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Turing conditions for pattern forming systems on evolving manifolds.

Robert A Van Gorder1, Václav Klika2, Andrew L Krause3

  • 1Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, 9054, New Zealand. rvangorder@maths.otago.ac.nz.

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This summary is machine-generated.

This study presents general conditions for diffusion-driven instabilities in reaction-diffusion systems on evolving domains. These findings simplify analyzing pattern formation in dynamic biological and chemical systems.

Keywords:
Evolving spatial domainsPattern formationReaction–diffusionTuring instability

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Area of Science:

  • Mathematical Biology
  • Chemical Kinetics
  • Theoretical Ecology

Background:

  • Pattern formation in reaction-diffusion systems is crucial for developmental biology, spatial ecology, and chemistry.
  • Analyzing instabilities on growing or time-dependent domains is complex due to time-varying base states and non-autonomous perturbation structures.

Purpose of the Study:

  • To derive general conditions for the onset and structure of diffusion-driven instabilities in reaction-diffusion systems on time-evolving domains.
  • To provide a versatile and implementable framework applicable to various domain evolution laws and system complexities.

Main Methods:

  • Utilizing the time-evolution of the Laplace-Beltrami spectrum of the domain.
  • Employing functions that describe domain evolution to establish conditions for instability onset.
  • Formulating sufficient conditions as differential inequalities.

Main Results:

  • General conditions for diffusion-driven instabilities on evolving domains were obtained.
  • The derived conditions generalize existing criteria for Turing instabilities on static domains.
  • Demonstrated applicability to domains with rapid changes and oscillatory homogeneous states (Turing-Hopf instabilities).

Conclusions:

  • The developed framework offers a unified approach to studying pattern formation on dynamic domains.
  • Provides versatile and straightforward criteria for predicting instabilities in complex systems.
  • The methodology is extendable to higher-order spatial systems, highlighting its broad applicability.