionary and adaptive o v e r t h e web, such thar it c a n remain an o n g o i n g a c t i v i t y e v e n w h e n t h e c u r r e n t p r o j e c t has reached its goals and a c e r t a i n d e c i s i o n h a s been made.

The Practice of Modeling

395

N o m a t t e r h o w good a n d a p p r o p r i a t e a d e c i s i o n , an o p e n s y s t e m t e n d s t o c h a n g e a n d e v o l v e , a n d d e c i s i o n s will e v e n t u a l l y n e e d t o be reassessed a n d a d a p t e d t o n e w d e v e l o p m e n t s a n d n e w d a t a . T h e w e b p r e s e n c e ot s t a k e h o l d e r s a n d t h e n previous efforts as part o f a P M p r o j e c t , t o g e t h e r with m o d e l i n g coots a n d d a t a chac h a v e b e e n d e v e l oped a n d r e s e a r c h e d , should r e m a i n a v a i l a b l e tor future a p p l i c a t i o n s . Fticure p r o j e c t s will t h e n n o t n e e d co start from s c r a t c h , as t h e r e will be a c c e s s co all che p r e v i o u s l y c o l l e c t e d information, and, even more importantly, t h e social capita, o f social n e t works a n d links d e v e l o p e d as part o f t h e p r e v i o u s P M a d v e n t u r e . A P M p r o j e c t b e c o m e s a kind o f o p e n - s o u r c e p r o j e c t with various s t a k e h o l d e r s c o n t r i b u t i n g t o ic in various roles. S o m e will be a d m i n i s t e r i n g t h e process a n d guiding ics progress, o t h e r s will be c o n t r i b u t i n g b i t s o f d a t a a n d k n o w l e d g e , o t h e r s will be d e v e l o p i n g m o d e l s a n d a n a l y t i c a l cools, w h i l e yec o c n e r s will b e writing d o c u m e n t a t i o n a n d d i s s e m i n a t i n g resulrs t o o t h e r i n t e r e s t e d parties. T h i s is very s i m i l a r t o t h e s t r u c t u r e o f m a n y o p e n - s o u r c e software projeccs, t h o u s a n d s o f w h i c h are a d m i n i s tered by S o u r c e F o r g e at htrp://www . s o u r c c f o r g e . n e t — a p o w e r s h o p tot o p e n s o f t w a r e development.

9.4

Conclusions M u c h h u m a n crearivicy is geared cowards m o v i n g e n e r g y a n d m a t e r i a l s r a t h e r chart information,

even

though

information

hasbecome

another

crucial

component

o f h u m a n welfare a n d l i v e l i h o o d - I n f o r m a t i o n , u n l i k e e n e r g y a n d m a t e r i a l s , is n o t s u b j e c t t o c o n s e r v a t i o n laws By c o p y i n g i n f o r m a t i o n from sources a n d disti i b u t i n g it t o n e w d e s t i n a t i o n s , we d o n o r lose i n f o r m a t i o n a t c h e s o u r c e . T h i s is what is k n o w n as n o n - r i v a l goods in e c o l o g i c a l e c o n o m i c s ( D a l y a n d Farley, 2 0 0 4 ) . A s with gravity, by using i n f o r m a t i o n we d o n o t d e c r e a s e t h e a b i l i t y o f o t h e r s t o use it- N e v e r t h e l e s s , exchange

of i n f o r m a t i o n

is rescricted by p a t e n t

law, as well

as by i n s t i t u t i o n a l ,

culcural a n d t r a d i t i o n a l hurdles t h a t c r e a t e p r o t e c t i v e barriers h i n d e r i n g t h e free flow1 of

chiS

v a l u a b l e c o m m o d i t y - In this way, we are m a k i n g it e x c l u d a b l e . It is n o t

surprising t h a t p r i v a t e c o m p a n i e s are o f t e n r e l u c t a n t

t o s h a r e d a t a a n d software,

b e c a u s e it c a n i m p a c t t h e i r profits in a c o m p e t i t i v e m a r k e t . U n f o r t u n a t e l y ,

barri-

ers tt) i n f o r m a t i o n e x c h a n g e a r e also s i g n i f i c a n t in t h e a c a d e m i c c o m m u n i t y , w h e r e r h e l o n g - s t a n d i n g e m p h a s i s o n p u b l i c a t i o n a n d ( p e r h a p s u n w a r r a n t e d } fear o f misuse of released d a t a a n d sofeware h a v e i n h i b i t e d free a n d o p e n e x c h a n g e . a n d t e n u r e p.c a c a d e m i c of p e e r - r e v i e w e d

institutions

is still

p u b l i c a t i o n s a n d success

largely d e p e n d e n t in s e c u r i n g grant

upon

Promotion che v o l u m e

and contract

funds-

A s a result, a c a d e m i c s c i e n t i s t s h a v e l i t t l e o r n o i n c e n t i v e t o spend t h e t i m e a n d effort that is r e q u i r e d t o d o c u m e n t a n d d i s s e m i n a t e t h e i r d a t a and/or i h e i r m o d e l s and

code

for che g r e a t e r

good

o f t h e research

community.

This

p r o b l e m is

e x a c e r b a t e d by che fact t h a t g r a n t a n d c o n t r a c t f u n d i n g f o r research, rarely provides d i r e c t support for d o c u m e n t a t i o n a n d d i s s e m i n a t i o n a c t i v i t i e s . T h e issue is p a r t i c u larly a c u t e w h e n it c o m e s t o s h a r i n g t h e s o u r c e c o d e o f m o d e l s a n d d a t a analysis software - e v e n if a scientist o r e n g i n e e r is a m e n a b l e co s h a r i n g c h e c o d e , t h e e f f o r t required t o p r o v i d e d o c u m e n t a t i o n t o m a k e it useful is o f t e n v i e w e d as a n Insurmountable obstacle. F u n d i n g a g e n c i e s worldwide s e e m enhance

communication

a n d promote

co r e c o g n i z e c l e a r l y t h e pressing n e e d t o open

exchange

o f data

and information

a m o n g s c i e n t i s t s a n d b e t w e e n a c a d e m i c a n d p r i v a t e i n s c i t u t i o n s via t h e I n t e r n e t .

396

Systems Science and Modeling for Ecological Economics } lie N a t i o n a l S c i e n c e F o u n d a t i o n , for e x a m p l e , h a s initiated several n e w m a j o r research

initiatives that a r e a i m e d a t d e v e l o p i n g and/or explicitly requiring

enhanced communication

T h e s e i n i t i a t i v e s include N E O N

(National

this

Ecological

O b s e r v a t o r y N e t w o r k ) , C L E A N E R ( C o l l a b o r a t i v e L a r g e - S c a l e E n g i n e e r i n g Analysis N e t w o r k for E n v i r o n m e n t a l R e s e a r c h ) , C U A H S I the Advancement

o f Hydro logical S c i e n c e

( C o n s o r t i u m o f U n i v e r s i t i e s for

I n c . ) and O R I O N

(Ocean

Research

I n t e r a c t i v e O b s e r v a t o r y N e t w o r k ) , t o n a m e just a lew. T h e E u r o p e a n U n i o n has funded such o p e n - s o u r c e projects as H a r m o n - I ! a n d S e a m l e s s . A l l o f these initiatives e m b r a c e t h e idea t h a t d e v e l o p i n g t h e infrastructure needed t o allow free a n d o p e n e x c h a n g e o f large v o l u m e s o f data a n d i n f o r m a t i o n will he c r u c i a l for m a k ing rapid scientific a d v a n c e m e n t s in t h e future

F o r e x a m p l e , t h e success of c u r r e n t

efforts t o d e v e l o p Earth observatories in both terresrnal (e.g. N E O N S a n d m a r i n e (e.g O R I O N ) e n v i r o n m e n t s will b e critically d e p e n d e n t upon t h e successful develo p m e n t o f this infrastructure, because these observatories will have to c o l l e c t , proi_ess a n d d i s s e m i n a t e large v o l u m e s of data a n d assimilate them i n t o models m a timely m a n n e r . Ihe

challenges

we f a c e

in c r e a t i n g

a n e w tesearch

paradigm

S u b s t a n t i a l i m p r o v e m e n t s in hardware (e.g. n e t w o r k and c o m p u t i n g

are many.

infrastructure)

and software ( e . g . database m a n i p u l a t i o n software a n d data-assimilating

numerical

m o d e l s ) , a n d a m u c h higher level o f s t a n d a r d t i a r i o n of d a t a formats, will be required. N e w m e a n s for carrying out r e a l - t i m e data processing a n d a u t o m a t e d data quality c o n t r o l will also have t o b e developed

