Fractional reaction-diffusion equation for species growth and dispersal

Reaction-diffusion equations are commonly used to model the growth and spreading of biological species. The classical diffusion term implies a Gaussian dispersal kernel in the corresponding integro-difference equation, which is often unrealistic in practice. In this paper, we propose a fractional reaction-diffusion equation where the classical second derivative diffusion term is replaced by a fractional derivative of order less than two. The resulting model captures the faster spreading rates and power law invasion profiles observed in many applications, and is strongly motivated by a generalised central limit theorem for random movements with power-law probability tails. We then develop practical numerical methods to solve the fractional reactiondiffusion equation by time discretization and operator splitting, along with Partially supported by NSF grants DMS-0139927 and DMS-0417869 and the Marsden Fund administered by the Royal Society of New Zealand. Boris Baeumer Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9001, New Zealand Tel.: +643-479-7763 Fax: +643-479-8427 E-mail: [email protected] Mihály Kovács Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9001, New Zealand Tel.: +643-479-7889 Fax: +643-479-8427 E-mail: [email protected] Mark M. Meerschaert Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9001, New Zealand Tel.: +643-479-7889 Fax: +643-479-8427 E-mail: [email protected] 2 Baeumer, Kovács, Meerschaert some existing methods from the literature on anomalous super-diffusion. In the process, we establish the mathematical relationship between the discrete time integro-difference and continuous time reaction-diffusion analogues of the model, along with error bounds. Our general approach also applies to other alternative non-Gaussian dispersal kernels, and it identifies the analogous continuous time evolution equations for those models.