General conditions for Turing and wave instabilities in reaction-diffusion systems
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 ∞, 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
Versions
Related
- Chemotaxis-induced linear instabilities and pattern formation in a reaction-diffusion model
- Identifying Non-Ideal Reaction-Diffusion Systems Unable to Maintain Diffusion Out-of-Equilibrium
- Computation of Turing bifurcation normal form for n-component reaction-diffusion systems
- A Structure-Preserving Deep Galerkin Method for Nonlinear Coupled Reaction–Diffusion Systems
- Amplitude equations for wave bifurcations in reaction-diffusion systems
BibTeX
@misc{villarsepulveda2026general,
title = {{General conditions for Turing and wave instabilities in reaction-diffusion systems}},
author = {Edgardo Villar-Sepúlveda and Alan R. Champneys},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28036},
eprint = {2609.28036},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28036},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - General conditions for Turing and wave instabilities in reaction-diffusion systems
AU - Edgardo Villar-Sepúlveda
AU - Alan R. Champneys
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28036
UR - https://arxiv.org/abs/2609.28036
AB - 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 ∞, 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
ER -