Instabilities and Patterns in Coupled Reaction-Diffusion Layers

TL;DR

This paper analyzes how coupled reaction-diffusion layers exhibit various instabilities and pattern formations, including Turing bifurcations with tunable length scales, using linear stability analysis and numerical simulations.

Contribution

It provides a comprehensive stability analysis of coupled reaction-diffusion layers, identifying multiple bifurcation scenarios and demonstrating control over pattern scales.

Findings

Identification of eight primary bifurcation scenarios including Turing-Turing bifurcations.

Demonstration of tunable length scales via inter-layer coupling.

Numerical simulation of square patterns with specific length scale ratios.

Abstract

We study instabilities and pattern formation in reaction-diffusion layers that are diffusively coupled. For two-layer systems of identical two-component reactions, we analyze the stability of homogeneous steady states by exploiting the block symmetric structure of the linear problem. There are eight possible primary bifurcation scenarios, including a Turing-Turing bifurcation that involves two disparate length scales whose ratio may be tuned via the inter-layer coupling. For systems of $n$-component layers and non-identical layers, the linear problem's block form allows approximate decomposition into lower-dimensional linear problems if the coupling is sufficiently weak. As an example, we apply these results to a two-layer Brusselator system. The competing length scales engineered within the linear problem are readily apparent in numerical simulations of the full system. Selecting a $\sqrt{2}$:1 length scale ratio produces an unusual steady square pattern.