However, we believe t h a t o n e of t h e great-

est c h a l l e n g e s we face in this e n d e a v o r is butldirfa t h e c o m m u n i t y - m o d e l i n g a n d i n f o r m a t i o n - s h a l i n g culture t h a t will he required for success. H o w d o we get e n g i neers and s c i e n t i s t s t o put aside their traditional modes of doing business? H o w d o we provide t h e i n c e n t i v e s t h a t will be required to m a k e these c h a n g e s h a p p e n ' H o w do we get o u r c o l l e a g u e s t o see t h a t t h e benefits o f sharing resources far o u t w e i g h t h e c o s t s 1 T i m e l y s h a n n g o f data a n d i n f o r m a t i o n is in the best interests n o t o n l y of t h e research c o m m u n i t y , hut also of che scientist why is doing t h e sharing - substantial additional henefits will be derived through n e w c o n t a c t s , c o l l a b o r a t i o n s a n d a c k n o w l e d g e m e n t t h a t a r e fostered hy open e x c h a n g e . N u m e r o u s e x a m p l e s a t t e s t ro this fact. T h e real c h a l l e n g e we face is getting our c o l l e a g u e s t o t e c o g n t : e t h e p o t e n tial benefits thac c a n be derived from adopting a c o m m u n i t y - m o d e l i n g a n d inlormat i o n - s h a r i n g c u l t u r e . J o a d d i t i o n , we need to dispel t h e unwarranted fears t h a t manys c i e n t i s t s a n d e n g i n e e r s harbor: that they w j l he " s c o o p e d " if they release t h e i r d a t a too soon or b l a m e d if t h e r e is a hug tn their code. Finally, we n e e d t o a c c e p t rhe f a c t that releasing u n d o c u m e n t e d o r poorly d o c u m e n t e d software ts preferable to n o t releasing it ac all-

Further reading Tfie enci of finrmw nniiprions is Qpalyted by Diamond, J. (2005), in Collars s Hiw Societies Choose to Fat! nr Sutieeif, Penguin, 592 pp. There is a g?cruAt\g b«K of literature an human behavicrr You eii|f read Jfittfc aijOut c.xr.piex. economicfcehmwrm W.R. Arthur. S.N. Duilaut. and D. Lnnt. 1997. The economy as an evolving complex system 11 Santa Fe Institute Studies m ifit' Science oj CiOTipfextcy, Vol XXVIU Addison-'Wesley; K.L. JudJ, and L Te^itsion, 2006. Handbook ot Computational Economics Volume II: Agent-Based Computational Economics- Elsevier B.V., Kirmsn. A,P., Whom or what does the representative individual represent' Journai of Economic Pm pec rives, 1992.

The Practice of Modeling

397

6 ( 2 ) : pp 1 1 7 - 1 3 6 . Then; is a u'liole .Journal of Economic Bdvuiur & Orficimwticwi pablisfaJ by S/JTingLT. An inierfirin^ analysis of human behauor from a psychologist's viewportt is found in Whybrow. P C . ( 2 0 0 6 ) American Mania When More Is Not Enongli. W. W. Norton, .352 pp To reari more dhotit clhrmie change you can start uiifi ine pfijjes from ihe Union af Cimeerned Scientists nt http://www.LC5.us3.org/global_wriiiriing/. Here you will find all ike bask

flaw. For

muTe ui-depth analysis, £0 to ihe Real Climate hlog tu httpV/realclimmc ora'. it has urtic'dS fat till lei els, storting from very bask facts up to quite sophisticated

and technical discussions oI particular

problems and issues. The history of climate change research i.s described by James Fleming, 2007. Intimate Universality: Local and Global Themes

in the History of Weather and Climate.

Sciancc History Publications.

284 pp. Some

m i p u i t a n i issues r e l a t e d

to model failures are discussed

in

a

Jvjsinnn

paper by

tun

do ten

authors:

B.S. Mcintosh, C . Giupponi. A . A . Voinov, C Smith, K B. Matthews, M. MonticiH". M J . Kolktnan, N Grossman, M. van Irrersum, L). Haase. A. Haase, J. Mysiak, J.C.J. Grout, S. Sieber, P. Verweij, N. Qinnn, P. Wacger, N. Gubcr, D Hepting, H. Scholtent A. Sulisu, H van Delden, E. Gaddis, H Assaf, 200S. Brid£in[t rhe gap; between design and use: developing rcx>U to support environmental management and policy In lakenian, A . ('lien, S., A Ruzoh. A A Yi nnov. (Eds ) .Suite of the an and (unites in Environmental Modelling anil Software. Elsevier (in press) science means for us and how it changes some o]

The almost classic description of what post-normal our paradigms,

can be found in K m r o w i a , S and Ravcrz, J.R ( 1 9 9 3 ) Science for the post-nor-

mal age. Futures 25 ( 7 ) , 7 3 9 - 7 5 5 . Kasenur. R , lager. ]•• laeger. C C and Gardner. M T (eds.) (20C3)

Public pan id/union in

justainabiliry science: A Iwndbook Cambridge University Press, Cambridge. 316 pp. - Thrj is on important colleciion of papers on {Jtirttcifxitory resefl'e.h There is nut very much about modeling in there, but it does lay out some very impciant

principles far stakeholder

mwAvement and public

participation. For mere details about HubNet - the participatory northwestern.edii/nerlogo/docs/hubner.html

modeling comjxmeni

t:f NeiLugo

see hirp://ccl.

There is also rt link to a collection of /jflflici|xi£u';>

pw/ecrs thill use this tool: http://ccl.ncirthwesiern.edu/partsinis.html Ofhei Ta'euint papers on picriicipiitirry modeling art; Korlmacher, K. S., 20C1 T h e politics of participation in watershed modeling. Environmental Management 2 7 ( 2 ) , 1 6 1 - 1 7 6 . Beirele, T.G. and Cayfoid, I ( 2 0 0 2 ) . Etemoeracy in pracrice: Public participation in cnvironmenral decisions Resources fur rhe Future, Washington, D.C.. Carr, D S . and Halvorsen. K (20C1) An evaluation of rhrce democratic, community-based approaches to ciliien participation. Survey!, conservations with community groups, and community dinners. Society rmri Nntiocl Resources, 14: 1 0 7 - 1 2 6 . Duram, L.A. and Brown. K G . ( 1 9 9 9 ) . Assessing public participation iu U . S . watershed initiatives. Society and Natural Resources, 12: 4 5 5 - 4 6 7 . Gough, C . E . and Darier,

( 2 0 C ) ) Contexts of ciri;en participatiun

Public pariicipwion in

suscarnablliry science. In: B kasemu, C-G. Jaeger. 1 Jager and M.T. Gardener (eds ), Public participation in sustainahilir-f science: A handbook Cambridge University Press, Cambridge p. >16 The Solomons Harbor project is described m Gaddis, E. Vladich, H , and Voinov, A (2007). Participator,' modeling and the dilemma of diffuse nitrogen management in a residential watershed. Etivi'tm. .Model. Snfiware. 22. 5, 6 1 9 - 6 2 9 . For more information about the S: Albans project see U S D A ( 1 9 9 1 } . Sr. Albans Bay Rural Glean Water Program. United Stares Department of Agnailrure, Vermont W j r e r Resources Research Genter; Hyde, K., Kamman, N , Smdr:er, E. ( 1 9 9 4 ) . History of phosphorus loading to Sr. Albans Bay, 1 8 5 0 - 1 9 9 0 . Lake Chaniplain Basin Program

Techieal Report No. 7B; Brown

398

S y s t e m s Science and Modeling for Ecological Economics Uaddis, E.J., Voiriov, A , ( 2 0 0 8 ) . Participataty modeling ul phosphorus reduction scenarios in a mixed use watershed in Vermont (in preparation). Apfelbmini Cuumy.

SI

(1995).

Rule of L a n d s c a p e

uri L '^Ewpi n! unof'f Mfma^ettii'm

Conference

in Stormwater

Management;

In:

National

E n t a n c i m Uvbdn Watershed Mcina^em^nc or rhe 1 .ocal

and Stale Leit'is, C h i c a g o , Illinois EPA/625/IV95/003

Shared Vision P&irmiiig f S V P i ideas hat<® keen cdvtAared inside the Army

Corps of Engineers, but

rarely gnr fafifehed in i w t m ' i w e d literature. /wnirnif)' some of their i-erv first repcru rcit'vam to ideas of participatory

re.warch go brtcls more

the: are

i0y

than 3 0 years-. Wagnei T.F., O r t o l a n d o L

( 1 9 7 6 ) . Testing an iterative, open pioce^s for water resources planning. Fort Bel voir, Va.: L'.S. AT my Engineer Institute lor W a t e r Resources 6 6 pp ( 1 W R c o n t r a c t report no. 7 6 - 2 ) : W a g n e r T P . , O r t o h n d o L. ( 1 9 7 5 ) . Analysis ol New T e c h n i q u e s tor Public Involvement in W a t e r Planning. W a t e r Resources Bulletin, V I ! , N 2, pp..329-344- An important recent paper' on S V P - Palmer, R . A . . Cardwell, H.H., L o n e . M . A . , W e r i c k , W , ( 2 0 0 8 ) . Disciplined Planning, Structured Participation, and C o l l a b o r a t i v e Modeling - Applying Shared: Vision Planning to W a t e r Resources Decision Waking, sail in iffietii at this tunc

A S C E

J W a t e r Resources M a n a g e m e n t and Planning, is

In (lie memu^ile, to find more apnut circles o{ influence,

ice Voinov.

