Front Dynamics in Reaction-Diffusion Systems with Levy Flights: A Fractional Diffusion Approach

Abstract

The use of reaction-diffusion models rests on the key assumption that the underlying diffusive process is Gaussian. However, a growing number of studies have pointed out the presence of anomalous diffusion, and there is a need to understand the dynamics of reactive systems in the presence of this type of non-Gaussian diffusion. Here we present a study of front dynamics in reaction-diffusion systems where anomalous diffusion is due to asymmetric Levy flights. Our approach consists of replacing the Laplacian diffusion operator by a fractional diffusion operator, whose fundamental solutions are Levy α-stable distributions. Numerical simulation of the fractional Fisher-Kolmogorov equation and analytical arguments show that anomalous diffusion leads to the exponential acceleration of fronts and a universal power law decay, x −α , of the tail, where α, the index of the Levy distribution, is the order of the fractional derivative.

Fronts in anomalous diffusion–reaction systems

“…is an asymmetry parameter [39,40]. The propagation of fronts in a bistable superdiffusive system governed by the asymmetric model (2.6) with the reaction function (2.3) has been considered in recent studies [34,41].…”

“…[49]. del Castillo-Negrete et al [39] investigated the phenomenon of the front acceleration in the framework of a fractional superdiffusion-reaction equation with a fully asymmetric superdiffusion operator, (2.12) which corresponds to equation (2.6) with θ = −1 and a rescaled spatial variable, for a front between the stable phase, u(−∞) = 1, and an unstable phase, u(∞) = 0. del Castillo-Negrete et al considered the evolution of a small disturbance governed by a linearized equation…”

Mesoscopic description of reactions for anomalous diffusion: a case study

“…[15,16], where front propagation was discussed for symmetric and asymmetric Lévy flights, respectively, see also Ref. [17] and Ref.…”

Fractional Reaction-Diffusion Equations

“…Del-Castillo-Negrete, Carreras, and Lynch (2002) studied the front propagation and segregation in a system of reaction-diffusion equations with cross-diffusion. Recently, del-Castillo-Negrete, Carreras, and Lynch (2003) discussed the dynamics in reaction-diffusion systems with non-Gaussian diffusion caused by asymmetric Lévy flights and solved the following model…”