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arXiv · 1201.4417

Instabilities and Patterns in Coupled Reaction-Diffusion Layers

Anne J. Catlla·Amelia McNamara·Chad M. Topaz
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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.

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Anne J. Catlla, Amelia McNamara, Chad M. Topaz. 2012-01-21. Instabilities and Patterns in Coupled Reaction-Diffusion Layers. https://doi.org/10.1103/physreve.85.026215

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