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(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 9.5.3. Does the result agree with that obtained from the linear approximation?
2. dx/dt = x(1 - 0.5/) dy/dt = /(-0.25 + 0.5x)
4. dx/dt = x(1.125 - x - 0.5y) dy/dt = y(-1 + x)
Chapter 9. Nonlinear Differential Equations and Stability
(d) Determine the prey-predator amplitude ratio for other solutions of the system (2), that is, for solutions satisfying other initial conditions. Is the ratio independent of the initial conditions?
? 8. (a) Find the period of the oscillations of the prey and predator populations, using the
approximation (24), which is valid for small oscillations. Note that the period is independent of the amplitude of the oscillations.
(b) For the solution of the nonlinear system (2) shown in Figure 9.5.3 estimate the period as well as possible. Is the result the same as for the linear approximation?
(c) Calculate other solutions of the system (2), that is, solutions satisfying other initial
conditions, and determine their periods. Is the period the same for all initial conditions?
? 9. Consider the system
where a and b are positive constants. Observe that this system is the same as in the example in the text if a = 1 and b = 0.75. Suppose the initial conditions are x(0) = 5 and y(0) = 2.
(a) Let a = 1 and b = 1. Plot the trajectory in the phase plane and determine (or estimate) the period of the oscillation.
(b) Repeat part (a) for a = 3 and a = 1/3, with b = 1.
(c) Repeat part (a) for b = 3 and b = 1/3, with a = 1.
(d) Describe how the period and the shape of the trajectory depend on a and b.
? 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), valid near the critical point, show that X = c/y and y = a/a. _ _
(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/y and a/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.
(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?
11. Suppose that the predator-prey equations (1) of the text describe foxes (y) and rabbits (x) in a forest. A trapping company is engaged in trapping foxes and rabbits for their pelts. Explain why it is reasonable for the company to conduct its operation so as to move the population of each species closer to the center (c/y, a/a). When is it best to trap foxes? Rabbits? Rabbits and foxes? Neither?
Hint: See Problem 6. An intuitive argument is all that is required.
12. Suppose that an insect population x is controlled by a natural predator population y according to the model (1), so that there are small cyclic variations of the populations about the critical point (c/y, a/a). Suppose that an insecticide is employed with the goal of reducing the population of the insects, and suppose that this insecticide is also toxic to the predators; indeed, suppose that the insecticide kills both prey and predator at rates proportional to their respective populations. Write down the modified differential equations, determine the new equilibrium point, and compare it to the original equilibrium point.
To ban insecticides on the basis of this simple model and counterintuitive result would certainly be ill-advised. On the other hand, it is also rash to ignore the possible genuine existence of a phenomenon suggested by such a model.
dx/dt = ax[1 ó (y/2.)], dy/dt = by[-1 + (x /3)],
x(t) dt, y = - y(t) dt,
9.6 Liapunovís Second Method
13. As mentioned in the text, one improvement in the predator-prey model is to modify the equation for the prey so that it has the form of a logistic equation in the absence of the predator. Thus in place of Eqs. (1) we consider the system
dx/dt = x (a ó a x ó ay), dy/dt = y(-c + y x),
where a, a, a, c, and y are positive constants. Determine all critical points and discuss their nature and stability characteristics. Assume that a/a ^ c/y. What happens for initial data x = 0, y = 0?
9.6 Liapunovís Second Method
In Section 9.3 we showed how the stability of a critical point of an almost linear system can usually be determined from a study of the corresponding linear system. However, no conclusion can be drawn when the critical point is a center of the corresponding linear system. Examples of this situation are the undamped pendulum, Equations (1) and (2) below, and the predator-prey problem discussed in Section 9.5. Also, for an asymptotically stable critical point it may be important to investigate the basin of attraction; that is, the domain such that all solutions starting within that domain approach the critical point. Since the theory of almost linear systems is a local theory, it provides no information about this question.