The average sizes of the prey and predator populations are defined as x = 1 T _ A+T A

Chapter 9, Problem 10

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The average sizes of the prey and predator populations are defined as x = 1 T _ A+T A x(t) dt, y = 1 T _ A+T A y(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/. N b. For the solution of the nonlinear system (2) shown in Figure 9.5.3, estimate x and y as well 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 not have a formula for the integrand. G c. Calculate other solutions of the system (2)---that is, solutions satisfying other initial conditions---and determine x and y for these solutions. Are the values of x and y the same for all solutions? In 11 and 12, we consider the effect of modifying the equation for the prey x by including a term x2 so that this equation reduces to a logistic equation in the absence of the predator y. deals with a specific system of this kind, and takes up this modification to the general Lotka-Volterra system. The system in is another example of this type. 1