Global dynamics of competition models with nonlocal dispersals I: Symmetric kernels

TL;DR

This paper analyzes the global behavior of two-species competition models with symmetric nonlocal dispersal kernels, establishing conditions under which local stability implies global stability and exploring the effects of spatially varying competition and mixed dispersal strategies.

Contribution

It provides new theoretical results on the global dynamics of competition models with symmetric nonlocal dispersals, including cases with spatially dependent coefficients and mixed dispersal strategies.

Findings

Local stability of semi-trivial states often implies global stability.

When both semi-trivial states are unstable, a positive stable state exists.

Results extend to models with location-dependent coefficients and mixed dispersal strategies.

Abstract

In this paper, the global dynamics of two-species Lotka-Volterra competition models with nonlocal dispersals is studied. Under the assumption that dispersal kernels are symmetric, we prove that except for very special situations, local stability of semi-trivial steady states implies global stability, while when both semi-trivial steady states are locally unstable, the positive steady state exists and is globally stable. Moreover, our results cover the case that competition coefficients are location-dependent and dispersal strategies are mixture of local and nonlocal dispersals.