Pattern formation in superdiffusion predator–prey-like problems with integer- and noninteger-order derivatives

Abstract

This paper focuses on the modeling and application of fractional derivative to model the interactions between two different species in which the one named predator depends on the other called prey solely for survival. The interaction between predator and prey has been one of the most intriguing and interesting subjects in applied mathematical biology and ecology. In the models, the classical reaction–diffusion equations subject to the Neumann boundary conditions are formulated on a finite but large domain x ∈ [0, L] by replacing the second-order spatial derivatives with the fractional Laplacian operator of order 1 < α ≤ 2, which is classified as superdiffusion process. We examine the resulting coupled reaction–diffusion models for linear stability analysis and derive conditions under which the spatial patterns is evolved. In a view to understand our theoretical findings, the species spatial interactions is described in one and two dimensions. Through numerical experiments, we observe that a number of patterns can arise, including Turing spots, spiral-like structures, and seemingly complex spatiotemporal distributions.