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Numerical Analysis and Applications

2016 year, number 3

Mathematical study of two-variable systems with adaptive numerical methods

Kolade M. Owolabi1,2
1University of the Western Cape Private, Bag X17, Bellville 7535, South Africa
2Federal University of Technology, Akure PMB 704, Akure, Ondo State, Nigeria
Keywords: модель хищник-жертва, ЭВР-методы, нелинейный, образование структур, реакция-диффузия, устойчивость, зависящие от времени дифференциальные уравнения в частных производных (ДУЧП), неустойчивость по Тьюрингу, predator-prey model, ETD methods, nonlinear, pattern formation, reaction-diffusion, stability, time-dependent PDE, Turing instability


In this paper, we consider reaction-diffusion systems arising from two-component predator-prey models with Smith growth functional response. The mathematical approach used here is twofold, since the time-dependent partial differential equations consist of both linear and nonlinear terms. We discretize the stiff or moderately stiff term with a fourth-order difference operator, advance the resulting nonlinear system of ordinary differential equations with a family of two competing exponential time differencing (ETD) schemes, and analyze them for stability. A numerical comparison of these two methods for solving various predator-prey population models with functional responses is also presented. Numerical results show that the techniques require less computational work. Also in the numerical results, some emerging spatial patterns are unveiled.