GLOBAL ATTRACTOR OF COUPLED DIFFERENCE EQUATIONS AND APPLICATIONS TO LOTKA-VOLTERRA SYSTEMS C. V.

GLOBAL ATTRACTOR OF COUPLED DIFFERENCE EQUATIONS AND APPLICATIONS TO LOTKA-VOLTERRA SYSTEMS C. V. PAO Received 22 April 2004 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 in-vestigate 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 Differenceequationsappearasdiscretephenomenainnatureaswellasdiscreteanalogues of differential equations which model various phenomena in ecology, biology, physics, chemistry,economics,andengineering.Therearelargeamountsofworksintheliterature thataredevotedtovariousqualitativepropertiesofsolutionsofdifferenceequations,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 +kf (1) un,vn,un−s1,vn−s2, vn =vn−1 +kf (2) un,vn,un−s1,vn−s2 (n =1,2,...), (1.1) un =φn n ∈I1 , vn =ψn 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 58 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 differen-tial system dt = f (1) u,v,uτ1,vτ2, dt = f (2) u,v,uτ1,vτ2 (t > 0), (1.3) u(t) =φ(t) −τ1 ≤t ≤0 , v(t) =ψ(t) −τ2 ≤t ≤0 , 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 follows (cf. [7, 11, 12, 15, 20]): (a) the cooperative system: un =un−1 +kα(1)un 1−a(1)un +b(1)vn +c(1)vn−s2 vn =vn−1 +kα(2)vn 1+a(2)un −b(2)vn +c(2)un−s (n =1,2,...), (1.4) un =φn n ∈I1, vn =ψn n ∈I2; (b) the competition system: un =un−1 +kα(1) 1−a(1)un −b(1)vn −c(1)vn−s2 vn =vn−1 +kα(2) 1−a(2)un −b(2)vn −c(2)un−s1 (n =1,2,...), (1.5) un =φn n ∈I1 , vn =ψn n ∈I2 ; (c) the prey-predator system: un =un−1 +kα(1) 1−a(1)un −b(1)vn −c(1)vn−s2 vn =vn−1 +kα(2) 1+a(2)un −b(2)vn +c(2)un−s1 (n =1,2,...), (1.6) un =φn n ∈I1 , vn =ψn n ∈I2 . In the systems (1.4), (1.5), and (1.6), un and vn represent the densities of the two popula-tion species at time nk(≡n∆t), k is a small time increment, and for each l =1,2,α(l),a(l), b(l), and c(l) are positive constants representing the various reaction rates. There are huge amounts of works in the literature that dealt with the asymptotic be-havior of solutions for differential and difference systems with time delays, and much of C. V. Pao 59 the discussions in the earlier work are devoted to differential systems, including various Lotka-Volterra-type equations (cf. [2, 3, 5, 7, 8, 12, 15, 19, 20]). Later development leads to various forms of difference equations, and many of them are discrete analogues of dif-ferential equations (cf. [2, 3, 4, 5, 6, 8, 9, 10, 11, 19]). In recent years, attention has also beengiventofinite-differenceequationswhicharediscreteapproximationsofdifferential equations with the effect of diffusion (cf. [14, 15, 16, 17, 18]). In this paper, we consider the coupled difference system (1.1) for a general class of reaction functions (f (1), f (2)), and our aim is to show the existence and uniqueness of a global positive solution and the asymptotic behavior of the solution with particular emphasison the global attraction of a positive equilibrium solution. The results for the general system are then applied to each of the three Lotka-Volterra models in (1.4)–(1.6) where some easily verifiable conditions on the rate constants a(l), b(l), and c(l), l = 1,2, are obtained so that a unique positive equilibrium solution exists and is a global attractor of the system. The plan of the paper is as follows. In Section 2, we show the existence and uniqueness of a positive global solution to the general system (1.1) for arbitrary Lipschitz continu-ous functions (f (1), f (2)). Section 3 is concerned with some comparison theorems among solutions of (1.1) for three different types of quasimonotone functions. The asymptotic behavior of the solution is treated in Section 4 where sufficient conditions are obtained for ensuring the global attraction of a positive equilibrium solution. This global attrac-tion property is then applied in Section 5 to the Lotka-Volterra models in (1.4), (1.5), and (1.6) which correspond to the three types of quasimonotone functions in the general system. 