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Each of Problems 1 through 5 can be interpreted as describing the interaction of two species
with population densities x and y. In each of these problems carry out the following steps.
(a) Draw a direction field and describe how solutions seem to behave.
(b) Find the critical points.
(c) For each critical point find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system; classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable.
(d) Sketch the trajectories in the neighborhood of each critical point.
(e) Draw a phase portrait for the system.
(f) Determine the limiting behavior of x and y as t and interpret the results in terms of
the populations of the two species.
1. dx/dt = x (1.5 — 0.5 y)
dy/dt = y(—0.5 + x)
3. dx/dt = x (1 — 0.5x — 0.5 y)
dy/dt = y(—0.25 + 0.5x)
5. dx/dt = x (— 1 + 2.5x — 0.3 y — x2) dy/dt = y(—1.5 + x)
6. In this problem we examine the phase difference between the cyclic variations of the predator and prey populations as given by Eqs. (24) of the text. Suppose we assume that K > 0 and that t is measured from the time that the prey population x is a maximum; then ô = 0. Show that the predator population y is a maximum at t = n/2*fac = T/4, where T is the period of the oscillation. When is the prey population increasing most rapidly, decreasing most rapidly, a minimum? Answer the same questions for the predator population. Draw a typical elliptic trajectory enclosing the point (c/y, a/a), and mark these points on it.
7. (a) Find the ratio of the amplitudes of the oscillations of the prey and predator populations about the critical point (c/y, a/a), using the approximation (24), which is valid for small 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 9.5.3. Does the result agree with that obtained from the linear approximation?
> 2. dx/dt = x (1 — 0.5y)
dy/dt = y(—0.25 + 0.5x)
> 4. dx/dt = x (1.125 — x — 0.5 y)
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 — and b are positive constants. Observe that this system is the same as in the example in the text if — = 1 and b = 0.75. Suppose the initial conditions are x(0) = 5 and y(0) = 2.
(a) Let — = 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 — = 3 and — = 1/3, with b = 1.
(c) Repeat part (a) for b = 3 and b = 1/3, with — = 1.
(d) Describe how the period and the shape of the trajectory depend on — 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. _ _
(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, 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). When is it best to trap foxes? Rabbits? Rabbits and foxes? Neither?