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9.5 Predator-Prey Equations
FIGURE 9.5.3 Variations of the prey and predator populations with time for the system (2).
The general system (1) can be analyzed in exactly the same way as in the example. The critical points of the system (1) are the solutions of
x (a — a y) = 0, y(—c + ó x) = 0,
that is, the points (0, 0) and (c/y, a/a). We first examine the solutions of the corresponding linear system near each critical point.
In the neighborhood of the origin the corresponding linear system is
The eigenvalues and eigenvectors are Ã1 = a, g(1) =
so the general solution is
Ã2 = -C, ª =
= Ñ1Û eat + C2u' e ^
Thus the origin is a saddle point and hence unstable. Entrance to the saddle point is along the y-axis; all other trajectories depart from the neighborhood of the critical point.
Next consider the critical point (c/y, a/a). If x = (c/y) + u and y = (a/a) + v, then the corresponding linear system is
ó a/a 0
The eigenvalues of the system (17) are r = ii^/ac, so the critical point is a (stable) center of the linear system. To find the trajectories of the system (17) we can divide the second equation by the first to obtain
dv dv/dt (ya/a)u
du du/dt (ac/y)v'
Chapter 9. Nonlinear Differential Equations and Stability
Y2—u du + a2cv dv = 0.
Y 2—u2 + a2cv2 = k,
where k is a nonnegative constant of integration. Thus the trajectories of the linear system (17) are ellipses, just as in the example.
Returning briefly to the nonlinear system (1), observe that it can be reduced to the single equation
where C is a constant of integration. Again it is possible to show that the graph of Eq. (22), for fixed C, is a closed curve surrounding the critical point (c/y, —/a). Thus this critical point is also a center for the general nonlinear system (1).
The cyclic variation of the predator and prey populations can be analyzed in more
These equations are good approximations for the nearly elliptical trajectories close to the critical point (c/y, —/a). We can use them to draw several conclusions about the cyclic variation of the predator and prey on such trajectories.
1. The sizes of the predator and prey populations vary sinusoidally with period 2n/y—c. This period of oscillation is independent of the initial conditions.
2. The predator and prey populations are out of phase by one-quarter of a cycle. The prey leads and the predator lags, as explained in the example.
3. The amplitudes of the oscillations are Kc/y for the prey and —+jcK/a+Ja for the predator and hence depend on the initial conditions as well as on the parameters of the problem.
4. The average populations of predator and prey over one complete cycle are c/y and —/a, respectively. These are the same as the equilibrium populations; see Problem 10.
Cyclic variations of predator and prey as predicted by Eqs. (1) have been observed in nature. One striking example is described by Odum (pp. 191-192); based on the records of the Hudson Bay Company of Canada, the abundance of lynx and snowshoe hare as indicated by the number of pelts turned in over the period 1845-1935 shows
dy = dy/dt = y( c + y x) dx dx/dt x(a - ay)
Equation (21) is separable and has the solution
— ln y - a y + c ln x - y x = C,
detail when the deviations from the point (c/y, —/a) are small and the linear system (17) can be used. The solution of the system (17) can be written in the form
u = — K cos(v—c t + ô),
where the constants K and ô are determined by the initial conditions. Thus
9.5 Predator-Prey Equations
a distinct periodic variation with period of 9 to 10 years. The peaks of abundance are followed by very rapid declines, and the peaks of abundance of the lynx and hare are out of phase, with that of the hare preceding that of the lynx by a year or more.
The Lotka-Volterra model of the predator-prey problem has revealed a cyclic variation that perhaps could have been anticipated. On the other hand, the use of the Lotka-Volterra model in other situations can lead to conclusions that are not intuitively obvious. An example that suggests a possible danger in using insecticides is given in Problem 12.
One criticism of the Lotka-Volterra equations is that in the absence of the predator the prey will grow without bound. This can be corrected by allowing for the natural inhibiting effect that an increasing population has on the growth rate of the population; for example, the first of Eqs. (1) can be modified so that when y = 0, it reduces to a logistic equation for x (see Problem 13). The most important consequence of this modification is that the critical point at (c/y, a/a) moves to (c/y, a/a — î c/ay) and becomes an asymptotically stable point. It is either a node or a spiral point depending on the values of the parameters in the differential equations. In either case, other trajectories are no longer closed curves but approach the critical point as t ^ro.