Chapter 7.4, Problem 10P

The average size of the prey and predator populations are defined as

$\begin{array}{cc}\bar{x}=\frac{1}{T}{\displaystyle {\int }_A^{A+T}x(t)dt,}& \bar{y}=\frac{1}{T}{\displaystyle {\int }_A^{A+T}y(t)dt,}\end{array}$

respectively, where $T$ is the period of a full cycle, and $A$ is any nonnegative constant.

a) Using the approximation $(22)$, which is valid near the critical point, show that $\bar{x}=c/\gamma$ and $\bar{y}=a/\alpha$.

b) For the solution of the nonlinear system $(2)$ shown in Figure 7.4.3, estimate $\bar{x}$ and $\bar{y}$ as well as you can. Try to determine whether $\bar{x}$ and $\bar{y}$ are given by $c/\gamma$ and $a/\alpha$, respectively, in this case.

Hint: Consider how you might estimate the value of an integral even though you do not have a formula for the integrand.

c) Calculate other solutions of the system $(2)$, that is, solutions satisfying other initial conditions, and determine $\bar{x}$ and $\bar{y}$ for these solutions. Are the values of $\bar{x}$ and $\bar{y}$ the same forall solutions?