Chapter 7.4, Problem 7P

a) Find the ratio of the amplitudes of the oscillations of the prey and predator populations about the critical point $(c/\gamma ,a/\alpha )$, using the approximation $(22)$, which is valid forsmall oscillations. Observe that the ratio is independent of the initial conditions.

b) Evaluate the ratio found in part (a) for the system $(2)$.

c) Estimate the amplitude ratio for the solution of the nonlinear system $(2)$ shown in Figure 7.4.3. Does the result agreewith that obtained from the linear approximation?

d) Determine the prey-predator amplitude ratio for othersolutions of the system $(2)$, that is, for solutions satisfyingother initial conditions. Is the ratio independent of the initialconditions?