Competition between stable equilibria in reaction-diffusion systems: the influence of mobility on dominance

TL;DR

This study analyzes how small changes in species mobility affect the dominance of stable equilibria in reaction-diffusion models, with applications to ecological competition and insights into mobility's role in species advantage.

Contribution

It provides a first-order computation of front speed variation due to diffusion symmetry breaking and applies it to ecological models, revealing conditions where mobility influences species dominance.

Findings

Mobility increase can be advantageous or disadvantageous depending on parameters.

In the Lotka-Volterra model near bistability, increased mobility favors the species.

Geometric interpretations clarify the influence of mobility on stability.

Abstract

This paper is concerned with reaction-diffusion systems of two symmetric species in spatial dimension one, having two stable symmetric equilibria connected by a symmetric standing front. The first order variation of the speed of this front when the symmetry is broken through a small perturbation of the diffusion coefficients is computed. This elementary computation relates to the question, arising from population dynamics, of the influence of mobility on dominance, in reaction-diffusion systems modelling the interaction of two competing species. It is applied to two examples. First a toy example, where it is shown that, depending on the value of a parameter, an increase of the mobility of one of the species may be either advantageous or disadvantageous for this species. Then the Lotka-Volterra competition model, in the bistable regime close to the onset of bistability, where it is shown that an increase of mobility is advantageous. Geometric interpretations of these results are given.