Eur. Phys. J. Plus (2018) 133: 15
https://doi.org/10.1140/epjp/i2018-11863-9

Regular Article

Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative

¹ Institute for Groundwater Studies, Faculty of Natural, Agricultural Sciences University of the Free State, 9300, Bloemfontein, South Africa
² Department of Mathematical Sciences, Federal University of Technology, PMB 704, Ondo State, Nigeria

^* e-mail: mkowolax@yahoo.com

Received: 20 October 2017
Accepted: 2 January 2018
Published online: 18 January 2018

Abstract

In this paper, we model an ecological system consisting of a predator and two preys with the newly derived two-step fractional Adams-Bashforth method via the Atangana-Baleanu derivative in the Caputo sense. We analyze the dynamical system for correct choice of parameter values that are biologically meaningful. The local analysis of the main model is based on the application of qualitative theory for ordinary differential equations. By using the fixed point theorem idea, we establish the existence and uniqueness of the solutions. Convergence results of the new scheme are verified in both space and time. Dynamical wave phenomena of solutions are verified via some numerical results obtained for different values of the fractional index, which have some interesting ecological implications.