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arXiv · 2609.28036

General conditions for Turing and wave instabilities in reaction-diffusion systems

Edgardo Villar-Sepúlveda·Alan R. Champneys
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Abstract

Necessary and sufficient conditions are provided for a diffusion-driven instability of a stable equilibrium of a reaction-diffusion system with $n$ components and a diagonal diffusion matrix. These can be either Turing or wave instabilities. Known necessary and sufficient conditions are reproduced for there to exist diffusion rates that cause a Turing bifurcation of a stable homogeneous state in the absence of diffusion. The method of proof here though, which is based on a study of dispersion relations in the contrasting limits in which the wavenumber tends to zero and to $\infty$, gives a constructive method for choosing diffusion constants. The results are illustrated on a model for the dispersion of malaria, a 3-component FitzHugh-Nagumo-like model proposed to study excitable wavetrains, and for two different coupled Brusselator systems with 4 components

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Edgardo Villar-Sepúlveda, Alan R. Champneys. 2026-09-23. General conditions for Turing and wave instabilities in reaction-diffusion systems. https://doi.org/10.1007/s00285-023-01884-x

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