Nonstandard finite difference methods for a convective predator-prey pursuit and evasion model

Abstract

One standard and two nonstandard finite difference methods are used to solve a convective predator-prey model consisting of cross-diffusion terms for which no exact solution is known. The model involves a system of strongly coupled nonlinear parabolic partial differential equations. Through some numerical experiments, we observe that reasonable numerical solutions are obtained when the temporal step size satisfies k≤0.0005 with spatial step size h=0.1. We construct a nonstandard method using ideas from Anguelov et al. and the scheme is denoted by NSFD1. We obtain four conditions for NSFD1 to replicate the positivity property of the continuous model, but these conditions consist of many parameters. Therefore, an unconditionally positivity-preserving scheme is considered which is denoted as NSFD2. However, we found that NSFD2 is in general not consistent. We obtain conditions under which NSFD2 is both consistent and bounded and some numerical results are presented when these conditions are satisfied.