Numerical simulations of multicomponent ecological models with adaptive methods

Theoretical Biology and Medical Modelling, Jan 2016

Background 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. Some of its emergence phenomena include spatiotemporal patterns, oscillating solutions, multiple steady states and spatial pattern formation. Methods Many 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. Results When 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. Conclusion The 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. AMS Subject Classification: Primary 70K05, 65M20, Secondary 70K20, 74H65.

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Numerical simulations of multicomponent ecological models with adaptive methods

Owolabi and Patidar Theoretical Biology and Medical Modelling Numerical simulations of multicomponent ecological models with adaptive methods Kolade M. Owolabi Kailash C. Patidar Background: 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. Some of its emergence phenomena include spatiotemporal patterns, oscillating solutions, multiple steady states and spatial pattern formation. Methods: Many 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. Results: When 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. Conclusion: The 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 wellbalanced 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. Coexistence; Competitive; Exponential time differencing; Predator-prey; Mutualism; Multistep methods; Reaction-diffusion systems; Stability analysis Background 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. The most popular and well-known predator-prey model is named after the two scientists, Alfred Lotka (1880– 1949) and Vito Volterra (1860–1940). Lotka and Volterra in their earlier research work apply the model to address the interacting population systems called predator–prey. Our numerical study in this paper is aimed at reflecting the types of interactions which we describe as predation (a process where one species of organisms called predator depends solely on the other known as the prey, for survival), competition (a situation whereby two or more different species of organisms struggle for the available resources; definitely, we expect the growth rate of each population to decrease) and lastly, the mutualism or symbiosis (organisms coexist without negatively affecting each other and hence, species growth rate is increased) [ 2, 11, 12, 23, 32, 33, 37, 39, 43, 48, 49 ]. 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 predator-prey system has received tremendous attraction over the years but has been represented mainly in terms of ordinary differential equations, which modelled the species distribution. The Dynamics of the Lotka-Volterra predator-prey model are quite interesting. However, this model is structurally unstable since a small perturbation of the equations often results to a drastic change in the dynamical system. For this reason, the presence of diffusion mechanism takes place though it changes the behavior of the whole model to coupled partial differential equations, called reaction-diffusion system. With the introduction of diffusion, the analysis of the whole system remain tactical in the literature [ 46, 47 ], and therefore, numerical approximations are quite often used to simulate these types of models. Predator-prey systems have been studied by many researchers in various forms (...truncated)


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Kolade Owolabi, Kailash Patidar. Numerical simulations of multicomponent ecological models with adaptive methods, Theoretical Biology and Medical Modelling, 2016, pp. 1, 13, DOI: 10.1186/s12976-016-0027-4