The existence of periodic travelling waves for singularly perturbed predator-prey equations via the Conley Index

Abstract

This paper is concerned with the existence of periodic travelling wave solutions of reactiondiffusion equations, which arise as models of predator-prey interactions in mathematical ecology. The relevant equations are somewhat similar to the Fitz-Hugh-Nagumo equations, and our results apply to this latter system as well. In studying the Fitz-Hugh-Nagumo equations, Conley ([ 3 I), initiated a new topological approach to this problem. His method depends on the construction of an isolating neighborhood (cf. Section 2A) about a “singular” periodic solution. Associated with such a neighborhood is a topological invariant, the Conley Index, which, when nontrivial, implies the existence of a smooth solution near the singular one. The new aspects of the present discussion are in the method used to construct the isolating neighborhood (in IFi4), and in the computation of the index. With regard to the isolating neighborhood, our construction is global in that a parameter taken to be small in Conley’s equations, is now allowed to assume large values. In order to compute the index, we “continue” our equations to those studied by Conley. Actually, we go one step further and continue Conley’s equations to the Van der Pol equations crossed with a pair of linear equations admitting a repelling critical point. The index for this latter system is easily computed and turns out to be non-trivial. It follows from the invariance of the index under continuation (21, that the desired