Title: Mathematical analysis and numerical simulation of a fractional reaction-diffusion system with Holling-type III functional response

Authors: Kolade M. Owolabi

Addresses: Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa; Department of Mathematical Sciences, Federal University of Technology, PMB 704, Akure, Ondo State, Nigeria

Abstract: In recent years, many investigators have questioned the use of convectional diffusion equation to model many physical or real life situations. As a result, fractional space derivatives have been proposed to model anomalous diffusion or related processes, where a particle plume spreads at inconsistent rate with the classical Brownian motion model. By replacing the second derivative in the classical diffusion model with fractional derivative, results to enhance a process known as superdiffusion. A high-dimensional predator-prey reaction-diffusion system with Holling-type III functional response, where the usual second-order derivatives give place to a fractional derivative of order α with 1 < α ≤ 2. Analysis of the main equation guides in the correct choice of parameter values. We established the condition for local and global stabilities. We also show that the system undergoes a Hopf bifurcation subject to a small perturbation of the steady-state solution. The complexity of fractional derivative at some instances of order α for the superdiffusive scenario is demonstrated with some numerical experiments in one, two and three dimensions. The effectiveness of the numerical method is demonstrated through numerical simulations to confirm the theoretical results.

Keywords: Fourier spectral method; Runge-Kutta method; fractional reaction-diffusion; Hopf bifurcation; oscillations; Holling-type III; predator-prey; stability analysis.

DOI: 10.1504/IJMMNO.2019.098788

International Journal of Mathematical Modelling and Numerical Optimisation, 2019 Vol.9 No.2, pp.196 - 219

Received: 08 Mar 2018
Accepted: 21 Apr 2018

Published online: 02 Apr 2019 *

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