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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, ó/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 = óx[1 - (y/2)], dy/dt = by[-1 + (x/3)],
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 Ó x ó ay), dy/dt = y(óc + ů x),
where a, Ó, a, c, and ů are positive constants. Determine all critical points and discuss their nature and stability characteristics. Assume that a/Ó ^ 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.
In this section we discuss another approach, known as Liapunovís5 second method or direct method. The method is referred to as a direct method because no knowledge of the solution of the system of differential equations is required. Rather, conclusions about the stability or instability of a critical point are obtained by constructing a suitable auxiliary function. The technique is a very powerful one that provides a more global type of information, for example, an estimate of the extent of the basin of attraction of a critical point. In addition, Liapunovís second method can also be used to study systems of equations that are not almost linear; however, we will not discuss such problems.
Basically, Liapunovís second method is a generalization of two physical principles for conservative systems, namely, (i) a rest position is stable if the potential energy is a local minimum, otherwise it is unstable, and (ii) the total energy is a constant during any motion. To illustrate these concepts, again consider the undamped pendulum (a conservative mechanical system), which is governed by the equation
2 +óT sin ‚ = 0. (1)
The corresponding system of first order equations is
dx dy g
dt = y dt = ó sď'x' (2)
5Alexandr M. Liapunov (1857-1918), a student of Chebyshev at St. Petersburg, taught at the University of Kharkov from 1885 to 1901, when he became academician in applied mathematics at the St. Petersburg Academy of Sciences. In 1917 he moved to Odessa because of his wifeís frail health. His research in stability encompassed both theoretical analysis and applications to various physical problems. His second method formed part of his most influential work, General Problem of Stability of Motion, published in 1892.
Chapter 9. Nonlinear Differential Equations and Stability
where x = ‚ and y = dQ/ dt. If we omit an arbitrary constant, the potential energy U is the work done in lifting the pendulum above its lowest position, namely,
U(x, y) = mgL(1 - cos x); (3)