Numerical patterns in reaction–diffusion system with the Caputo and Atangana–Baleanu fractional derivatives

Abstract

In this paper, we numerically study a nonlinear time-fractional reaction-diffusion equation involving the Caputo and Atangana–Baleanu fractional derivatives of order α ∈ (0, 1). A novel algorithm known as the Laplace Adams–Bashforth method is formulated for the approximation of these derivatives. In the simulation framework, a tri-tropic food chain system is considered in which the classical time-derivatives are replaced with non-integer order derivatives. Mathematical analysis of the main system is examined for both stability and Hopf-bifurcations to occur. Numerical simulation results show the existence of chaotic behaviours and spatiotemporal oscillations as well as the emergence of some Turing patterns (such as, spots and stripes) in two-dimensional space.