Global attractor of coupled difference equations and applications to Lotka-Volterra systems
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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. 1. 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). In this paper, 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 un = φn
n ∈ I1 ,
vn = ψn
(n = 1,2,...),
(1.1)
n ∈ I2 ,
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
Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:1 (2005) 57–79 DOI: 10.1155/ADE.2005.57
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Global attractor of difference equations
integers given by
I1 ≡ − s1 , −s1 + 1,...,0 ,
I2 ≡ − s2 , −s2 + 1,...,0 .
(1.2)
System (1.1) is a backward (or left-sided) difference approximation of the delay differential system du dv = f (1) u,v,uτ1 ,vτ2 , = f (2) u,v,uτ1 ,vτ2 (t > 0), dt dt u(t) = φ(t) − τ1 ≤ t ≤ 0 , v(t) = ψ(t) − τ2 ≤ t ≤ 0 ,
(1.3)
where uτ1 = u(t − τ1 ), vτ2 = v(t − τ2 ), and τ1 and τ2 are positive constants representing the time delays. In relation to the above differential system, the constant k in (1.1) plays the role of the time increment ∆t in the difference approximation and is chosen such that s1 ≡ τ1 /k and s2 ≡ τ2 /k are positive integers. Our consideration of the difference system (1.1) is motivated by some Lotka-Volterra models in population dynamics where the effect of time delays in the opposing species is taken into consideration. The equations for the difference approximations of these model problems, referred to as cooperative, competition, and prey-predator, respectively, involve three distinct quasimonotone reaction functions, and are given as
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