Chapter 7.4, Problem 6P

In this problem, we examine the phase difference between the cyclic variations of the predator and prey populations asgiven by Eqs. (22) of this section. Suppose we assume that $K>0$ and that $t$ is measured from the time that the prey population $(x)$ is a maximum; then $\varphi =0.$ Show that the predator population $(y)$ is a maximum at $t=\pi /(2\sqrt{ac})=T/4$, where $T$ is the period of the oscillation. When is the prey population increasing most rapidly? decreasing most rapidly? A minimum? Answer the same questions for the predator population. Draw a typical elliptic trajectory enclosing the point $(c/\gamma ,a/\alpha )$, and mark these points on it.