Global attractor of a coupled finite difference reaction diffusion system with delays

In the study of asymptotic behavior of solutions for reaction diffusion systems, an important concern is to determine whether and when the system has a global attractor which attracts all positive time-dependent solutions. The aim of this paper is to investigate the global attraction problem for a finite difference system which is a discrete approximation of a coupled system of two reaction diffusion equations with time delays. Sufficient conditions are obtained to ensure the existence and global attraction of a positive solution of the corresponding steady-state system. Applications are given to three types of Lotka-Volterra reaction diffusion models, where time-delays may appear in the opposing species.