A., W.E. C o x , a n d H E- Gardwell ( 2 0 0 7 ) . Pilot C o l l a b o r a t i v e M o d e l i n g Study tor Regulatoiy Issues on rhe James River. World O u r N;itur;il Habitat.

Eiii-'ironnvniriJ and Waro 1 Resources

Congress 2 0 0 7 : Restoring

A S C E .

The imfwittfttf? of good visuals is hard ;o ecerestrnlflle. To find some e.vtitm^ ideas un hctw to prepare jour iisuak check out the book series by Edward Tufte: 1 9 0 0 - Eiit'twrimg Jnfoimition; 1997 - Visuo/ Explanations; 2 0 0 1 - The Visual Diipkiv of Quanntclnf! In forma Don; 2 0 0 6 - Beautiful

Evidence,

Graphics Press. Some key umteis

on open

source

arid general puMe license ideas are Bruce Perens,

Richard

S t a l l m a n and Erie R a y m o n d , .Much of di£ir work Mil te found on lhe webP e i e n v IV, 1 9 9 S T h e open source definition htcp7/perens com/Articles/OSD.htm 1 Perens, B., 2 0 0 6 a . Software Patents vs. F i c e S o i r u a i e

http://perens.eom/Articlei/Pa tents,

litml .Pircns. B., 2 0 0 6 b. T h e Problem o'l Software Patents in Standards, http: il pe reus, ETA TI/Articles/ Paten t Farming, lit in 1-2 R a y m o n d , E.. 2 0 0 0

A Brief History of H a c k e r d o m .

http://www.cath.org/-esr/writings/cathe-

dial-bazaar/hacker-history/index.hrml Kay mum!,

E..

2000a.

Homestead mg

rhe Noosphcre.

h rtp7/ «'»• w.c a to. of ft/-esr/writings/

c a t h ed ra I -ba :a ar/homest e ad i rig,' There is cisu a book % Erie R a y m o n d , 1999. The Cathedral

and the Bazaar.

Levy, S . , 1984, Hackers. Anchur/Dbubleday, New York

This Jigs mro che history of the phenom-

e n a l of hackers

asd tufwt ihe\ are

For mnre about

and Braithivaite J.. 2 0 0 2 . In.format it in Feudalism

O'Reilly, 2 6 8 pp.

m format ion and ffrnpeny

rights see Drahos, P.

Who O n u s the Knowledge Economy' New York.

T h e New Press. The history oj copyright and the Didevor-Otiji/Iovcei controversy is de.\i.rJ.\\i by Walker. D. ( 2 0 0 0 ) . H^tri of ihe Entinfitenmeiic

Ccpvnght

r/u' F rench Revolution and /nformation Rei'oIution

http://i|ui3;etxix.quintessenr.ar/pipeimail/cdiiToruai/200;'necember/000054.1itml For iTiore t : k ; u ! this

Post, R o h e i t . Editor, 1 9 9 1 . Uiu and the Older of Culture.

Berkeley:

University of C a l i f o r n i a Piess, http://ark.cdl ib.or^ark:/l3030/lt9q[2nb69.J/ Other relevant fiuWicamini are: Tuomi, llkka, 2 0 0 4 . Knowledge s'-hann^ and the itlea of die public jormm,

L N E S C O 21st Century Dialogues, Duiidrnj World R'nonfe^e Soiaeiies, J o i n t Research

C e n t r e , Institute fnr Prnspective Technological Studies, S e o u l , Korea, N a t i o n a l A c a d e m y ol Engineering, 2 0 0 3 , The impact of academic Press, Washington-

research

on industrial perfor"uiiKc.

National A c a d e m i c s

The Practice of Modeling

O n parents sec': Howard. J . 2 0 0 5 . The emerging intellectual products

business of knowledge

transfer : creating

399 value /rom

ami s e n u e s . C a n b e r r a : Australian G o v e r n m e n t Department of Education,

S c i e n c e and Training. Newscom,

2005.

T h e open-source

patent

conundrum.

http:.//news.com.com,'2102• 1 0 ? I _

3 - 5 5 5 7 3 4 0 html-'tag=st.util.punt Newsforge,

2004.

H P m e m o forecasts M S patent attacks o n free software

riewsfutRt.com/article.pi'sid = 04/07/19/2.315200

hrrp://www.

To C o n c l u d e

Our ignorance

is not so vast as our failure to use what we know. M. King Hubbert

T h e r e was o n c e a t i m e w h e n h u m a n s w e r e few, weak a n d v u l n e r a b l e , o n a large, h o s t i l e p l a n e t . T h e y e n d e a v o r e d n o t t o s u c c u m b , n o t t o adapt t o t h e e n v i r o n m e n t , but

instead t o try s o m e t h i n g d i f f e r e n t o n t h e e v o l u t i o n a r y

trail. T h e y

began

to

c h a n g e t h e e n v i r o n m e n t . T h e c l e a r a n d o b v i o u s goal was t o grow, t o g a i n power, t o t a k e c o n t r o l . In t h e b e g i n n i n g , this was a b a t t l e w i t h n o c l e a r w i n n e r s .

Sometimes

h u m a n s s u c c e e d e d , and would d e v e l o p i n t o m i g h t y c i v i l i z a t i o n s , a n d t h e i r n u m b e r s grew a l o n g w i t h t h e i r power t o h a r n e s s r h e e n v i r o n m e n t

B u t t h e n s o m e t h i n g would

go w r o n g , c i v i l i z a t i o n s would c o l l a p s e , h u m a n power would d i m i n i s h , a n d t h e y would h a v e to start a g a i n s o m e w h e r e else. In aggregate, it was a m o r e or less e q u a l b a t t l e u n t i l s o m e t h i n g really r e m a r k a b l e c h a n g e d t h e world. H u m a n s l e a r n e d to h a r n e s s fossil energy. S u d d e n l y t h e y b e c a m e masters o f past worlds, of t h e e n e r g y that h a d a c c u m u l a t e d o v e r m i l l e n n i a in t h e past a n d was stored t h e r e , w a i t i n g for t h e right m o m e n t to c o m e . S u d d e n l y , t h e n e w e v o l u t i o n a r y path b e c a m e really fueled. H u m a n s a c h i e v e d t h e p o w e r and t h e luxury t o a l l o w s o m e ol t h e i r best m i n d s just co t h i n k ; chey n o longer n e e d e d co h u n c , or co sow, or t o build. W i t h t h e power o f c o n c e n t r a t e d old e n e r g y it was n o p r o b l e m t o provide t h e s e m i n d s w i t h all t h e y n e e d e d in t e r m s o f food, c l o t h i n g or s h e l t e r

T h e y could spend their

e n t i r e lives t h i n k i n g , i n v e n t i n g , d e s i g n i n g , c o m i n g up w i t h new, betcer s o l u t i o n s for the new alternative human evolution. T h a t

is w h e n h u m a n e v o l u t i o n ,

'advance-

m e n t ' really t o o k off, and p o p u l a t i o n b e g a n to a d v a n c e in huge leaps. L o c a l p o c k e t s of c i v i l i z a t i o n b e c a m e u n i t e d o n a g l o b a l s c a l e inco o n e t e c h n o c r a t i c c i v i l i z a t i o n , a n d t h e goal still r e m a i n e d t h e s a m e - to e x p a n d , grow, e m p o w e r .

401

402

Systems Science and Modeling for Ecological Economics A n d s o t h e h u m a n p o p u l a t i o n grew, b o t h in rerms o f its n u m b e r s a n d in t e r m s o f its rates of c o n s u m p t i o n . C u r r e n t l y we really a r e at a t u r n i n g p o i n t : a paradigm shift is badlv n e e d e d . T h e r e a r e t h r e e r e a s o n s for this: • Climate Change. •

Resource depletion and peak oil;

•

Globalization. C l i m a t e c h a n g e is h a p p e n i n g already and its

c h a n g e is likely to a c c e l e r a t e . W e find n u m e r o u s e v i d e n c e s for t h a t A r e c e n t study has s h o w n rhar 1 5 0 years ot records show trends toward fewer days ol i c e r o v e r T r e n d s in i c e duration in 6 5 waterb o d i e s across t h e G r e a t Lakes region ( M i n n e s o t a , Wisconsin,

Michigan, Ontario

and N e w York)

during a period ol rapid c l i m a t e warming ( 1 9 7 5 2 0 0 4 ) show t h a t average i c e duration decreased by 5 . 3 days per decade. Average t e m p e r a t u r e s from fall through spring in this region increased by 0 . 7 degrees Celsius. T h e average n u m b e r of days with s n o w decreased by 5 . 0 days p e r d e c a d e , a n d t h e average s n o w d e p t h o n t h o s e days d e c r e a s e d hy 1.7 c e n t i m e t e r s pei decade. 1 There shrinking

is

mounting

glaciers.

