10.The average sizes of the prey and predator populations are defined as ¯x=1T∫A+TAx(t)dt,¯y=1T∫A+TAy(t)dt,x¯=1T∫AA+Txtdt,y¯=1T∫AA+Tytdt, 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/γx¯=c/γ and ¯y=a/αy¯=a/α. b.For the solution of the nonlinear system (2) shown in Figure 9.5.3, estimate ¯xx¯ and ¯yy¯ as well as you can. Try to determine whether ¯xx¯ and ¯yy¯ are given by c/γ and a/α, 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 ¯xx¯ and ¯yy¯ for these solutions. Are the values of ¯xx¯ and ¯yy¯ the same for all solutions? In Problems 11 and 12, we consider the effect of modifying the equation for the prey x by including a term −σx2 so that this equation reduces to a logistic equation in the absence of the predator y. Problem 11 deals with a specific system of this kind, and Problem 12 takes up this modification to the general Lotka-Volterra system. The systems in Problem 3 is another example of this type.
10.The average sizes of the prey and predator populations are defined as
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/γx¯=c/γ and ¯y=a/αy¯=a/α.
b.For the solution of the nonlinear system (2) shown in Figure 9.5.3, estimate ¯xx¯ and ¯yy¯ as well as you can. Try to determine whether ¯xx¯ and ¯yy¯ are given by c/γ and a/α, 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 ¯xx¯ and ¯yy¯ for these solutions. Are the values of ¯xx¯ and ¯yy¯ the same for all solutions?
In Problems 11 and 12, we consider the effect of modifying the equation for the prey x by including a term −σx2 so that this equation reduces to a logistic equation in the absence of the predator y. Problem 11 deals with a specific system of this kind, and Problem 12 takes up this modification to the general Lotka-Volterra system. The systems in Problem 3 is another example of this type.
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