Stability and Hopf Bifurcation of Steady State Solutions of a Singularly Perturbed Reaction-Diffusion System

Abstract

This paper presents a stability analysis of steady state solutions of a singularly perturbed reaction-diffusion system which arises as a model for predator-prey interactions. This is the first illustration of the application of a topological invariant for systems of boundary value problems called the stability index, recently introduced by C. Jones and the author, which counts the multiplicity of unstable eigenvalues of the linearized equations. The main results are as follows: (i) It is shown that the spectrum $\sigma $ of the linearized operator is approximated by the spectrum $\sigma _R $ of a certain reduced operator defined in the small parameter limit. (ii) Under additional assumptions on the parameters $\sigma _R $ is characterized in two cases. In the first case stability is obtained for all interval sizes L, while in the second case, countably many Hopf bifurcations occur as L is increased.