Bifurcation and Pattern Formation in an Activator-Inhibitor Model with Non-local Dispersal

Abstract

In this paper, by approximating the non-local spatial dispersal equation by an associated reaction-diffusion system, an activator-inhibitor model with non-local dispersal is transformed into a reaction-diffusion system coupled with one ordinary differential equation. We prove that, to some extent, the non-locality-induced instability of the non-local system can be regarded as diffusion-driven instability of the reaction-diffusion system for sufficiently small perturbation. We study the structure of the spectrum of the corresponding linearized operator, and we use linear stability analysis and steady-state bifurcations to show the existence of non-constant steady states which generates non-homogeneous spatial patterns. As an example of our results, we study the bifurcation and pattern formation of a modified Klausmeier-Gray-Scott model of water-plant interaction.

Keywords: Bifurcation; Non-local dispersal; Pattern formation; Reaction–diffusion-ODE system; Spectrum; Water–plant model.