Front and Turing patterns induced by Mexican-hat-like nonlocal feedback

TL;DR

This paper explores how Mexican-hat-shaped nonlocal feedback influences front propagation and pattern formation in a bistable reaction-diffusion system, revealing control over velocity and the emergence of Turing patterns.

Contribution

It demonstrates how nonlocal spatial coupling can control front velocity and induce Turing patterns in a reaction-diffusion model, combining stability analysis and numerical simulations.

Findings

Front velocity can be controlled by nonlocal coupling strength.

Spatially periodic Turing patterns can be induced beyond certain coupling thresholds.

Coexistence of homogeneous and patterned solutions observed.

Abstract

We consider the effects of a Mexican-hat-shaped nonlocal spatial coupling, i.e., symmetric long-range inhibition superimposed with short-range excitation, upon front propagation in a model of a bistable reaction-diffusion system. We show that the velocity of front propagation can be controlled up to a certain coupling strength beyond which spatially periodic patterns, such as Turing patterns or coexistence of spatially homogeneous solutions and Turing patterns, may be induced. This behaviour is investigated through a linear stability analysis of the spatially homogeneous steady states and numerical investigations of the full nonlinear equations in dependence upon the nonlocal coupling strength and the ratio of the excitatory and inhibitory coupling ranges.