These

evidence

of

processes

a r e occur-

rapidly

ring faster in t h e Polar R e g i o n s . T h e A r c t i c is e x p e c t e d t o b e c o m e a n e w p e r m a n e n t sea r o u t e from t h e A t l a n t i c t o t h e Pacific, i c e in G r e e n l a n d is disappearing A tropical v i m s h a s caused a n e p i d e m i c i n Italy, w h e n several hundreds o f cases o f c h i k u n gunya, a form o f d e n g u e fever n o r m a l l y found in t h e Indian O c e a n region, h a v e b e e n registered in C a s t i g l i o n e di C e r v i a in N o r t h e r n

Italy. I n this

case t h e disease was spread by insects: tiger m o s quitoes, w h o c a n n o w thrive in a w a r m i n g Europe. Tiger m o s q u i t o e s a r e n o w found across s o u t h e r n

Figure C.1

Europe and e v e n in F r a n c e a n d S w i t z e r l a n d . The

drought

conditions

in

south-east-

ern A u s t r a l i a s e e m t o b e p e r m a n e n t

now. For

e l e v e n years in a row t e m p e r a t u r e s h a v e above normal

The Shrinking Ice Cover m Greenland

been

S y d n e y ' s n i g h t s a r e its w a r m e s t

s i n c e records were first k e p t 1 4 9 years a g o S y d n e y h a d ts w e t t e s t year s i n c e 1 9 9 8 . r e c e i v i n g 1 4 9 9 m i l l i m e t e r s , well a b o v e t h e l o n g - t e r m a v e r a g e ol 1 2 1 5 . M u c h of ic was c o a s t a l , rain thac fell at t h e w r o n g t i m e f o r farmers, s o a k e d i n t o d r o u g h t - p a r e hod soils o r e v a p o r a t e d during s c o r c h i n g days. S y d n e y h a d its s t o r m i e s t year s i n c e 1 9 6 3 . wich 3 3 t h u n d e r s t o r m s , h i s t o r i c a v e r a g e

28.

' http://www nsf.gOv/di5COVtries/dlsc_summ.jsp !cntn_id , s I ] D 9 6 7 & g o v D e l = G S K S F _ I

To Conclude

403

T h e list ot these c h a n g e s c a n he c o n t i n u e d . C o r a l reefs are b l e a c h e d and are degrading. H u r r i c a n e s h a v e b e c o m e m o r e powerful and f r e q u e n t . Floods and droughts are b e c o m i n g

more severe. M o s t disturbing are t h e n u m e r o u s positive

feedback-

e f f e c t s involved in t h e a b o v e , a n d that drive t h e c l i m a t i c m a c h i n e of this p l a n e t . A c c o r d i n g to t h e I n t e r n a t i o n a l Panel on C l i m a t e C h a n g e ( I P C C ) it is "very u n l i k e l y " that we will avoid the c o m i n g era of " d a n g e r o u s c l i m a t e c h a n g e " .

Most

likely we should e x p e c t water shortages, c r o p failures, disease, d a m a g e s from e x t r e m e w e a t h e r e v e n t s , c o l l a p s i n g infrastructures, and b r e a k d o w n s in t h e d e m o c r a t i c process. O u r first e x p e r i e n c e o f r e - e n g i n e e r i n g t h e planet s e e m s 10 be producing q u i t e ugly results U n i n t e n t i o n a l l y we may h a v e triggered t o o m a n y positive feedbacks that t e n d to get out o f c o n t r o l . If we can't stop it - we will need to adapt to it. A n y adaptat i o n will requiie additional resources. U n f o r t u n a t e l y t h e r e s o u r c e base also does nor look very promising. As we h a v e already seen t h e r e is m o u n t i n g e v i d e n c e t h a t oil reserves a t e a p p r o a c h i n g t h e threshold w h e n e x t r a c t i o n will c o n s u m e a l m o s t as m u c h e n e r g y as energy produced. It b e c o m e s meaningless to produce oil as an energy source after t h a t . A t t h e s a m e t i m e t h e r e is growing d e m a n d , especially in S o u t h - E a s t A s i a .

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The growing price of oil. This time there seems to be no other reason except that supply

cannot catch up with demand In the twenty-first c e n t u r y oil prices have gone up o v e r 8 0 0 % . T h e r e was a previous price spike in the 1970s, but at that t i m e it was a deliberate decision o f O P E C t o decrease oil e x p o r t s to get a price h i k e T h e r e is n o such policy pursued today, yet prices are steadily growing. W h y is that? W e have entered t h e era where supply c a n n o longer keep pace with d e m a n d . Supply is stagnating, while d e m a n d c o n t i n u e s to grow.

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405

i n t e r d e p e n d e n t system. T h e top 15 W o r l d oil producers d e l i v e r o v e r 6 3 m i l l i o n barrels o f oil per day. A t t h e s a m e t i m e the top 15 oil e x p o r t e r s s h i p m o r e than 3 9 million barrels o f oil per day, m e a n i n g that a l m o s t 2/3 o f all oil produced is d e s t i n e d t o s o m e o t h e r l o c a t i o n , in many cases traveling many miles across t h e o c e a n s . Most of t h e developed c o u n t r i e s are d e p e n d e n t o n foreign energy supplies. A l m o s t all c o u n t r i e s d e p e n d on food imports. S o m e t i m e s as m u c h as 7 0 % of f o o d supply has to he delivered. W h i l e in developed c o u n t r i e s foreign imports are largely for e x o t i c and luxury items, in s o m e o f the M i d d l e East a n d A f r i c a n c o u n t r i e s t h e y are a necessity. E v e n lor m a n y c o n v e n t i o n a l items we see t h a t trade flows c i r c l e the Earth in m a n y cases going in b o t h d i r e c t i o n s , as is t h e case w i t h , bay. oranges. F i n a n c i a l flows further c o n n e c t t h e W o r l d

A n e s t i m a t e d 150 million migrants

worldwide have sent s o m e U S $ 3 0 0 billion to t h e i r families in d e v e l o p i n g c o u n t r i e s during 2 0 0 6 through more t h a n 1.5 billion separate f i n a n c i a l transactions.

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Flows of imports and exports of many food items go in both directions.

A t this point we are n o t looking at positive and n e g a t i v e impacts of globalization W h a t is important is to realize that this system is in place, a n d that as a result, rhe world is c o m p l e t e l y i n t e r c o n n e c t e d . Local crises will spread around swiftly; o v e r c o n Suniption in l h e developed c o u n t r i e s will not be c o n t a i n e d only to t h e areas o f those c o u n t r i e s . Just like depletion o f oil reserves in, say. the 4 8 states of t h e U S A will not s t o p o i l c o n s u m p t i o n in t h e country, c l i m a t e c h a n g e triggered hy greenhouse gas emissions is not going to lie limited o n l y to t h e l o c a t i o n s where these gases are emitted. T h e e n v i r o n m e n t s t h a t we have c r e a t e d are facing c o n s i d e r a b l e risks, and t h e safety net. o n c e provided by t h e favorable natural e n v i r o n m e n t o n planet Earth seems t o he eroding. S i n c e h u m a n s h a v e taken c o n t r o l , to s h a p e t h e e n v i r o n m e n t t o our o w n use rather t h a n adapt to w h a t was offered, we n o w h a v e a fiduciary responsibility for t h e results o f o u r efforts. In many cases t h e natural e n v i r o n m e n t s that were there t o provide h u m a n s with resources needed a n d to a b s o r b t h e waste a n d pollution t h a t h u m a n s c r e a t e d , are n o longer in place. F u r t h e r m o r e , t h e y could n e v e r provide the c a r r y i n g capacity needed to m a i n t a i n t h e c u r r e n t size of t h e h u m a n p o p u l a t i o n at the c o m f o r t levels t h a t it has b e c o m e a c c u s t o m e d to. T h e paradigm shift, if it c o m e s , needs to he based o n an u n d e r s t a n d i n g o f how systems work, o f h o w we got h e r e , a n d what t h e i n d i r e c t and delayed responses of t h e system c a n he. T h e o n e resource t h a t does n o t seem t o h a v e any limits is information

