Dynamics of Pattern Formation Process in Fractional-Order Super-Diffusive Processes: a Computational Approach

Abstract

This paper explores the suitability of space fractional-order reaction-diffusion scenarios to model some emergent pattern formation in predator-prey models. Such fractional reaction-diffusion equations are obtained on the basis of a continuous-time random walk approach with spatial memory and local kinetic reaction. The classical space second-order derivative is changed by the fractional Laplacian case. We employ the Fourier spectral method to numerically approximate the fractional Laplacian and advance in time with the novel ETDRK4 method. In other to obtain guidelines on the correct choice of parameters when numerically simulating the full reaction-diffusion models, the local dynamics of the systems are considered. The biological wave scenarios of solutions are verified by presenting some numerical results in two dimensions to mimic some spatiotemporal dynamics such as spots, stripes and spiral patterns which has a lot of ecological implications.