Global Attractor of Coupled Difference Equations and Applications to Lotka-volterra Systems

This paper is concerned with a coupled system of nonlinear difference equations which is a discrete approximation of a class of nonlinear differential systems with time delays. The aim of the paper is to show the existence and uniqueness of a positive solution and to investigate the asymptotic behavior of the positive solution. Sufficient conditions are given to ensure that a unique positive equilibrium solution exists and is a global attractor of the difference system. Applications are given to three basic types of Lotka-Volterra systems with time delays where some easily verifiable conditions on the reaction rate constants are obtained for ensuring the global attraction of a positive equilibrium solution.