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Diffusive instability in hyperbolic reaction-diffusion equation with different inertia.

Santu Ghorai1, Swarup Poria2, Nandadulal Bairagi1

  • 1Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Raja Subodh Chandra Mallick Road, Kolkata 700032, India.

Chaos (Woodbury, N.Y.)
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Summary
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This study investigates instabilities in hyperbolic reaction-diffusion systems with varying inertia. Wave instability is possible with different inertias, unlike in parabolic systems, and Turing instability is inertia-independent.

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Area of Science:

  • Mathematical Biology
  • Chemical Kinetics
  • Nonlinear Dynamics

Background:

  • Reaction-diffusion systems are fundamental models in biological and chemical pattern formation.
  • Hyperbolic systems incorporate inertia, offering distinct dynamics from traditional parabolic models.
  • Understanding instabilities is crucial for predicting pattern emergence.

Purpose of the Study:

  • To analyze wave, Turing, and Hopf instabilities in a 2D hyperbolic reaction-diffusion system with differing inertia.
  • To theoretically and numerically determine criteria for these instabilities.
  • To compare the behavior of hyperbolic systems with their parabolic counterparts.

Main Methods:

  • Theoretical analysis of stability criteria.
  • Numerical simulations of the reaction-diffusion system.
  • Application of the Schnakenberg system as a local interaction model.

Main Results:

  • Wave instability can occur in hyperbolic systems with different inertias and equal diffusivities, a scenario impossible in parabolic systems.
  • Wave instability is possible in two-species hyperbolic systems with identical inertia if diffusion coefficients differ.
  • Turing instability is independent of inertia, but local system stability is inertia-dependent.

Conclusions:

  • Inertia plays a significant role in wave instability in hyperbolic reaction-diffusion systems.
  • The distinct dynamics of hyperbolic systems allow for instabilities not observed in parabolic systems.
  • The findings provide insights into pattern formation mechanisms influenced by inertial effects.