title: 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 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.
authors.affiliation: Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8071, USA.
authors.forename: D
authors.initials: D
authors.lastname: del-Castillo-Negrete
authors.forename: B A
authors.initials: BA
authors.lastname: Carreras
authors.forename: V E
authors.initials: VE
authors.lastname: Lynch