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Diffusive instabilities in a hyperbolic activator-inhibitor system with superdiffusion.

Alain Mvogo1,2, Jorge E Macías-Díaz3, Timoléon Crépin Kofané4

  • 1Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Cameroon.

Physical Review. E
|May 20, 2018
PubMed
Summary
This summary is machine-generated.

Anomalous diffusion and inertial time significantly alter wave instabilities in hyperbolic reaction-diffusion systems. Increasing superdiffusion raises Turing instability wave numbers, while faster activator diffusion is crucial for wave instability.

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Area of Science:

  • Mathematical Biology
  • Non-linear Dynamics
  • Physical Chemistry

Background:

  • Reaction-diffusion systems are fundamental to pattern formation.
  • Anomalous diffusion, particularly superdiffusion, challenges classical models.
  • Hyperbolic systems introduce inertial effects, impacting wave propagation.

Purpose of the Study:

  • To analyze wave instabilities in hyperbolic activator-inhibitor systems with anomalous superdiffusion.
  • To determine the influence of superdiffusion exponent, diffusion ratio, and inertial time on instability conditions.
  • To derive analytical conditions for diffusion-driven wave instabilities.

Main Methods:

  • Linear stability analysis of a hyperbolic activator-inhibitor model.
  • Modeling anomalous superdiffusion using the 2D Weyl fractional operator (α∈[1,2]).
  • Numerical simulations to validate analytical predictions.

Main Results:

  • The wave number of Turing instability increases with the superdiffusion exponent (α).
  • Wave instability requires the activator to diffuse faster than the inhibitor.
  • Critical wave number decreases with α and increases with inertial time (τ), with τ_max=3.6.

Conclusions:

  • Anomalous diffusion and inertial time are critical factors influencing wave instabilities.
  • The findings provide insights into pattern formation in complex media.
  • This study extends reaction-diffusion theory to include fractional calculus and inertial effects.