Systems of Differential Equations

In Chapter 3 we saw how to model and analyze continuously changing quantities using differential equations. In many applications of interest there may be two or more interacting quantities — populations of two or more species, for instance, or parts of a whole, which depend upon each other. When the amount or size of one quantity depends in part on the amount of another, and vice versa, they are said to be coupled, and it is not possible or appropriate to model each one separately. In these cases we write models which consist of systems of differential equations. In this chapter, we will find that the quantitative and qualitative approaches we used to analyze individual differential equations in Chapters 3 and 4 extend in a more or less natural way to cover systems of differential equations. Extending them will require some basic multivariable calculus, principally the use of partial derivatives; we provide some support in the worked examples for the unfamiliar reader, but recommend using a calculus text for more complete reference.