Chaos, Solitons and Fractals 141 (2020) 110302 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Chaotic and spatiotemporal oscillations in fractional reaction-diffusion system Kolade M. Owolabi a,b,c,∗ , Berat Karaagac a,d a Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam b Department of Mathematical Sciences, Federal University of Technology, Akure, PMB 704, Ondo State, Nigeria c Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa d Department of Mathematics Education, Faculty of Education, Adıyaman University, Adıyaman, 2230, Turkey a r t i c l e i n f o Article history: Received 8 April 2020 Revised 25 August 2020 Accepted 14 September 2020 MSC: 34A34 35A05 35K57 65L05 65M06 93C10 Keywords: Activator-inhibitor model Caputo fractional derivative Predator-prey system Chaotic and spatiotemporal patterns Numerical simulations a b s t r a c t This paper focuses on the design and analysis of an efficient numerical method based on the novel im- plicit finite difference scheme for the solution of the dynamics of reaction-diffusion models. The work re- places the integer first order derivative in time with the Caputo fractional derivative operator. The dynam- ics of activator-inhibitor as encountered in chemistry, physics and engineering processes, and predator- prey models are two cases addresses in this study. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction-diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve cross-diffusion problem in two-dimensions. In the experimental results, a number of spatiotemporal and chaotic patterns that are related to Turing pattern are observed. It was discovered in the simulation experiments that the species predator-prey model distribute in almost same fashion, while that of the activator-inhibitor dynamics behaved differently regardless of the value of fractional order chosen. © 2020 Elsevier Ltd. All rights reserved. 1. Introduction The theory of diffusive waves in reaction-diffusion models is dated to 1930s, and it has provided explanation and modelling for many biological, chemical and other physical scenarios. Further- more, engineering applications of reaction-diffusion waves abound as exemplified in many combustion processes. Moreover, at the heart of the mathematical analysis of nonlinear dynamics, bifur- cation theory of ordinary and partial differential equations are dif- fusive waves. Over time, reaction-diffusion waves have produced many interesting nonlinear dynamics in mathematical biology and ecology, with pattern formation process in diffusive model being a most fascinating feature. The dynamics of emerging patterns is generally determined by a series of bifurcations. In reality and natural habitat, the population of different species like predator and prey coexist and are involved in different inter- ∗ Corresponding author at: Department of Mathematical Sciences, Federal Univer- sity of Technology, PMB 704, Akure, Ondo State, Nigeria. E-mail address: koladematthewowolabi@tdtu.edu.vn (K.M. Owolabi). species interactions. A more unswerving and trustworthy way to designate a community of species which interact in nonlinear fash- ion is expressed by system of equations. A two-variable reaction- diffusion system is an equation of the form ∂ u(x, t ) ∂ t = D u u + f (u, v) ∂ v(x, t ) ∂ t = D v v + g(u, v) (1.1) where D u and D v are the diffusion constants for species u(x, t) and v(x, t), respectively at position x and time t. The Laplacian opera- tor  is expressed in terms of the second order partial differential operator, which can easily be approximated by using the spectral methods or finite difference method. The totality of reactions are represented by the local kinetic functions f(u, v) and g(u, v). Nowadays, to adequately represent most physical phenomena, the idea of fractional derivatives which is the extension of mean- ing from classical-order differentiation and integration to fractional order integration and differentiation [28]. This concept of frac- tional calculus has been into practice for over many decades, and till date, see [6–8,25,27,31,37,38]. The idea of time-fractional heat https://doi.org/10.1016/j.chaos.2020.110302 0960-0779/© 2020 Elsevier Ltd. All rights reserved.