LaSalle corollary

A brief mathematical digression on Liapunov stability theory will set the stage for the results of this chapter. Those familiar with the LaSalle corollary to Liapunov stability theory, also called the invariance principle by some authors, can skip immediately to Section 3. 

The following theorem is a minor variation of the LaSalle corollary of Liapunov stability theory, taken from [WLu] (see also [H2], where F is required to be continuous on the closure of G). It also holds under less restrictive hypotheses than are required here. 

We hope the reader will appreciate the elegance and simplicity of the arguments supporting Theorem 3.2, which are based on the LaSalle corollary. In particular, a linearized stability analysis about each of the rest points of (3.3), required in Chapter 1, was completely avoided. A careful reading of the proof of Theorem 3.2 reveals that assumption (iii) on f is not crucial to the proof we will have more to say about this later. Finally, it should be noted that the assumption (iv) on f can be relaxed somewhat. It can be weakened to requiring only that f be locally Lipschitz continuous... 

An application of the LaSalle corollary yields the desired result. ... 

The reader will have noticed that the Liapunov function used in the proof of the theorem was not obvious on either biological or mathematical grounds. Its discovery by Hsu greatly simplified and extended earlier arguments given in [HHW]. This is typical of applications of the LaSalle corollary. Considerable ingenuity, intuition, and perhaps luck are required to find a Liapunov function. 

Similarly, our analysis of the variable-yield model in Chapter 8 is limited to two competing populations because we rely on the techniques of monotone dynamical systems theory. One would expect the main result of Chapter 8 to remain valid regardless of the number of competitors, just as it did for the simpler constant-yield model treated in Chapters 1 and 2. Perhaps the LaSalle corollary of Chapter 2 can be used to carry out such an extension, using arguments similar to those used in [AM] (described in Chapter 2). As noted in [NG], a structured model in which... 

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