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

Edgardo Villar-Sepúlveda1, Alan R Champneys2

  • 1Engineering Mathematics, University of Bristol, Ada Lovelace Building, Tankard's Cl, University Walk, Bristol, Somerset, BS8 1TW, UK. edgardo.villar-sepulveda@bristol.ac.uk.

Journal of Mathematical Biology
|January 28, 2023
PubMed
Summary

This study provides conditions for diffusion-driven instabilities in reaction-diffusion systems, enabling the selection of diffusion constants for Turing or wave instabilities. These findings are demonstrated on models like the FitzHugh-Nagumo system.

Keywords:
Diffusion-driven instabilityReaction–diffusionSpatio-temporal oscillationsTuring instabilityWave instability

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Area of Science:

  • Chemical kinetics
  • Mathematical modeling
  • Nonlinear dynamics

Background:

  • Reaction-diffusion systems are crucial for modeling spatially extended phenomena.
  • Understanding instabilities is key to predicting pattern formation and system behavior.
  • Previous work established conditions for Turing bifurcations but lacked constructive methods for parameter selection.

Purpose of the Study:

  • To establish necessary and sufficient conditions for diffusion-driven instabilities in n-component reaction-diffusion systems.
  • To develop a constructive method for selecting diffusion constants that induce Turing or wave instabilities.
  • To illustrate these conditions using established models.

Main Methods:

  • Analysis of dispersion relations in the limits of zero and infinite wavenumber.
  • Derivation of conditions for Turing and wave instabilities.
  • Application of the method to FitzHugh-Nagumo-like and Brusselator systems.

Main Results:

  • Necessary and sufficient conditions for diffusion-driven instabilities (Turing or wave) are derived.
  • A constructive approach for choosing diffusion constants is presented.
  • The method successfully reproduces known conditions for Turing bifurcations and provides new insights.

Conclusions:

  • The study offers a robust framework for analyzing and predicting instabilities in reaction-diffusion systems.
  • The developed method facilitates the design of systems exhibiting specific spatiotemporal patterns.
  • The findings are broadly applicable to various chemical and biological systems modeled by reaction-diffusion equations.