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Phys. Rev. E 74, 066118 (2006) [10 pages]

Kinetic equations for reaction-subdiffusion systems: Derivation and stability analysis

A. Yadav and Werner Horsthemke
Department of Chemistry, Southern Methodist University, Dallas, Texas 75275-0314, USA
Received 21 August 2006; published 28 December 2006

We derive general kinetic equations for reacting and subdiffusing entities based on a nonlinear continuous time random walk formalism proposed by Vlad and Ross [Phys. Rev. E 66, 061908 (2002)]. Reaction and diffusion processes are separable in a typical reaction-diffusion system, and their combined influence on the evolution of the density of a species is a simple sum. Our derivation shows that this is no longer true for subdiffusive entities undergoing reactions. The strong memory effects in the transport process, i.e., the non-Markovian nature of subdiffusion, results in a nontrivial combination of reactions and spatial dispersal, which we discuss in detail. We carry out a linear stability analysis of the derived reaction-subdiffusion system to understand the effects of memory on pattern formation. We find that the Turing instability persists in the subdiffusive system. However, the memory modifies the Turing threshold and the characteristics of the band of unstable modes close to this threshold.

©2006 The American Physical Society

URL: http://link.aps.org/doi/10.1103/PhysRevE.74.066118
DOI: 10.1103/PhysRevE.74.066118
PACS: 82.40.Ck; 05.40.Fb; 87.23.Cc; 05.60.-k
  • 82.40.Ck
    Pattern formation in chemical reactions with diffusion, flow and heat transfer
  • 05.40.Fb
    Random walks and Levy flights
  • 87.23.Cc
    Population dynamics and ecological pattern formation
  • 05.60.-k
    Transport processes
  • YEAR: 2006
KEYWORDS: reaction-diffusion systems, random processes, nonlinear dynamical systems, pattern formation, transport processes

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