In this and the subsequent chapters, I return to the milieu of the first 12 chapters, in which only dependence on time was at issue. However, whereas the first 12 chapters considered equilibrium issues almost exclusively, this chapter and the remaining chapters consider nonequilibrium phenomena. Here, the story is amazingly complicated and there is no sense in which it can be said that the interesting questions are all solved. Indeed, the complicated nonequilibrium dynamics that arise even from very simple models are still a wealthy source of very interesting mathematics. Meanwhile, similar appearing dynamics appears in real biological systems and the underlying causes are a subject of intense investigation.

I shall start by describing a predator-prey model that has a stable, time-dependent, periodic solution. The model presented provides one mathematical explanation for cyclic behavior. However, the model itself is not the main point of this chapter. Rather, you should focus on those aspects of the model that guarantee the existence of cyclic solutions, for those aspects are found in many other models. That is, there are certain generic properties of a differential equation that a priori imply that there are periodic solutions. In particular, the properties to notice are the existence of a basin of attraction in the phase plane that has inside a certain type of unstable equilibrium point.