Existence of Travelling Wave Solutions of Predator–Prey Systems via the Connection Index

Abstract

We consider a reaction–diffusion system which arises as a model of predator–prey interactions in mathematical ecology. The system admits four constant equilibria, two of which are stable. The main result is an existence theorem for travelling wave solutions which connect the two stable equilibria. Most previous efforts with regard to travelling waves for systems employ singular perturbations to locate the connecting orbit. The new aspect of the present discussion lies in devising some novel methods for “locating” the connecting solution in phase space which do not require that the equations be singularly perturbed. In certain cases, the main result holds for an arbitrary choice of diffusion coefficients.