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

The use of reaction-diffusion models rests on the key assumption that the 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 reactive systems in the presence of this type of non-Gaussian diffusion. Here we study 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 of order $\alpha$, whose fundamental solutions are Levy $\alpha$-stable distributions that exhibit power law decay, $x^{-(1+\alpha )}$. Numerical simulations of the fractional Fisher-Kolmogorov equation and analytical arguments show that anomalous diffusion leads to the exponential acceleration of the front and a universal power law decay, $x^{-\alpha }$, of the front’s tail.