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Diffusive instabilities in a hyperbolic activator-inhibitor system with superdiffusion

Alain Mvogo1,2,*, Jorge E. Macías-Díaz3,†, and Timoléon Crépin Kofané4,‡

  • 1Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Cameroon
  • 2Abdus Salam International Center For Theoretical Physics, P.O. Box 586, Strada Costiera 11, 34014 Trieste, Italy
  • 3Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes, Aguascalientes 20131, Mexico
  • 4Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Cameroon

  • *mvogal_2009@yahoo.fr
  • †jemacias@correo.uaa.mx
  • ‡tckofane@yahoo.com
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Phys. Rev. E 97, 032129 – Published 23 March, 2018

DOI: https://doi.org/10.1103/PhysRevE.97.032129

Abstract

We investigate analytically and numerically the conditions for wave instabilities in a hyperbolic activator-inhibitor system with species undergoing anomalous superdiffusion. In the present work, anomalous superdiffusion is modeled using the two-dimensional Weyl fractional operator, with derivative orders α∈ [1,2]. We perform a linear stability analysis and derive the conditions for diffusion-driven wave instabilities. Emphasis is placed on the effect of the superdiffusion exponent α, the diffusion ratio d, and the inertial time τ. As the superdiffusive exponent increases, so does the wave number of the Turing instability. Opposite to the requirement for Turing instability, the activator needs to diffuse sufficiently faster than the inhibitor in order for the wave instability to occur. The critical wave number for wave instability decreases with the superdiffusive exponent and increases with the inertial time. The maximum value of the inertial time for a wave instability to occur in the system is τmax=3.6. As one of the main results of this work, we conclude that both anomalous diffusion and inertial time influence strongly the conditions for wave instabilities in hyperbolic fractional reaction-diffusion systems. Some numerical simulations are conducted as evidence of the analytical predictions derived in this work.

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Phys. Rev. E 97, 032129– Published 23 March, 2018
  • Received 2 January 2018
  • Revised 1 March 2018
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