Moreover, by sharing i n f o r m a t i o n , we d o n o t subtract from it. If 1 h a v e a

b u c k e t o f popcorn and want t o share n with my n e i g h b o r s , I will h a v e to give t h e m s o m e ot the popcorn from rhe b u c k e t . A s a result, there will he less left for me. T h i s

420 Systems Science and Modeling for Ecological Economics is n o t che c a s e with i n f o r m a t i o n

If I s h a r e w i t h you w h a t 1 know, I d o n o t t h e n know-

less, prphaWy more, b e c a u s e w h i l e c o m m u n i c a t i n g I m i g h t u n d e r s t a n d my i n f o r m a t i o n better. If it is in o u r g e n e t i c h e r i t a g e t o grow, t o c o n s u m e m o r e , t o e x p a n d , t h e n p r o b a b l y t h e o n l y a r e a w h e r e we c a n d o it safely - is with i n f o r m a t i o n . T h e p l a n e t is l i m i t e d : t h e r e is o n l y t h a t m u c h of land, o i l , water, t i n , c o p p e r a n d gold. N o m a t t e r h o w efficiently we use it, if t h e r e are m o r e a n d m o r e users, we will e v e n t u a l l y run o u t o f t h e g o o d s

I n f o r m a t i o n is limitless. W e c a n e x p l o r e , r e s e a r c h ,

study, learn as m u c h as we wish. V e r n a d s k i i d r e a m t o f a svstem h e c a l l e d " n t x j s p h e r e " a b i o s p h e r e d i i v e n by h u m a n i n t e l l e c t , spirituality, k n o w l e d g e , a n d u n d e r s t a n d i n g . M o d e l s are a n i m p o r t a n t part o f this u n d e r s t a n d i n g . T h e y are building b l o c k s o f our world view, I h e m o d e l s c a n be s i m p l e o r c o m p l e x , c o n c e p t u a l o r n u m e r i c a l , f o r m a l or v e r b a l , but tor m o d e l s t o be good t h e y n e e d t o be based o n a c u l t u r e o f m o d e l i n g o n good m o d e l i n g practice

T h a t is w h a t we tried t o learn in this book

It we h a v e

c o m m o n standards fot o u r models, it will be easier for us t o c o m m u n i c a t e o u r unders t a n d i n g , to find c o m m o n ground, t o a v o i d conflict a n d m a k e t h e right decisions. T h e m o d e l i n g process c a n work as o u r s h a r e d f a c t - f i n d i n g a n d u n d e r s t a n d i n g e x p e r i e n c e dial leads us toward a s h a r e d v i s i o n ol t h e past, present a n d future. A n y dispute c a n be t r e a t e d as a c l a s h o f d i f f e r e n t models. S t a k e h o l d e r s c o n t r i b u t i n g t o a dispute r e s o l u t i o n e x e r c i s e c o m e t o t h e table with i h s i r d i f f e r e n t m o d e l s , qi.:alit a t i v e a n d q u a n t i t a t i v e , o f t h e s y s t e m at s t a k e

T h e dispute e v o l v e s b e c a u s e o f t h e

inconsistencies and controversies between t h e different models. I hypothesize that by h a r m o n i z i n g t h e i m p e l s f o r use in a c o m m o n f r a m e w o r k , m u c h o f t h e c o n f l i c t c a n he resolved. I n a way p a r t i c i p a t o r y m o d e l i n g is a m e c h a n i s m o f |Oint fact finding a n d u n d e r s t a n d i n g w h e n d a t a a n d k n o w l e d g e are s h a r e d a m o n g s t a k e h o l d e r s in a t t e m p t s to build a c o m m o n m o d e l . W h e n rhe p a r t i c i p a n t s m u t u a l l y e d u c a t e e a c h o t h e r a b o u t t h e m o d e l s t h e y use, a n d a r r i v e at a s h a r e d m o d e l o f a system t h e r e r e m a i n s less reason lor c o n f l i c t a n d dispute. A s t h e b o o k goes t o p r i n t , we are w i t n e s s i n g a burst o f t h e h o u s i n g b u b b l e in l h e U S A a n d a slide of t h e U S e c o n o m y towards r e c e s s i o n . For a systems s c i e n t i s t tins a c t u a l l y m a y be a p o s i t i v e t r e n d . T h e e c o n o m y ts well o v e r d u e t o slow d o w n , g i v i n g people pause t o r e c o n s i d e r s o m e of o u r priorities. H o w e v e r , i n s t e a d , a n o t h e r s t i m u l u s p a c k a g e is going t o he passed by t h e U S g o v e r n m e n t , simply p u t t i n g m o r e m o n e y in t h e h a n d s ol p e o p l e t o e n s u r e t h a t t h e y spend m o r e t o fuel f u r t h e r g r o w t h . T h e system is f u r t h e r forced i n t o o v e r d r i v e towards a c o l l a p s e . I n s t e a d o f i n v e s t i n g in e d u c a t i o n , in r e t r a i n i n g , in r e s e a r c h , in t h e future; a g a i n we a r e c h o o s i n g t o invest in c o n s u m p t i o n , for t h e p r e s e n t c o m m o n understanding...

i f we c o u l d o n l y s h a r e o u r m o d e l s a n d r e a c h a

Index

Activity diagrams, 4 6 - 7

Rrundiand Commission, 15

Adams method, 8 9

Bulirsch-Sroer method, 89

Adaptive modeling, 3 6 2 - 5 Advection, IOC. 1 0 2 - 4

C/C

Age groups (cohorts), 2 5 5 - 6 0

Calibration

Agent-based models, 24, 365 tools, 5 7 - 9 Albedo, I I

+ (compurer languages). 45. 50, 364

definition. 126 empirical vs process-bascd model, 1 2 3 - 4 o f model, 1 1 5 - 2 9 , I 50

Alternative energy, 2 9 1 - 3

Canada, population dynamics, 2 5 0 - 4

Analysis, see Model analyse

Carbon-dioxide rheory ol climate change, 3 5 7 - 8

Analvtic models, 24

Carrying capacity, 146. 273

A r c G I S , 243

C C M P (Chesapeake Community Model Program),

A r c I N F O , 39, 365

389

Arrays, in Stella, 1 6 1 - 2 , 165

C C S M (Community Climate System Model), 389

A S S E M B L E R (computer language), 50

C E R C L A (Comprehensive Environmental Response,

Australia, climate change, 402

Compensation and Liability Act of 1980), 3 3

Australian cajeput, 21 7

Chaos, 2 8 1 - 5

Avenue computer language, 365

C h e a p oil, 2 S 7 - 9 6 Chernobyl, 308

B A L S E C T , 50

Chesapeake Bay, warer quality, 2 4 4 - 5

Bank account model, 9 4 - 5 , 9 8

Chesapeake Community Model Program ( C C M P ) ,

B A S I C (comparer language), 50

.339

B A S I N S , 50, 2.32

C h i n a , decision-making, 3 6 1 - 2

Bi-flows, 2 1 0 . 211

C L E A N E R (Collaborative Laigc-Scale Engineering

Biofuel, 287

Analysis Nerwork tor Environmental Re-

Biofucl production, 362

search). 50. 395

Biological lime. 34

Climate change, 402

Bipolar see saw effect, 1 32

Australia, 402

Black-box models. 24. 4 0 - 1 . 60, 1 1 8 - 1 9

carbon-dioxide theory, 3 5 7 - 8

Blue crabs. 151

global, 1 1 - 1 2 , 294, 357

Bonnini's paradox, 43, I 36

global warming, 323, 3 5 7 - 8

Boundaries, 18

Intergovernmental Panel ( I P C C ) . 359, 3 6 0

Box models, 23, 9 9

Leipzig Declaration, 359

407

408

Index

Climate research, history, 3 5 9 - 6 1

Corporate rule, 2 7 2 - 8

C L I M B E R model, 1 3 1 - 2

Corporations, 2 7 2 - 8 . 3 5 8

C M A S (Community Modeling and Analysis

Cred i bil i ry of mode 1, defi n i rion, 1.3 5 - 6 Critical Natural Capital, 2 8 8 - 9

System), 389 Coal-to-Liquid production, 277—8

C S M P , 50

Cobb-Douglas production function, 14

C S T M (National Community Sediment Transport Model), 3 8 9

Coexistence state, 146, 147 Collaborative Large-Scale Engineering Analysis Network for Environmental Research

C U A H S I (Consortium o f Universiries for the Advancement of Hydrological Sciences, Inc.), 395

