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Abstract
BackgroundThe study of dynamic relationship between a multi-species models has gained a huge amount of scientific interest over the years and will continue to maintain its dominance in both ecology and mathematical ecology in the years to come due to its practical relevance and universal existence. Some of its emergence phenomena include spatiotemporal patterns, oscillating solutions, multiple steady states and spatial pattern formation.MethodsMany time-dependent partial differential equations are found combining low-order nonlinear with higher-order linear terms. In attempt to obtain a reliable results of such problems, it is desirable to use higher-order methods in both space and time. Most computations heretofore are restricted to second order in time due to some difficulties introduced by the combination of stiffness and nonlinearity. Hence, the dynamics of a reaction-diffusion models considered in this paper permit the use of two classic mathematical ideas. As a result, we introduce higher order finite difference approximation for the spatial discretization, and advance the resulting system of ODE with a family of exponential time differencing schemes. We present the stability properties of these methods along with the extensive numerical simulations for a number of multi-species models.ResultsWhen the diffusivity is small many of the models considered in this paper are found to exhibit a form of localized spatiotemporal patterns. Such patterns are correctly captured in the local analysis of the model equations. An extended 2D results that are in agreement with Turing typical patterns such as stripes and spots, as well as irregular snakelike structures are presented. We finally show that the designed schemes are dynamically consistent.ConclusionThe dynamic complexities of some ecological models are studied by considering their linear stability analysis. Based on the choices of parameters in transforming the system into a dimensionless form, we were able to obtain a well-balanced system that is biologically meaningful. The accuracy and reliability of the schemes are justified via the computational results presented for each of the diffusive multi-species models.
Highlights
The study of dynamic relationship between a multi-species models has gained a huge amount of scientific interest over the years and will continue to maintain its dominance in both ecology and mathematical ecology in the years to come due to its practical relevance and universal existence
Exponential time differencing method We present a brief introduction to the derivation of exponential time differencing Runge-Kutta and multistep methods of Adams-type along with their stability regions
Stability analysis We investigate the stability of the fourth-order ETDADAMS4 (18) and Fourth-order exponential time differencing Runge-Kutta method (ETDRK4) (19) schemes by linearizing the nonlinear autonomous system [8, 19]
Summary
The study of dynamic relationship between a multi-species models has gained a huge amount of scientific interest over the years and will continue to maintain its dominance in both ecology and mathematical ecology in the years to come due to its practical relevance and universal existence. The study of reaction-diffusion problems in ecological context have gained a huge amount of scientific interest, due to their practical relevance and emergence of some interesting phenomena such as spatial patterns, oscillating solutions, phase planes, chaotic behaviours and multiple steady states to mention a few. A lot of research attention has been devoted to the study of population dynamics with regards to ecological interactions over the past few decades Such studies include the predator-prey system that describes the situation in which the existence of the species called the predator depends solely on the other species called the prey. The Dynamics of the Lotka-Volterra predator-prey model are quite interesting This model is structurally unstable since a small perturbation of the equations often results to a drastic change in the dynamical system. With the introduction of diffusion, the analysis of the whole system remain tactical in the literature [46, 47], and numerical approximations are quite often used to simulate these types of models
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