Mathematical Models and Environmental Change

Case Studies in Long Term Management

Douglas J. Crookes

• Publisher: Routledge
• Year: 2021
• eBook ISBN: 9781000539011
• Edition: 1
• Topic: Economics
• Subtopic: Ecology
• Index: Economics

This book demonstrates how mathematical models constructed in system dynamics modelling platforms, such as Vensim, can be used for long-term management of environmental change.

It is divided into two sections, with the first dedicated to theory, where the theory of co-evolutionary modelling and its use in the system dynamics model platform is developed. The book takes readers through the steps in the modelling process, different validation tools applicable to these types of models and different growth specification, as well as how to curve fit using numerical methods in Vensim. Section 2 comprises of a collection of applied case studies, including fisheries, game theory and wildlife management. The book concludes with lessons from the use of co-evolutionary models for long-term natural resource management.

The book will be of great interest to students and scholars of environmental economics, natural resource management, system dynamics, ecological modelling and bioeconomics.

1 Introduction

DOI: 10.4324/9781003247982-1

The aim of this book is to show how mathematical models constructed in system dynamics modelling platforms, such as Vensim, may be used for long-term management of environmental change. Environmental change is a change or disturbance in the natural environment, for example, through human disturbance or natural processes (Johnson et al. 1997). The concept denotes some form of change in the environment over time. When mathematically modelling these interacting processes, we are referring to non-linear simultaneous differential equations. In biological terms, these processes are referred to as co-evolution, whereas in the fisheries economics literature they are referred to as bioeconomic models. We will utilise the biological term in this book, since it is slightly more encompassing, but many of the models discussed here would fit into the bioeconomic category.

Models for the management of environmental change are ubiquitous. Following Nielsen et al. (2018), there are three types of models: those that provide short-term (tactical) advice, models for the medium term (management strategy evaluation, MSE) and models that provide long-term (strategic) advice. Although opinions vary over the exact time frames for each of these models, Shin et al. (2005) state that tactical models provide advice that is annual, and strategic models provide advice over a period of five to ten years. Therefore, it is concluded that MSE provides advice over a period of one to five years.

Although different models have different purposes and some are useful (Box and Draper 1987), here we are interested in models that provide strategic (long-term) advice. Which models would fall into this category? Although we are not concerned here with a comprehensive review of strategic models, Crookes and Blignaut (2016) found that results of a co-evolutionary model for steel were comparable to artificial neural networks (ANNs) over a ten-year forecast period, after which they digressed. This suggests that co-evolutionary models can be used for strategic advice (but we will further assess this claim during the course of this book).

The aim of this book is to show how co-evolutionary models may be used for long-term management of environmental change. What is co-evolution? Co-evolution occurs when two or more entities interact with each other over time (Conrad 1999). In ecology, this could be two or more species that interact with each other. In this case, these interactions often give rise to the well-known Lotka–Volterra system of equations: predator–prey (Lotka 1925; Volterra 1926, 1928, 1931), interspecific competition (Gause 1932; Gause and Witt 1935), mutualism (May 1982) and parasite–host interactions (Anderson and May 1982). In socioeconomic systems, one entity could be an economic agent (such as a hunter, a poacher or a fisher), and the other could be a biological entity (such as a fish stock). In this case, the equations give rise to so-called fisheries economics models, or (mathematical) bioeconomic models, as we alluded to previously (Clark 1990). Finally, interactions could be between two economic entities, giving rise to inter-sectoral dynamics (Crookes and Blignaut 2016) or interactions between different macroeconomic variables (such as the Goodwin model; Barbosa‐Filho and Taylor 2006).

A major constraint in the development of these models is that the biological and economic parameters are often unknown (Butterworth et al. 2010; Nielsen et al. 2018). Various methods have been proposed to overcome this: The non-linear least squares method is provided in packages such as EViews and MATLAB^® (Gatabazi et al. 2019). Linear programming is also employed based on the criterion of minimisation of the mean absolute percentage error (MAPE) (Wu et al. 2012). In this book, curve fitting (in other words, the process of deriving a graphical line or mathematical function that best matches a set of data observations) is employed using numerical methods (Markov chain Monte Carlo simulation) in the Vensim modelling platform (Eberlein and Peterson 1992). This method has the advantage of optimising over a wide range of unknown parameter values (Banerjee et al. 2014). The algorithm is a differential evolution/Markov chain hybrid method (Vrugt et al. 2008). Differential evolution belongs to the class of genetic algorithms. It is also possible to use the algorithm for Bayesian inference by specifying a prior distribution and likelihood function (Ter Braak 2006).

Wilensky and Reisman (2006) argue that when curve fitting is employed, it is important that these types of models are validated. The system dynamics modelling literature provides a means by which these types of models may be validated. Furthermore, co-evolutionary models are well suited for use in packages (Swart 1990). The approach used here is, therefore, to build and validate the models using the system dynamics simulation methodology (Sterman 2000).

Once the models have been built and validated, they can be employed for a variety of purposes. There are at least five uses of the models: (1) They can be used to inform the value of biological and economic parameters. This may be an end in itself. These parameters may be unknown in the literature, and curve fitting may be used to estimate these unknown values. (2) The models may contribute towards a better understanding of the behaviour of the system. The choice of model that provides the best fit of the historical data could provide information of the nature of the system and the behaviour of the entities within that system. (3) The models may be used to forecast future values of the entities in question. (4) Most of these model uses are in the context of exploitation. However, co-evolutionary models may also be used as input into game theoretic systems in order to identify conditions for co-operation. (5) Finally, these models may be used for scenario analysis. It involves interrogating the model, for example, through what-if type analysis. All of these uses (points 1–5) suggest that these types of models, when properly built and validated, could contribute towards providing advice for the management of natural resources and other economic and financial systems.