( C L E A N E R ) , 50, >95 Combinatorial optimization problem, 342

Curve lining, 126

Community Climate System Model ( C C S M ) , 389

Curve numbers, 2 2 7 - 9

Community modeling, 3 9 6

Cutler Reservou, Total Maximum Daily Load ( T M D L ) process, 3 7 2 - 3 , 3 7 7 , 3 8 0

ongoing projects, 3 8 9 - 9 0 vs open-sot.iicc software, 3 8 8 - 9 5

Daisy World model, 25

Community Modeling and Analysis System

Dansgaatd-Oeschger ( D - O ) events, 1 3 1 - 2

( C M A S ) , 389

Data assimilation, 132

Community models, 25

Data models (expeiunentcil models), 1 1 6 - 1 8

Companion modeling. 363

Dayjul variable, 20.3-4

Comparison

Decision support, W e b features, 3 9 3 - 4

quantitative mathematical. 117

Decision variables, 310

visual, 116

Decisionmaking

Compart mental models, 99 -101

complexity of, 3 6 1 - 2

Competition, 2 7 2 - 3

public participation ill, 362

Complexity, 3 Comprehensive Environmental Response, Compensation and Liability A n of 1980 ( C E R C L A ) . 33

DEM (Digital Elevation Model), 243 Demand-supply-price theory, 2 6 4 - 7 2 Demographics, 2 5 0 - 6 4

Computer languages, 5 0 See also Modeling software; and individual

DELAY function, 222

languages

Computer models, 24 Conceptual models, 1 8 - 2 1 . 22, 3 0 - 1 9 diagrammatic. 22 formalism in diagrams, 4 3 - 4

Demonstration, models for, 25 DeVrics solar cycles, 132 Diagrammatic models, see Conceptual models Diderot, Denis, 3 8 5 - 6 Difference equation. 8 6 Differential equations, 84, 86

spatial domain, 3 4 - 4 0

Diffusion, 1 0 0 - ! , 1 0 4 - 5

structural domain, 4 0 - 8

Digital Elevation Model ( D E M ) , 24 3

system components, 1 8 - 1 9

Diminishing marginal utility. 2 6 9 - 7 0

temporal domain, 3

Discounting. 3 2 3 - 4

verbal/descriptive, 2 1 - 2

Documental ion, 211

Condorr.et, Marquis de, 386

scientific models, 390

Conductivity, hydraulic (peicolation), 213, 2 1 4 - 1 5 ,

software, 395, 3 9 6

215-16

Domain ontologies, 4 8

Conservation, 295

Drought, in Australia. 402

Consortium of Universities for rhe Advancement of

Drugs, side-effects, 8 - 9

Hydrological Sciences, Inc. ( C U A H S 1 ) , 395 Constraints, 310, 311 Continuous models, 8 4 - 5 , 101 Control f'acrots. 3 1 0

Duration, 3 Dynamic modeling, 2 6 4 - 7 2 D Y N A M O . 50

Control t u n a ions, 19

Earth Summit. United Nations, 3 5 9

Control parameters, 3 1 0 , 311

Earth system, ecological-economic-social hierarchy,

Control variables, 3 1 0

14-15

Copyleft, 383

Earth System Modeling Framework ( E S M F ) , .389

Copyright, 3 8 3 , 3 8 5 - 6

Ecological economics, 249, 2 7 9 - 8 7

Cormas, 5C. 58

(vs.) environmental economics, 304

Index Ecological modeling, 3 9 0

European Union, open-source projects, 395

Econom tcs

Evaporation, 204, 2 0 5 - 9

ecological, 249, 2 7 9 - 8 7

409

goodness of fit, 207

vs environmenra!, 3 0 4

Everglades model, 240

i c e :!!SC Socio-cconmnios

Excel, 5 0 , 5 5 . 69.. 364, 378

Ecosystem mamigemenr, 59], )91

statistical tools, 120

problem characteristics. 392

ExpeiiiYiewal models (data models), 1 1 6 - 1 8 . 119

Ecwy^tcm moJels, 25

Exponential growth, in populations, 254 -5, 2 9 0

Ecosystem services, 299, 302—I

Exponential growth model 69. 8 1 - 3 , 9 8 - 9 , 108

I citegtared Valuation of Ecosystem Services and Tradeoffs ( i n V E S T ) . 304

Extend, 45, 50, 54 Extendable modeling systems, 4 9 - 5 0

Educatcd-guew approach. 124

Extendable models, 4 9 - 5 0 , 5 1 - 2

Education, models for, 15

"Eyehallmg", 116

Efficiency, increasing. 295 Elements

Feasibility condition^ 146 Feedback

relationships between, 9— 11

negative. 11

system as combination of, 7 - 8

positive, 1 1 - ! 2

Emergent functions, 12

Field capacity. 2 I 3

Emergent properties, 7. 12 EML (Environmental Markup Language),

1

Empirical models, 24, 40, 7 6 . 7 7 . 1 1 8 - 2 0 , [27

Field-related classification fit model, 2 4 - 5 Fish populations, 4 1 - 2 Fishpond model, t.35

calibration, 123 vs process-based models, 24, 40

conceptual model, 41 optimization, 3 2 4 - 3 8

Energy, 2 8 7 - 9 6

Flows

alternative energy, 2 9 1 - 3 consumption and economic development, 295

ot information, 9, 10

curbing ihe demand, © 4 - 5

material, 9 - 1 0

production, 287

positive/negative, 2 1 0 - 1 1

sources, 2 7 7 - 8

Sec tdso Inllow/outflow model; Stock-jiui-llow

Energy diagrams, 44

models, Stock and-tlow representation

Energy Return on Energy Invested ( E R O E I ) . 2 6 - 7

Flush tank model, 9 2 - 4

Energy Return on Energy Invested ( E R O E I ) index,

Food import dependency, 4 0 4 , 4 0 5 Forcing lunaions, 1 8 - 1 9

259-93 Environmental economics vs ecological economics, 504

Formalism for conceptual diagrams, 4 3 - 4

Environmental Markup Language (EML), 3 8 3

Forrester, Jay, 2 5 - 6 , 2 9 6

Environments I Protection Agency (EPA), 232, 246,

formalism for conceptual diagrams, 4 3 - 4 F O R T R A N (Computer language), 50

369 Environments, software packages, 30. 57 EPA ( U S Environmental Protection Agency), 232,

Hvdrologtcal Simulation Program ( H S P F ) , 232 Fossil energy, 401 Fossil fuels, 287

246, SW E P I C A (European Project for Ice Coring in Antarc-

Free Software Foundation (FSF), 5 8 2 - 3 "Freeze" flow, 2 IO1

tica), 132

F S F (Free Software Foundation), 3 8 2 - 3

Equilibrium in predator-prey system. 142. 153, 1 5 5 - 6 , 158

Function of system, 12

trivial/non-tnviai, 1 4 1 - 2 , 161

Functions

Equilibrium states, srable/unstable/ncutral, 9 5 - 7

control functions, 19

EROEI (Energy Return on Energy Invested) index,

emergent, 12

$89-93 Error model, I 1 6 - 1 8 qualitative/quantitative, 1 ) 6

forcing,

18-19

Futures, 301 - 4 17

E S M F (Earth System Modeling Framework), 3 8 9

landuse changes, 3 0 3 natural resources, 3 2 4

Euler method, 8 6 . 87, 8 8 European Project for Ice Coring in Antarctica ( E P I C A ) , 132

G C C (Global Climate Coalition), $59 G C M s (Global Circulation Models), 159

412

Index

Modeling

piocess-based, see proceu-basrd models

adaptive, 5 6 2 - 5

purpose of, 25

art ot, 2 9 - 7 8

rigid, 106

choosing :i tool, 3 6 8 - 7 0

simple vs complex, 24

coiivin unify, see C o m men fry modeling

simplei than reality, 1 - 2 , 5

companion.

simulation, .see simulation models

conclusions, 395

soft. 106

dynamic, 2 6 4 - 7 2

transparency, 369

ecological, 3 9 0

visual, 2

gmdf lines lor success. 3 7 5 - 8 1

visualization, 369

mediated. 3 6 3 open research, 3 9 0 - 1 paiticipatory, sec Participatory modeling ( P M !

why they don't woik, 3 5 5 - 6 2 Modified Universal Soil Loss Equation ( M U S L E ) , 233

practice ill, 3 5 5 - 9 9

Moisture content ot soil, 21 5

scientific, .390-1

Monod ( M i c h a d i s - M e n i e n ) function, 73, 75,

shared vision, 363 summary, 355

148 M o n t e Carlo method, 314, 3 4 5 , 3 4 5 - 6

Modeling languages, 4'), 50, 5 5 - 9

Mortality, 250.

