Chaotic and spatiotemporal oscillations in fractional reaction-diffusion system
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. Furthermore, engineering applications of reaction-diffusion waves abound as exemplified in many combustion processes. Moreover, at the heart of the mathematical analysis of nonlinear dynamics, bifurcation theory of ordinary and partial differential equations are diffusive 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-species interactions. A more unswerving and trustworthy way to designate a community of species which interact in nonlinear fashion is expressed by system of equations. A two-variable reaction-diffusion system is an equation of the formwhere Du and Dv are the diffusion constants for species u(x, t) and v(x, t), respectively at position x and time t. The Laplacian operator Δ 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 meaning from classical-order differentiation and integration to fractional order integration and differentiation [28]. This concept of fractional calculus has been into practice for over many decades, and till date, see [6], [7], [8], [25], [27], [31], [37], [38]. The idea of time-fractional heat or diffusion equations are utilized to describe transport scenarios with long memory where the diffusion rate is incompatible with the standard Brownian motion. In fractional form, the above reaction-diffusion equation becomeswhere ∂β/∂tβ is the Caputo-time-fractional derivative of order 0 < β ≤ 1, which is expressed asfor β ∈ (0, 1]. It should be mentioned that one obtains the standard case in (1.1) with .
Numerical solution of (1.2) usually constitutes a huge challenge, particularly when extended to high dimensions to investigate the possibility of pattern formation in Turing-like models. Therefore, most of the computational work reported are either based on one-dimensional fractional ordinary or partial differential equations.
Numerical technique for one-dimensional fractional reaction-diffusion of one or two species system is relatively not difficult and has, in the last few decades, preoccupied the interest of many scholars. However, formulation of fractional-order PDEs in high dimensions is under-researched and still poorly understood. Nevertheless, there exists a handful of previous studies considering high-dimensional fractional reaction-diffusion models for pattern formation [3], [22], [25], [27]. Hence, this work is motivated by such paucity of research and focuses on examining the formulation of numerical scheme based on the novel alternating direction implicit finite difference method for the solution of self- and cross-diffusion predator-prey model.
The rest of this paper is organized thus: in Section 2, the numerical algorithms for the approximation in time and space are given. In Section 3, we introduce two mathematical models that are largely found in application areas of sciences and engineering. The qualitative analysis of these models are also analyzed. Numerical experiments which reveal the impact of the fractional order parameter and sensitivity of the models to various initial conditions are discussed in Section 4. Conclusion is finally drawn.
Section snippets
Numerical algorithms for the fractional reaction-diffusion equation
In this section, various numerical methods based on the finite difference schemes will be suggested in one and two dimensions to numerically approximate the Caputo time-fractional reaction-diffusion equations.
Dynamical systems and their linear stability analysis
A fundamental objective of theoretical ecology and biology is to provide insight into the interactions between different species or organisms living in a given habitat. Empirical evidence has indicated that the spatial scale and structure of a particular habitat can greatly influence their population interactions. Investigating complex chemical or population dynamics is as old as (population) ecology, and began with the seminal papers of Lotka and Volterra [12], [13], [35], with a simple model
Numerical experiments
In this section, we shall examine the behaviour of the two dynamical systems discussed in Section 3 via some numerical simulation experiments mainly in two dimensions. All numerical computations are carried out with the use of Matlab 2013a package, running on a 64 bit Windows 8 CORE i3 computer, with 4 GB RAM and 2.50 GHz CPU.
Conclusion
The aim of this paper is to design a reliable numerical method based on the alternating direction implicit finite difference scheme to solve fractional reaction-diffusion system of predator-prey dynamics in two dimensions. The work replaces the classical partial time derivative with the Caputo fractional operator. As a case study, a predator-prey model with self- and cross-diffusion is examined for pattern formation. We realized that the cross-diffusive case could easily give rise to some new
Authors’ contributions
All authors contributed equally and significantly in writing this article.
Credit author statement
The initial draft was conceived and written by Kolade M. Owolabi.
The revision, proofreading and addition of dynamical model was carried out by Berat Karaagac.
All authors approved the final version.
Declaration of Competing Interest
Authors declare that they have no conflict of interest.
Acknowledgment
The authors appreciate the constructive remarks and suggestions of the anonymous referees that helped to improve the paper.
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