2. Existence and uniqueness of positive solution Before discussing the asymptotic behavior of the solution of (1.1) we show the existence and uniqueness of a positive solution under the following basic hypothesis on the func-tion (f (1), f (2)) ≡(f (1)(u,v,us,vs), f (2)(u,v,us,vs)). (H1) (i) The function (f (1), f (2)) satisfies the local Lipschitz condition f (l) u,v,u ,v − f (l) u0,v0,u0,v0 ≤K(l) |u−u0|+|v−v0|+us −u0+vs −v0 (2.1) for u,v,us,vs , u0,v0,u0,v0 ∈S×S, (l =1,2). (ii) There exist positive constants (M(1),M(2)), (δ(1),δ(2)) with (M(1),M(2)) ≥ (δ(1),δ(2)) such that for all (us,vs) ∈S, f (1) M(1),v,us,vs≤0 ≤ f (1) δ(1),v,us,vs f (2) u,M(2),us,vs ≤0 ≤ f (2) u,δ(2),us,vs when δ(2) ≤v ≤M(2), when δ(1) ≤u ≤M(1). (2.2) In the above hypothesis, S is given by S≡(u,v) ∈R2; δ(1),δ(2)≤(u,v) ≤ M(1),M(2) . (2.3) 60 Global attractor of difference equations To ensure the uniqueness of the solution, we assume that the time increment k satisfies the condition k K(1) +K(2)< 1, (2.4) where K(1) and K(2) are the Lipschitz constants in (2.1). Theorem 2.1. Let hypothesis (H1) hold. Then system (1.1) has at least one global solution (un,vn) in S. If, in addition, condition (2.4) is satisfied, then the solution (un,vn) is unique in S. Proof. GivenanyWn ≡(wn,zn) ∈S,weletUn ≡(un,vn)bethesolutionoftheuncoupled initial value problem 1+kK(1)un =un−1 +kK(1)wn + f (1) wn,zn,un−s1,vn−s2, 1+kK(2) vn =vn−1 +k K(2)zn + f (2) wn,zn,un−s1,vn−s2 (n =1,2,...), (2.5) un =φn n ∈I1 , vn =ψn n ∈I2 , where K(1) and K(2) are the Lipschitz constants in (2.1). Define a solution operator P : S→R2 by PWn ≡ P(1)Wn,P(2)Wn≡ un,vn Wn ∈S. (2.6) Then system (1.1) may be expressed as Un =PUn, Un = un,vn (n =1,2,...). (2.7) To prove the existence of a global solution to (1.1) it suffices to show that P has a fixed point in S for every n. It is clear from hypothesis (H1) that P is a continuous map on S which is a closed bounded convex subset of R2. We show that P maps S into itself by a marching process. Given any Wn ≡(wn,zn) ∈S, relation (2.6) and conditions (2.1), (2.2) imply that 1+kK(1) M(1) −P(1)Wn = 1+kK(1)M(1) −un−1 +k K(1)wn + f (1) wn,zn,un−s1,vn−s2 ≥ M(1) −un−1+kK(1) M(1) −wn+ f (1) M(1),zn,un−s1,vn−s2 − f (1) wn,zn,un−s1,vn−s2≥M(1) −un−1, 1+kK(2) M(2) −P(2)Wn (2.8) = 1+kK(2)M(2) −vn−1 +k K(2)zn + f (2) wn,zn,un−s1,vn−s2 ≥ M(2) −vn−1+kK(2) M(2) −zn+ f (2) wn,M(2),un−s1,vn−s2 − f (2) wn,zn,un−s1,vn−s2 ≥M(2) −vn−1 (n =1,2,...), C. V. Pao 61 whenever (un−s1,vn−s2) ∈S. This leads to the relation (1) (1) M(1) −un−1 1+kK(1) M(2) −P(2)Wn ≥ 1+kK(2) 1 (n =1,2,...). A similar argument using the second inequalities in (2.2) gives (1) (1) un−1 −δ(1) 1+kK(1) P(2)Wn −δ(2) ≥ 1+kK((2) (n =1,2,...), (2.9) (2.10) whenever (un−s1,vn−s2) ∈ S. Consider the case n = 1. Since (u1−s1,v1−s2) = (φ1−s1,ψ1−s2) and (u0,v0) = (φ0,ψ0) are in S, relations (2.9), (2.10) imply that (δ(1),δ(2)) ≤ (P(1)W1, P(2)W1) ≤(M(1),M(2)).ByBrower’sfixedpointtheorem,P≡(P(1),P(2))hasafixedpoint U1 ≡(u1,v1) in S. This shows that (u1,v1) is a solution of (1.1) for n =1, and (u1,v1) and (u2−s1,v2−s2 ) are in S. Using this property in (2.9), (2.10) for n = 2, the same argument shows that P has a fixed point U2 ≡ (u2,v2) in S, and (u2,v2) is a solution of (1.1) for n = 2 and (u3−s1,v3−s2 ) ∈ S. A continuation of the above argument shows that P has a fixed point Un ≡(un,vn) in S for every n, and (un,vn) is a global solution of (1.1) in S. To show the uniquenessof the solution,we considerany two solutions(un,vn), (u0 ,v0) in S and let (wn,zn) =(un −un,vn −vn). By (1.1), wn =wn−1 +kf (1) un,vn,un−s1,vn−s2− f (1) u0 ,v0,u0−s1,v0−s2, zn =zn−1 +k f (2) un,vn,un−s1,vn−s2 − f (2) u0 ,v0,u0−s1,v0−s2 (n =1,2,...), (2.11) wn =0 n ∈I1 , zn =0 n ∈I2 . The above relation and condition (2.1) imply that wn≤wn−1+kK(1) wn+zn+wn−s1+zn−s2, zn≤zn−1+kK(2) wn+zn+wn−s1+zn−s2 . Addition of the above inequalities leads to (2.12) wn+zn≤wn−1+zn−1 +k K(1)+K(2) wn+zn+wn−s1+zn−s2 (n =1,2,...). (2.13) Since wn =zn =0 for n =0,−1,−2,..., the above inequality for n =1 yields w1+z1≤k K(1) +K(2) w1+z1. (2.14) In view of condition (2.4), this is possible only when |w1|=|z1|=0. Using w1 =z1 =0 in (2.13) for n =2 yields w2+z2≤k K(1) +K(2) w2+z2. (2.15) ... - tailieumienphi.vn