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arXiv · 2409.06860

Designing reaction-cross-diffusion systems with Turing and wave instabilities

Edgardo Villar-Sepúlveda·Alan R. Champneys·Andrew L. Krause
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Abstract

General conditions are established under which reaction-cross-diffusion systems can undergo spatiotemporal pattern-forming instabilities. Recent work has focused on designing systems theoretically and experimentally to exhibit patterns with specific features, but the case of non-diagonal diffusion matrices has yet to be analysed. Here, a framework is presented for the design of general $n$-component reaction-cross-diffusion systems that exhibit Turing and wave instabilities of a given wavelength. For a fixed set of reaction kinetics, it is shown how to choose diffusion matrices that produce each instability; conversely, for a given diffusion tensor, how to choose linearised kinetics. The theory is applied to several examples including a hyperbolic reaction-diffusion system, two different 3-component models, and a spatio-temporal version of the Ross-Macdonald model for the spread of malaria.

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Edgardo Villar-Sepúlveda, Alan R. Champneys, Andrew L. Krause. 2025-08-24. Designing reaction-cross-diffusion systems with Turing and wave instabilities. https://arxiv.org/abs/2409.06860

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