Modeling process, 16-21

M S D D (Micro Sturmwatti Drainage Density)

an iterative process. 1 6 - 1 7 , 30. 49. 135 simple version, 4 8 - 9 Modeling sokware, 4 5 - 8 , 4 9 - 7 0 hieiaiehy of, 50 libraries, 50, 55 open-source, 6 0 Modeling systems, 50, 5 2 - 5

index, 373 Multi-Agent Simulator of Neigh burhoods ( M A S O N ) . 50, 58 Multi-scale Integrated Models of Ecosystem Services ( M I M E S ) , 25, 3C4 M U S L E (Modified Universal Soil Loss Equation), 233

extendable, 4 9 - 5 0 Model Maker. 50, 5 3 . 6 0

Natality. 250, 254

Models

National Academy ol Sciences, 360

affordability, 3 6 9 agent-based, see Ageni-based models analytic vjj computer. 24 classifications, 2 1 - 5 common standaids, 4 0 6 comparison, 1 16

National Climatic Data Center ( N C D C ) , 201, 205 National Community Sediment-Transport Model ( C S T M ) , 389 National Ecological Observatory Network ( N E O N ) . 395, 3 9 6

complexity, 125

National Research Council, 365

conceptual, w Conceptual models

National Science foundation, research initiatives,

continuous, 84—S, 101

395

definition, 1

Natural Capital Project. 304

documentation, see Documentation

Natural Resources Conservation Service i N R C S ) ,

empirical, see Empii ical models extendable, 4 9 - 5 0 , 5 1 - 2 extendibiluy, 369

227 Nature Conservancy and World Wildlife Fund, 304

flexihiiiry, 3 6 9

Navigator, 382

form of presentation, 2 1-2

N C D C (National Climatic Data C e n t e r ) ,

formal, 23

201,205

(Delusiveness, 368

Negative feedbacks 11

individual-based models, 24

NEON (National Ecological Observatory Network),

mathematical, ice mathematical models

395, 396

mechanistic, 40

Neptune. 5

method of formulation. 24

Net Logo. 50, 59, .365

modularity, 368

Non-linear systems, 139, 194

physical. 3—4, 2Z-3

Non-rival goods, 3 9 5

point models, 23

"Noospbere", 4 0 6

Index

N R C S (Natural Resources Conservation Service), 227

O S S . see Open-.source .software crimes

Numerical methods of solving equations, 86

O W L , 48

O A S I S (Operational Analysis and Simulation of

PAR (photoactive radiation), 206, 2 0 7 - 8

Interlaced Systems), 50. 5 1 - 2 Obiecrive function-; (goal functions), I 24. > 0 9 - I C ,

413

Parabolic function, 76 Paradigm shift, 4 0 5 Paradigms, in modeling packages, 61

310—11 O c e a n Research Interactive Observatory Network

Parameters, 1 8 - 1 9 , 112 ei itical values, 11 4

( O R I O N ) , 396 O C L (Operation Control Language), 5 1 - 2

determination of. I 9 - 2 C I tee, 124

Oil cheap oil, 2 8 7 - 9 6

Partial differential equations, 105

Mciiiurn-Term Oil Market Report, 289

Participation

peak oil, 2 7 7 , 2 8 7 , 2 9 4 , 4 0 3

daia collection and availability, 366

prices and world events, 4 0 3

developing policy alternatives, 3 6 6

reserves, 4 0 3

tennis of, 3 6 5 - 6

Open reseaich modeling, 3 W - 3

inteqireting results, 366

Open Source Definition, 383

model selection and development, .365

Open Source Development Network ( O S D N ) ,

scenario development, 3 6 6 Participatory modeling ( P M ) , 3 5 5 , 3 6 2 - 5 , 392,

31S3-4

394-5, 406

Open Standards, 383

flow chart for, 374

Open systems, 130-1 Open-source software ( O S S ) , 6 0 vs community modeling, 5 8 8 - 9 5 Open-source software ( O S S ) paradigm, 355, 370, 382

successful, criteria tor, 3 6 6 - 7 Particle seconds, 102 Patent farming, ,387 Patents, 3 8 6 - 7

4

Patuxenr Landscape Model ( P L M ) , 2 37, 239.

Internet software, 3 8 3 - 4

338 4 0

O p e n M I , 50, 57 OpenOfficc, 50, 55

Peak oil, 277, 2B7, 294, 403

OpenStarLugo, 59

Pearson moment product correlation coefficient, 1 i 7

Operation Control Language ( O C L ) , 5 1 - 2

Percolation (hydraulic conductivity), 213, 2 1 4 - 1 5 ,

Operational Analysis and Simulation ot Integrated Systems ( O A S I S ) , 50. 5 1 - 2

215-16 Periwinkle snails, 151

Optimality principles, 3 4 7 - 5 2

Phase plane, 14.3

Optimisation. 124, 125, 307-5.3

Phase poi traits, 106, 107

algorithms, 3 1 1 - 1 5 , 347

Phosphorus, St Albans Ray Watershed, 371

combinatorial problem, 342

Photoactive radiation ( P A R ) , 206, 2 0 7 - 8

controls, 308

Physical models, 3 4 , 2 2 - 3

fishpond model. 3 2 4 - 3 8

Plant growth, 3 4 7 - 8

goal (objective), 3 0 8

PLM (Paruxent Landscape Model), 237, 239. 3.38-40

introduction, 3 0 7 - 1 5 landscape, 3 3 8 - 4 7

Pluto, 5

Monte Carlo method, 314

PM, see Participatory modeling

numerical, algotithm of, 34?

Point models, 23

objective function, 124, 3 0 9 - 1 0 . 3 1 0 - 1 1

Pollution assimilation, 4 0 4

optimality principles, 3 4 7 - 5 2

Pollution icduction/dilution, 198

process, 311, 312

Population dynamics, Canada, 2 5 0 - 4

of real-life systems, 3 0 8 - 9

Population growth, 1 0 - 1 1 , 402

resource management, 3 1 5 - 2 4 summary, 307

exponential, 2 5 4 - 5 , 1 9 0 Population models, 24, 88

O R I O N (Ocean Research Interactive Observatory Network), 395, 396

Population pyramids, 255 Population senescence, 2 5 9

O S D N (Open Source Development Network), .383-4

Porosity of soil, 215, 222 Positive feedback. 1 1 - 1 2

I

414

Index

Potlareb, 384 Power;-!. 11. 45, 50, 53 Precipitation, 201 Predator-prey model, 70, 88 Predator-prey system. 139-95 classic model, 140-6 classic model modifications, 1 4 6 - 5 0 conclusions, 194 equilibrium in, 142, 153. 155-6, 158 sensitivity analysis, 145 Simile model, 166—80 S H E model, 180-94 spatial model, L61-94 summary, I 3 9 - 4 0 Predictive models, 25 Predictor-corrector methods, 8 9 Price, 2 6 4 - 7 2 Process (simulation) models, 364, 378 Process-based models, 24, 4 0 - 1 , 7 4 - 7 , 120-7 calibration, 1 2 3 - 4 vs empirical models, 24, 4 0 Psychology. 356 Public domain, 3 8 5 - 6 Python (computer language), 45

excludable and rival. 1 9 8 , 2 8 9 non-renewable, 404 population growth and, 2 8 1 - 7 renewahle/noivrenewable, 287 substicutability, 297 Retention. 227 Retention ponds, 24 I --2 R H E S S y J (Regional Hydio-Ecological Simulation System), 231, 216 Rigid models. 106 River network, 241 Runge-Kutta method, 87, 8 8 - 9 , 91 S-sbaped function. 75 S t Albans Bay Watershed, 3 7 0 - 1 , 376, 3 7 6 - 7 , 179, 380, 381 S A M T (Spatial Analysis Modeling Tool), 50, 57 Scales spatial, .3 temporal, 3 Scaling issues, 5 Scenario analysis, 21 Scenarios. J 3 3 - 4 , 307 Scientific modeling, 3 9 0 - i Scoping model, 200

R : value, 117, 120

Scotts Miracle-Gro, 275

Radiation,solar (photoactive), 206, 2 0 7 - 8

Sea ice. 358

Ram barrels, 242

Seamiess. 395

Rain gardens, 242

Self-competition, 2 7 2 - 3

Rainfall, 198-9

Senescence, ot population. 2 5 9

RAN (Redesigning the American Neighborhood)

Sensitivity analysis, 2 0 - 1 . 1 1 1 , 1 1 2 - 1 5

project, 235, 24.3, 3 7 1 - 2 , 3 7 5 , 3 7 8 Races, 43 REcutsive Porous Agent Simulation Toolkit (Re past), 50, 58, 59 Redesigning the American Neighborhood ( R A N ) project, 235, 243. 3 7 1 - 7 , 375, 3 7 5 Reductionist!!. 8 Regional Hydro-Ecological Simulation System (RHESSys), 2 3 3 , 2 3 6

predator-prey model, 145 Shared vision modeling, 363 Shared Vision Planning, .362,37 3, 377 Sheet flow, 2 3 1 - 4 Silicon Valley. 386 SimCity computer game. 50, 51 Simile, 45, 50. 53, 3 0 4 . 3 6 4 , 365 predator-prey system model, 1 6 6 - 8 0 S I M S A R , 50

