Solving systems of differential equations

Differential equations sometimes occur in systems consisting of two or more interlinked differential equations. A classic example is a simple model of the populations of competing species. This is a simple model of competing species labeled (the prey) and (the predators) given by the following equations:

The first equation dictates the growth of the prey species , which, without any predators, would be exponential growth. The second equation dictates the growth of the predator species , which, without any prey, would be exponential decay. Of course, these two equations are coupled; each population change depends on both populations. The predators consume the prey at a rate proportional to the product of their two populations, and the predators grow at a rate proportional to the relative abundance of prey (again the product of the two populations).

In this recipe, we will analyze a simple system of differential...