Global attractor of coupled difference equations and applications to Lotka-Volterra systems

Cv Pao

doi:10.1155/ade.2005.57

Abstract

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.

Highlights

Difference equations appear as discrete phenomena in nature as well as discrete analogues of differential equations which model various phenomena in ecology, biology, physics, chemistry, economics, and engineering
There are large amounts of works in the literature that are devoted to various qualitative properties of solutions of difference equations, such as existence-uniqueness of positive solutions, asymptotic behavior of solutions, stability and attractor of equilibrium solutions, and oscillation or nonoscillation of solutions
We investigate some of the above qualitative properties of solutions for a coupled system of nonlinear difference equations in the form un = un−1 + k f (1) un, vn, un−s1, vn−s2, vn = vn−1 + k f (2) un, vn, un−s1, vn−s2 (n = 1, 2, . . . ), un = φn n ∈ I1, vn = ψn n ∈ I2, (1.1)

Summary

Introduction

Difference equations appear as discrete phenomena in nature as well as discrete analogues of differential equations which model various phenomena in ecology, biology, physics, chemistry, economics, and engineering. There are large amounts of works in the literature that are devoted to various qualitative properties of solutions of difference equations, such as existence-uniqueness of positive solutions, asymptotic behavior of solutions, stability and attractor of equilibrium solutions, and oscillation or nonoscillation of solutions (cf [1, 4, 11, 13] and the references therein). We investigate some of the above qualitative properties of solutions for a coupled system of nonlinear difference equations in the form un = un−1 + k f (1) un, vn, un−s1 , vn−s2 , vn = vn−1 + k f (2) un, vn, un−s1 , vn−s2 Where f (1) and f (2) are, in general, nonlinear functions of their respective arguments, k is a positive constant, s1 and s2 are positive integers, and I1 and I2 are subsets of nonpositive

58 Global attractor of difference equations integers given by

Existence and uniqueness of positive solution

Comparison theorems

Asymptotic behavior

Applications to Lotka-Volterra systems

Methods