Global stability of nonhomogeneous equilibrium solution for the diffusive Lotka-Volterra competition model

TL;DR

This paper proves the global stability of positive equilibrium solutions in a spatially heterogeneous diffusive Lotka-Volterra competition model, considering multiple species and resource variations, using a novel Lyapunov functional approach.

Contribution

It introduces a new Lyapunov functional method to establish global stability of nonhomogeneous equilibria in complex spatial ecological models.

Findings

Positive equilibrium is globally stable in two-species systems with spatial heterogeneity.

Existence and stability of equilibria are shown for models with multiple species.

A new Lyapunov method is developed for non-constant equilibrium stability analysis.

Abstract

A diffusive Lotka-Volterra competition model is considered for the combined effect of spatial dispersal and spatial variations of resource on the population persistence and exclusion. First it is shown that in a two-species system in which the diffusion coefficients, resource functions and competition rates are all spatially heterogeneous, the positive equilibrium solution is globally asymptotically stable when it exists. Secondly the existence and global asymptotic stability of the positive and semi-trivial equilibrium solutions are obtained for the model with arbitrary number of species under the assumption of spatially heterogeneous resource distribution. A new Lyapunov functional method is developed to prove the global stability of a non-constant equilibrium solution in heterogeneous environment.