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

Edgardo Villar-Sepúlveda1, Alan R Champneys1, Andrew L Krause2

  • 1School of Engineering Mathematics and Technology, University of Bristol, Ada Lovelace Building, Tankard's Cl, University Walk, Bristol, BS8 1TW, United Kingdom.

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|September 11, 2025
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Summary
This summary is machine-generated.

This study introduces a framework for designing reaction-cross-diffusion systems to create specific spatiotemporal patterns. It analyzes non-diagonal diffusion matrices, enabling control over Turing and wave instabilities for pattern formation.

Keywords:
Diffusion-driven instabilityReaction-diffusionSpatiotemporal oscillationsTuring instabilitywave instability

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Area of Science:

  • Mathematical Biology
  • Chemical Kinetics
  • Nonlinear Dynamics

Background:

  • Reaction-diffusion systems are crucial for understanding pattern formation in biological and chemical systems.
  • Previous research focused on diagonal diffusion, limiting the analysis of complex pattern-forming instabilities.
  • Non-diagonal diffusion matrices introduce cross-diffusion effects that require novel theoretical frameworks.

Purpose of the Study:

  • To establish general conditions for spatiotemporal pattern formation in reaction-cross-diffusion systems.
  • To develop a framework for designing n-component systems exhibiting Turing and wave instabilities with a specific wavelength.
  • To analyze the impact of non-diagonal diffusion matrices on pattern formation.

Main Methods:

  • Development of a theoretical framework for analyzing reaction-cross-diffusion systems.
  • Methodology for selecting diffusion matrices based on reaction kinetics to induce specific instabilities.
  • Methodology for selecting linearised kinetics based on a given diffusion tensor.
  • Application of the framework to various models, including hyperbolic systems and epidemiological models.

Main Results:

  • A general framework is presented for designing reaction-cross-diffusion systems with desired pattern-forming instabilities.
  • The study demonstrates how to choose diffusion matrices or kinetics to control Turing and wave instabilities.
  • The framework is successfully applied to diverse systems, validating its applicability.

Conclusions:

  • The presented framework provides a systematic approach to designing reaction-cross-diffusion systems for specific spatiotemporal patterns.
  • Understanding non-diagonal diffusion is key to unlocking a broader range of pattern-forming phenomena.
  • This work advances the theoretical and practical design of complex pattern-forming systems.