Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach
D. del-Castillo-Negrete, B. Carreras, and V. Lynch (ORNL)
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 prevalence
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 the
presence of asymmetric Levy flights. Our approach consists of
replacing the Laplacian diffusion operator by a fractional diffusion
operator, whose fundamental solutionsare Levy 
-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, 
of the tail, where
the index of the Levy distribution, is the order of the fractional
derivative.
Reference:
"Front dynamics in reaction-diffusion systems with Levy flights: A fractional diffusion approach."
D. del-Castillo-Negrete, B. A. Carreras, and V. Lynch