The structure of the book is as follows: In the next chapter, we provide a review of different models of exploitation. Then, we discuss the simulation modelling technique, highlighting steps in the modelling approach and validation. In Chapter 4 we elaborate on the method of curve fitting, showing how to implement numerical simulation methods in Vensim. We highlight a number of different applications. One is forecasting populations. An application to rhino management is considered in Chapter 5. This technique emphasises the use of different models and comparing the dynamics thereof to determine which behaviour is most realistic.

Next, the co-evolutionary models are used in a modified prisoner’s dilemma game in order to determine under what conditions co-operation is possible. This is the topic of Chapter 6. Chapter 7 provides an example of these co-evolutionary models used for providing tactical management advice, using the case of the African penguin. This example highlights how incorporating environmental stochasticity may enhance the realism of these models. Finally, Chapter 8 concludes with a discussion of the findings.

References

1. Anderson, R.M. and May, R.M., 1982. Coevolution of hosts and parasites. Parasitology, 85(Pt 2), pp.411–426.
2. Banerjee, S., Carlin, B.P., and Gelfand, A.E., 2014. Hierarchical Modeling and Analysis for Spatial Data (2nd edn). London, UK: Chapman & Hall.
3. Barbosa‐Filho, N.H. and Taylor, L., 2006. Distributive and demand cycles in the US economy—A structuralist Goodwin model. Metroeconomica, 57(3), pp.389–411.
4. Box, G.E.P. and Draper, N.R., 1987. Empirical Model-Building and Response Surfaces. New York: John Wiley & Sons.
5. Butterworth, D.S., Bentley, N., De Oliveira, J.A., Donovan, G.P., Kell, L.T., Parma, A.M., Punt, A.E., Sainsbury, K.J., Smith, A.D., and Stokes, T.K., 2010. Purported flaws in management strategy evaluation: Basic problems or misinterpretations? ICES Journal of Marine Science, 67(3), pp.567–574.
6. Clark, C.W., 1990. Mathematical Bioeconomics: The Optimal Management of Renewable Resources (2nd edn). New York: Wiley.
7. Conrad, J.M., 1999. Resource Economics. New York: Cambridge University Press.
8. Crookes, D.J. and Blignaut, J.N., 2016. Predator-prey analysis using system dynamics: An application to the steel industry. South African Journal of Economic and Management Sciences, 19(5), pp.733–746.
9. Eberlein, R.L. and Peterson, D.W., 1992. Understanding models with Vensim™. European Journal of Operational Research, 59(1), pp.216–219.
10. Gatabazi, P., Mba, J.C., Pindza, E., and Labuschagne, C., 2019. Grey Lotka–Volterra models with application to cryptocurrencies adoption. Chaos, Solitons & Fractals, 122, pp.47–57.
11. Gause, G.F. 1932. Experimental studies on the struggle for existence. Journal of Experimental Biology, 9(4), pp. 389–402.
12. Gause, G. and Witt, A., 1935. Behavior of mixed populations and the problem of natural selection. The American Naturalist, 69(725), pp.596–609.
13. Johnson, D.L., Ambrose, S.H., Bassett, T.J., Bowen, M.L., Crummey, D.E., Isaacson, J.S., Johnson, D.N., Lamb, P., Saul, M., and Winter-Nelson, A.E., 1997. Meanings of environmental terms. Journal of Environmental Quality, 21(5), 581–589. doi:10.2134/jeq1997.00472425002600030002x
14. Lotka, A.J., 1925. Elements of Physical Biology. Baltimore, MD: Williams and Wilkins.
15. May, R.M., 1982. Mutualistic interactions among species. Nature, 296(5860), pp.803–804.
16. Nielsen, J.R., Thunberg, E., Holland, D.S., Schmidt, J.O., Fulton, E.A., Bastardie, F., Punt, A.E., Allen, I., Bartelings, H., Bertignac, M., and Bethke, E., 2018. Integrated ecological–economic fisheries models—Evaluation, review and challenges for implementation. Fish and Fisheries, 19(1), pp.1–29.
17. Shin, Y.J., Rochet, M.J., Jennings, S., Field, J.G., and Gislason, H., 2005. Using size-based indicators to evaluate the ecosystem effects of fishing. ICES Journal of Marine Science, 62(3), pp.384–396.
18. Sterman, J.D., 2000. Business Dynamics. Systems Thinking and Modelling for a Complex World. Boston, MA: Irwin McGraw-Hill.
19. Swart, J., 1990. A system dynamics approach to predator‐prey modeling. System Dynamics Review, 6(1), pp.94–99.
20. Ter Braak, C.J., 2006. A Markov Chain Monte Carlo version of the genetic algorithm Differential Evolution: Easy Bayesian computing for real parameter spaces. Statistics and Computing, 16(3), pp.239–249.
21. Volterra V., 1926. Fluctuations in the abundance of a species considered mathemat...

Table of contents

• Cover
• Half-Title
• Series
• Title
• Copyright
• Contents
• Acknowledgements
• 1 Introduction
• 2 Models of co-evolution
• 3 Co-evolutionary models and system dynamics modelling
• 4 Using numerical methods to estimate unknown parameters in co-evolutionary models
• 5 Co-evolutionary models and rhino management
• 6 Co-evolutionary models and the prisoner’s dilemma game
• 7 Co-evolutionary models and oceans governance: The case of the African penguin (Spheniscus demersus)
• 8 Discussion and conclusions
• Index