Register of Ecological Models ( R E M ) , 51

Simulated Annealing algorithm, 315

REM (Register of Ecological Models), 51

Simulation models, 116, 126, 129, 3 6 4 , 3 7 8

Repast (REcutsive Porous Agent Simulation

Simulmk, 45, 50, 55

Toolkit), 50, 58, 59 Research scientists, teamwork, 391 Resolution, 3, 3 1 - 2

S M E (Spatial Modeling Environment), 50, 57, 59, 237, 365 predator-prey system. 1 8 0 - 9 4

computer display, 15--6

Snow, see Ice and Snow

grain size and form. 35—40-

Social Security funds. 260-1

photographs, 35

Socio-economics, 2 4 9 - 3 0 6

spatial, 3 5 - 4 0 spalial vs temporal. 35

demographics, 2 5 0 - 6 4 summary, 2 4 9 - 5 0

Resource management, optimization, .315-24

Soft models, 106

Resources

Software, 364 collaborative research, 3 8 7 - 8

depletion, 403

Index development:, 3 8 7 - 8

Subsystems

documentation, 395, 396

death of, I 5

lot modeling, 2 6

See also Hierarchies tn systems

open-source projects, 395

Superfund, 33

systems dynamics software, 4 5

Supersystems, see Hierarchies in systems

,S<3£ also Modeling software; O p e n - s o u r c e software

Supply and demand, 2 6 4 - 7 2

( O S S ) paradigm; and individual

415

software

names

Surface roughness, 2 3 0 , 2 3 3 - 4 Sustainability, 2 7 8 - 8 7

Soil classification, 2 2 9

Sustainability of systems, 15, 16

held capacity, 2 1 5

S V P ( S h a r e d Vision P l a n n i n g ) , 362, 3 7 3 , 3 7 7

Hydrologic Soil Groups ( H S G ) , 2 2 9

Swarm, 50, 58, 59

infiltration in, 2 13

S W A T (Soil and W a t e r Assessment T o o l ) model,

Modified Universal Soii Loss Equation ( M U S L E ) , 233 moisture concent, 2 15 porosity, 2 1 5 , 222 Soil and W a t e r Assessment Tool ( S W A T ) model, 2.32-3,236

2 3 2 - 3 , 236 S Y S L , 50 Systems, 6 - 1 2 c o m p o n e n t renewal, 15 definition, 6 essential features, 6

Solar ( p h o t o a c t i v e ) radiation, 2 0 6 , 2 0 7 - 8

example of student as, 13

S o l o m o n s Harbour W a t e r s h e d , 3 7 0 , 377. 378,

hierarchies in, 1 2 - 1 6

3 7 9 - 8 0 , 3 8 0 - 1 , 381 S O N C H E S , 50

main dimensions, 1 7, 2 9 as a whole, 7, 8

Source Forge, 395

Systems diagrams, 5 4 - 5

S p a c e , 17, 2 3 - 4

Systems dynamics

in m a t h e m a t i c a l terms, 9 9 - 1 0 5 S p a c e (spatial) scales, 1 0 1 - 2 Spatial Analysis Modeling Tool ( S A M T ) , 5 0 . 57 Spatial model, predatot-prey system, 1 6 1 - 9 4

models, 24, 26 software, 45 tools, 5 2 - 4 Systems thinking, 2 5 - 7

Spatial Modeling E n v i r o n m e n t ( S M E ) , 50, 57, 59, 237,365 predator-prey system, 1 8 0 - 9 4

Temporal scales, 3 TerraCycle, 275

Spatial models, 23

Thalidomide, 9

Spatial scales, 3

T h e i l ' s measure of forecast quality, 11 7

Spreadsheets, 50, 5 5 - 6 , 6 9

" T h i n k globally - A c t locally", 1 5 - 1 6

Stakeholders, 3 6 3 - 7 , .375-81

Tiger mosquitoes, 4 0 2

See also Participation Stanford Watershed Model, 2 3 2 StarLogo, 50, 365

T i m e , 17, 23, 31 dilferent m different systems, 3 3 in m a t h e m a t i c a l terms, 8 1 - 9 9

S t e e l function. 7 6

T i m e (temporal) scales, 1 0 1 - 2

S t e l l a , 26, 4 4 - 5 , 50, 52, 6 0 , 6 1 , 3 6 4

Time-steps, 8 1 - 3 . 8 5 - 7 , 8 9 , 9 I , 144

arrays, 162 building blocks in, 6 2

T M D L (Total M a x i m u m Daily Load) process, Cutler Reservoir, 3 7 2 - 3 , 3 7 7 , 3 8 0

equations in, 8 0 - 1

Tolerance, 320-1

introduction to, 6 1 - 7 8

T O P M O D E L , 233

S t o c k - a n d - l l o w models, 24 S t o c k -and-flow representation, 46, 61 Stormwater, 2 4 0 - 1 Structural stability, 1 0 6 - 7 Structure

Total M a x i m u m Daily Load ( T M D L ) process, C u t l e r Reservoir, 3 7 2 - 3 , 3 7 7 , 3 8 0 T R - 5 5 (Hydrology of S m a l l W a t e r s h e d s ) , 2 2 7 , 2 2 9 , 233,235-6 Transpiration, 2 1 3 - 1 4 , 2 1 7 , 2 1 9 , 242

in m a t h e m a t i c a l terms, 1 0 5 - 8

Trend lines, 1 1 8 - 1 9

of system, 12, 1 7 - 1 8 , 2 4

Trial-and-error method, 124

S t u d e n t , as e x a m p l e of a system, I 3

Trophic chains, 139, 1 5 0 - 6 1 , 162

Sublimation, 210, 212

Trophic functions, 141, 1 4 7 - 8

Subsidies, for coal-to-liquid production, 2 7 8

Trophic levels, even/odd, 15 1 - 3

416

Index

U M L (Universal Modeling Language). 4 5 - 7 Uncertainties, 1 2 6 - 7 , 136 Understanding, models for, 25 United Nations, Eaitfi Summit, 3 5 9 Universal hell function, 76 Universal Modeling Language ( U M L ) , 4 5 - 7 Universities, intellectual property rights, 387 U S Federal Agencies, 228 U S Geological Survey ( U S G S ) , 115 watershed classification system, 2 4 6

WATer and Environmental Research Systems ( W A T E R S ) Network, 390 Water Evaluation and Planning system ( W E A P ) , 50.51 Water Framework Directive, European Commission, 363 W A T E R S (WATei and Environmental Research Systems) Network. 390 Watershed management. 2 4 5 - 6 , 3 9 1 - 2 W e b features, 3 9 3 - 4 Watersheds

Validation (verification), 131, 132, 13.3, 137

Hunting Creek optimization model, 3 3 9 - 4 1 , 3 4 2 - 4

Validation process, 21

Hydrological U n i t Classification ( H U C ) system,

Variables, 18 Vensim. 45, 50, 53, 297 Verification (validation), 131, 132. I 53, 137 Virtual Reality Markup Language ( V R M L ) , 38.3

246 Hydrology of Small Watersheds ( T R - 5 5 ) , 2 2 7 , 229,233,2.35-6 sheet flow, 23.3-4

Visual comparison, 116

spatial characteristics. 233

Visual models, 2

sutlace toughness. 230, 23.3-4

Visual Paradigm, 47

rime of concentration, 233, 2 3 4 - 5

Volterra model, 1 3 9 - 4 0

travel time, 233

V R M L (Virtual Reality Markup Language), 383

W E A P (Water Evaluation and Planning system),

Water, 1 9 7 - 2 4 7

Web, 3 9 2 - 3

50, 51 conclusions, 2 4 4 - 6

access to, 393

groundwater, 199. 223, 224, 2 3 0 - 1 , 245

features, 3 9 3 - 4

horizontal water flow, 205, 2 1 9

vs Internet, 393

importance of, 1 9 7 - 8 molecules, 7

See also Internet Wetland areas, 204, 2 1 7 - 1 8

runoff, 233

W h i t e box models, 4 0

in the saturated layer, 199, 2 1 9 - 2 3

" T h e W h o l e is more than the sum of parts", 15

summary, 197-8

Wind tunnels. 3

surface water, 199, 2 0 0 - 9

Wolves, in Yellowstone National Park, 162

in the unsaturated layer, 199, 2 1 2 - 1 8 , 2 2 1 - 3

World3 model, 296, 2 9 8

vertical water flow. 212, 2 1 4 - 1 5 , 2 1 9 - 2 1 See also Hydrologic models; Watersheds

Yellowstone National Park, wolves in, 162

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