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 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 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 3-component FitzHugh–Nagumo-like model proposed to study excitable wavetrains, and for two different coupled Brusselator systems with 4-components.

Amplitude equations for wave bifurcations in reaction–diffusion systems

“…The paper [23] points to another possible Turing-like instability, a (Turing) wave bifurcation, which would occur when the critical wavenumber corresponds to a pure-imaginary pair of eigenvalues. Whereas, when projected to an appropriate finite-dimensional subspace, the Turing bifurcation is typically a pitchfork bifurcation, the wave bifurcation is essentially a Hopf bifurcation.…”

“…In a recent paper [23], we developed necessary and sufficient conditions for an n-component reaction-diffusion system to undergo a Turing bifurcation, which corresponds to the transition through a parameter value at which there is a critical wavenumber with a zero temporal eigenvalue. An estimate for the amplitude of the ensuing spatial pattern, and indeed whether it is stable or not, can be captured by weakly nonlinear normal-form analysis.…”

“…But, it would be fair to say that this kind of pattern-forming instability has not received such a comprehensive treatment as the case of the stationary Turing patterns. One reason for this lack may be the fact that the minimum number of components for a reaction-diffusion system to feature a Turing instability is two, whereas at least three are required in the case of a wave bifurcation, see [23]. Thus, most canonical models of reaction-diffusion systems, such as the Brusselator, Gray-Scott model, etc having only two components, are incapable of developing wave instabilities.…”

“…Hence, we will also seek general formulae for these coefficients, both for the cases of standing and travelling waves. Moreover, all of the algorithms have been implemented in software, both Python and Mathematica, which are made freely available [24] for users to perform the necessary calculations on their own examples.…”

The appearance of standing wave structures in the reaction medium during the diffusion development of the chain reaction process

“…Accordingly, researchers as E. Villar-Sepúlveda & A.R. Champneys are devoting themselves to studying the spatial arrangement and dynamic behavior of radicals in standing wave structures with the main goal of improving reactor design and increasing its operational safety [5]. The focus is on understanding the specific conditions and underlying mechanisms that lead to the formation of these distinctive structures.…”