The average sizes of the prey and predator populations are

Chapter 9, Problem 10

(choose chapter or problem)

The average sizes of the prey and predator populations are defined asx = 1TA+TAx(t) dt, y = 1TA+TAy(t) dt,respectively, where T is the period of a full cycle, and A is any nonnegative constant.(a) Using the approximation (24), which is valid near the critical point, show that x = c/and y = a/.(b) For the solution of the nonlinear system (2) shown in Figure 9.5.3, estimate x and y aswell as you can. Try to determine whether x and y are given by c/ and a/, respectively,in this case.Hint: Consider how you might estimate the value of an integral even though you do nothave a formula for the integrand.(c) Calculate other solutions of the system (2)that is, solutions satisfying other initialconditionsand determine x and y for these solutions. Are the values of x and y the samefor all solutions?In 11 and 12, we consider the effect of modifying the equation for the prey x byincluding a term x2 so that this equation reduces to a logistic equation in the absence ofthe predator y. deals with a specific system of this kind, and takesup this modification to the general LotkaVolterra system. The systems in 3 and4 are other examples of this type.