Analysis and pattern formation scenarios in the superdiffusive system of predation described with Caputo operator

Abstract

The concept of a fractional derivative is introduced to the predator-prey system to describe the species anomalous superdiffusive process. To achieve this, a new class of predator-prey model with the Beddington-DeAngelis functional response is formulated in the sense of the Caputo fractional order operator. This work aims to give a mathematical basis for computational studies of a two-variable fractional reaction-diffusion system in one and two dimensions from biological and numerical perspectives. As a result, some details of the local and global dynamics of the reaction-diffusion system are provided by using the idea of the linear stability analysis and well-known dynamical systems theory to derive conditions on the parameters which can guarantee biologically meaningful equilibria also serve as a guide in ensuring the correct choice of parameters when numerically experimenting with the solutions of the full fractional reaction-diffusion model. The behavior of the new dynamical system is examined for both diffusive and non-diffusive systems at different instances of fractional order.