Chaos, Solitons and Fractals 127 (2019) 146–157 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Modeling and simulation of nonlinear dynamical system in the frame of nonlocal and non-singular derivatives Kolade M. Owolabi a,b,∗ , Edson Pindza c,d a Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa b Department of Mathematical Sciences, Federal University of Technology, Akure, PMB 704, Ondo State, Nigeria c Achieversklub School of Cryptocurrency and Entrepreneurship, 1 Sturdee Avenue, Rosebank 2196, South Africa d Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 002, South Africa a r t i c l e i n f o Article history: Received 4 June 2019 Revised 19 June 2019 Accepted 26 June 2019 MSC: 34A34 35A05 35K57 65L05 65M06 93C10 Fractional reaction-diffusion Holling-type III response Nonlocal and nonsingular kernels Stability analysis Numerical simulation a b s t r a c t This paper considers mathematical analysis and numerical treatment for fractional reaction-diffusion sys- tem. In the model, the first-order time derivatives are modelled with the fractional cases of both the Atangana-Baleanu and Caputo-Fabrizio derivatives whose formulations are based on the notable Mittag- Leffler kernel. The main system is examined for stability to ensure the right choice of parameters when numerically simulating the full model. The novel Adam-Bashforth numerical scheme is employed for the approximation of these operators. Applicability and suitability of the techniques introduced in this work is justified via the evolution of the species in one and two dimensions. The results obtained show that modelling with fractional derivative can give rise to some Turing patterns. © 2019 Elsevier Ltd. All rights reserved. 1. Introduction Over the years, the dynamical study of two or more interact- ing species model has received a lot of scientific attention in ar- eas of applied sciences and technology. Mathematical systems of biological processes are often formulated in terms of the differ- ential or difference equations. The reason for this special class of model is that biological or ecological processes are dynamical in nature, they often change with respect to time and space or some- times, stage of development. As a result, the steps involved in the modeling processes require a deep understanding of the underly- ing mathematical theory and tools for differential equations. Of particular interest here is the dynamical study of predator- prey model which has recurring interest in areas of combus- tion theory, population ecology and chemical kinetics [23,25]. The predator-prey concept is widely applied to describe the spatial ∗ Corresponding author at: Mathematics and Applied Mathematics, Mathematics Department, FUTA, Akure, Nigeria. E-mail addresses: mkowolax@yahoo.com, kmowolabi@futa.edu.ng (K.M. Owolabi). relationship between two different species. Its application has recorded a lot of success especially in the formation of tempo- ral and spatial patterns such as spots, stripes and mitotic-spots or mixture of stripes and spot in accordance to Turing systems as seen on most wildlife animals like tiger, cheetah, snakes and zebra among several others. Most of these dynamical models that display a range of interesting phenomena exist in the form of time-dependent reaction-diffusion systems. Recently, a lot of researchers have sug- gested that modeling a real-life phenomena with fractional-order derivative gives a better representation and accurate results when compared to the classical case [2,3,13,14,16,20–22]. It provides more realistic interpretation of physical phenomena. Also, many dynamics in interdisciplinary fields of research have been mod- eled by the concept of fractional differential equations, examples include the diffusion waves, traffic flow, earthquake modeling, vis- coelastic materials, gyro system, hydrology and groundwater pro- cesses, pattern formation in biological systems, chaos in finance, blow-up phenomena and wave propagation in nonlocal media [2,12,24,29,30,35,36,40,44]. In addition, differential equations with https://doi.org/10.1016/j.chaos.2019.06.037 0960-0779/© 2019 Elsevier Ltd. All rights reserved.