Mathematical Modelling of Natural Phenomena

Research Article

Travelling Waves in Near-Degenerate Bistable Competition Models

E.O. Alzahrani, F.A. Davidson c1 and N. Dodds

Division of Mathematics, Dundee University, Dundee DD1 4HN, Scotland UK.

Abstract

We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species is assumed not to diffuse. We then consider travelling wave solutions that connect the two stable semi-trivial states of the non-degenerate system. Next, a general energy function for the full system is introduced. Using this and the limiting arguments, we are able to determine the wave direction for small diffusion coefficient ratios. The results obtained only require knowledge of the system kinetics.

(Online publication July 27 2010)

Key Words:

  • competition;
  • reaction-diffusion;
  • free energy;
  • bistable;
  • travelling waves

Mathematics Subject Classification:

  • 35K57;
  • 35C07;
  • 35B65

Correspondence:

c1 Corresponding author. E-mail: fdavidso@maths.dundee.ac.uk

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