A Positivity-Preserving Nonstandard Finite Difference Scheme for Parabolic System with Cross-Diffusion Equations and Nonlocal Initial Conditions

Marc E. Songolo

Abstract


An important number of ecological phenomena can be modeled using nonlinear diffusion partial differential equations. This paper considers a system of cross-diffusion equations with nonlocal initial conditions. Such equations arise as steady-state equations in an age-structured predator-prey model with diffusion. We use the nonstandard finite difference method developed by Mickens. These types of schemes are made by the following two rules: first, renormalization of step size for the denominator function of representations of derivatives, and secondly, nonlocal representations of nonlinear terms. We obtained a scheme that preserves the positivity of solutions. Furthermore, this scheme is explicit and functional relationship is obtained between time, space, and age step sizes.

Keywords


Cross-diffusion equations; finite difference methods; nonlocal initial conditions; nonstandard finite difference schemes; positivity of solutions; predator-prey model.

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