An Unconditionally Stable Numerical Scheme for a Competition System Involving Diffusion Terms

Abstract

A system of difference equations is proposed to approximate the solution of a system of partial differential equations that is used to model competing species with diffusion. The approximation method is a new semi-implicit finite difference scheme that is shown to mimic the dynamical properties of the true solution. In addition, it is proven that the scheme is uniquely solvable and unconditionally stable. The asymptotic behavior of the difference scheme is studied by constructing upper and lower solutions for the difference scheme. The convergence rate of the numerical solution to the true solution of the system is also given.