Revealing new dynamical patterns in a reaction-diffusion model with cyclic competition via a novel computational framework

TL;DR

This paper introduces a novel computational framework that uncovers previously unknown dynamical patterns and wave propagation phenomena in classical reaction-diffusion models with spatial diffusion, enhancing understanding of biological pattern formation.

Contribution

The authors develop an adaptive finite element method with a new a posteriori error estimate, enabling the discovery of new pattern types in reaction-diffusion models.

Findings

Discovery of new wave propagation patterns in cyclic competition models

Identification of highly regular spatial patterns not previously documented

Demonstration of the framework's effectiveness in 2D and 3D simulations

Abstract

Understanding how patterns and travelling waves form in chemical and biological reaction-diffusion models is an area which has been widely researched, yet is still experiencing fast development. Surprisingly enough, we still do not have a clear understanding about all possible types of dynamical regimes in classical reaction-diffusion models such as Lotka-Volterra competition models with spatial dependence. In this work, we demonstrate some new types of wave propagation and pattern formation in a classical three species cyclic competition model with spatial diffusion, which have been so far missed in the literature. These new patterns are characterised by a high regularity in space, but are different from patterns previously known to exist in reaction-diffusion models, and may have important applications in improving our understanding of biological pattern formation and invasion theory. Finding these new patterns is made technically possible by using an automatic adaptive finite element method driven by a novel a posteriori error estimate which is proven to provide a reliable bound for the error of the numerical method. We demonstrate how this numerical framework allows us to easily explore the dynamical patterns both in two and three spatial dimensions.