Pattern selection in reaction diffusion systems

TL;DR

This paper introduces a mass minimization principle to predict pattern selection in reaction-diffusion systems, providing new physical insights into the dynamics and final configurations of Turing patterns.

Contribution

It reveals that peaks act as point sinks influenced by diffusive fluxes, and that the preferred pattern minimizes the total diffusive mass, advancing understanding beyond the onset of instability.

Findings

Peaks behave as point sinks influenced by diffusive fluxes.

Final pattern peaks are those that minimize diffusive mass.

Mass minimization explains pattern selection far from onset.

Abstract

Turing's theory of pattern formation has been used to describe the formation of self-organised periodic patterns in many biological, chemical and physical systems. However, the use of such models is hindered by our inability to predict, in general, which pattern is obtained from a given set of model parameters. While much is known near the onset of the spatial instability, the mechanisms underlying pattern selection and dynamics away from onset are much less understood. Here, we provide new physical insight into the dynamics of these systems. We find that peaks in a Turing pattern behave as point sinks, the dynamics of which are determined by the diffusive fluxes into them. As a result, peaks move towards a periodic steady-state configuration that minimizes the mass of the diffusive species. We also show that the preferred number of peaks at the final steady-state is such that this mass is minimised. Our work presents mass minimization as a simple, generalisable, physical principle for understanding pattern formation in reaction diffusion systems far from onset.