Analysis and Simulations of a Nonlocal Gray-Scott Model

TL;DR

This paper extends the Gray-Scott reaction-diffusion model by incorporating nonlocal diffusion, proves the existence of solutions, and uses numerical simulations to study pattern formation under this new framework.

Contribution

It introduces a nonlocal extension to the Gray-Scott model, proves existence of solutions, and explores pattern formation through numerical simulations.

Findings

Nonlocal diffusion influences pattern formation.

Existence of small-time weak solutions established.

Numerical simulations reveal effects of nonlocality on pulses.

Abstract

The Gray-Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. We consider an extension of this model in which the spread of the different chemicals is assumed to be nonlocal, and can thus be represented by an integral operator. In particular, we focus on the case of strictly positive, symmetric, $L^1$ convolution kernels that have a finite second moment. Modeling the equations on a finite interval, we prove the existence of small-time weak solutions in the case of nonlocal Dirichlet and Neumann boundary constraints. We then use this result to develop a finite element numerical scheme that helps us explore the effects of nonlocal diffusion on the formation of pulse solutions.