Stability of travelling waves for a three-component reaction-diffusion system

This paper investigates a reaction-diffusion system modeling three competing species, with diffusion in one space dimension. The main result is the existence and asymptotic stability of a travelling wave connecting two stable equilibria. The results are obtained in a small parameter regime where two of the species diffuse very slowly. Singular perturbation methods are used to locate the connecting solution, and the existence for positive values of the perturbation parameter is established via the connection index of Conley. The stability result applies geometric and topological methods developed by Alexander, Gardner, and Jones, to the analysis of the linearized equations about the travelling wave. This paper addresses some of the difficulties in extending these existence and stability methods to systems of more than two equations