Mathematical Modeling in the Social and Life Sciences

CHAPTER 4

Ecological Models: Interacting Species

There are craft standards in both mathematics and ecology and the ideal interdisciplinary study simultaneously enhances our understanding of the empirical world and constitutes an example of elegant craftsmanship by both ecological and mathematical standards. That is a difficult set of criteria, but there is no reason to believe that science at its best is easy.

—Lawrence B. Slobodkin

I. Introduction

The previous chapter developed some models for population growth of a single species inhabiting an environment in which the amount of resources never changed and the numbers of other species also remained fixed. Although such a situation may sometimes be approached in laboratory experiments, an effective mathematical model should not ignore the fluctuations of the other important variables in an ecosystem.

In this chapter we present several models that attempt to represent the population dynamics that can occur in a system when two or more species interact with each other in the same environment. As is the case with most of the material in this book, we only consider relatively simple models. We will look in detail at two particular models that were the classic beginnings of mathematical ecology. They form the bases on which scientists construct more sophisticated models.

Those readers unfamiliar with partial derivatives and the other basic ideas of the calculus of several variables should read Appendix IV before tackling Section III. ...