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

Source: [Designing reaction-cross-diffusion systems with Turing and wave instabilities](<https://www.sota2.com/research/paper/68fa4b761e209131ce9462df>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-28T13:55:00.501Z

Authors: Edgardo Villar-Sep\\'ulveda, Alan R. Champneys, Andrew L. Krause

Published: 2024-09-10T20:50:53.000Z

## 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.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2409.06860>)

[PDF](<https://arxiv.org/pdf/2409.06860.pdf>)