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

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 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, e conomics, 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 u n = u n−1 + kf (1)  u n ,v n ,u n−s 1 ,v n−s 2  , v n = v n−1 + kf (2)  u n ,v n ,u n−s 1 ,v n−s 2  (n = 1,2, ), u n = φ n  n ∈ I 1  , v n = ψ n  n ∈ I 2  , (1.1) where f (1) and f (2) are, in general, nonlinear functions of their respective arguments, k is a positive constant, s 1 and s 2 are positive integers, and I 1 and I 2 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 I 1 ≡  − s 1 ,−s 1 +1, ,0  , I 2 ≡  − s 2 ,−s 2 +1, ,0  . (1.2) System (1.1) is a backward (or left-sided) difference approximation of the delay differen- tial system du dt = f (1)  u,v,u τ 1 ,v τ 2  , dv dt = f (2)  u,v,u τ 1 ,v τ 2  (t>0), u(t) = φ(t)  − τ 1 ≤ t ≤ 0  , v(t) = ψ(t)  − τ 2 ≤ t ≤ 0  , (1.3) where u τ 1 = u(t − τ 1 ), v τ 2 = v(t − τ 2 ), and τ 1 and τ 2 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 s 1 ≡ τ 1 /k and s 2 ≡ τ 2 /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: u n = u n−1 + kα (1) u n  1 − a (1) u n + b (1) v n + c (1) v n−s 2  v n = v n−1 + kα (2) v n  1+a (2) u n − b (2) v n + c (2) u n−s 1  (n = 1,2, ), u n = φ n  n ∈ I 1  , v n = ψ n  n ∈ I 2  ; (1.4) (b) the competition system: u n = u n−1 + kα (1)  1 − a (1) u n − b (1) v n − c (1) v n−s 2  v n = v n−1 + kα (2)  1 − a (2) u n − b (2) v n − c (2) u n−s 1  (n = 1,2, ), u n = φ n  n ∈ I 1  , v n = ψ n  n ∈ I 2  ; (1.5) (c) the prey-predator system: u n = u n−1 + kα (1)  1 − a (1) u n − b (1) v n − c (1) v n−s 2  v n = v n−1 + kα (2)  1+a (2) u n − b (2) v n + c (2) u n−s 1  (n = 1,2, ), u n = φ n  n ∈ I 1  , v n = ψ n  n ∈ I 2  . (1.6) In the systems (1.4), (1.5), and (1.6), u n and v n 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) ,andc (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 been given to finite-difference equations which are discrete approximations of differential 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 emphasis on 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) ,andc (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 t he 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)forthreedifferent types of quasimonotone functions. The asymptotic behavior of the solution is treated in Section 4 where sufficient conditions are obtained for ensuring the g lobal 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,u s ,v s ), f (2) (u,v,u s ,v s )). (H 1 ) (i) The function ( f (1) , f (2) ) satisfies the local Lipschitz condition   f (l)  u,v,u s ,v s  − f (l)  u  ,v  ,u  s ,v  s    ≤ K (l)  |u − u  | + |v − v  | +   u s − u  s   +   v s − v  s    for  u,v,u s ,v s  ,  u  ,v  ,u  s ,v  s  ∈ ᏿ × ᏿ ,(l = 1,2). (2.1) (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 (u s ,v s ) ∈ ᏿, f (1)  M (1) ,v,u s ,v s  ≤ 0 ≤ f (1)  δ (1) ,v,u s ,v s  when δ (2) ≤ v ≤ M (2) , f (2)  u,M (2) ,u s ,v s  ≤ 0 ≤ f (2)  u,δ (2) ,u s ,v s  when δ (1) ≤ u ≤ M (1) . (2.2) In the above hypothesis, ᏿ is given by ᏿ ≡  (u,v) ∈ R 2 ;  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 (H 1 ) hold. Then sy stem (1.1) has at least one global solution (u n ,v n ) in ᏿. If, in addition, condition (2.4) is satis fied, then the solution (u n ,v n ) is unique in ᏿. Proof. Given any W n ≡ (w n ,z n ) ∈ ᏿,weletU n ≡ (u n ,v n ) be the solution of the uncoupled initial value problem  1+kK (1)  u n = u n−1 + k  K (1) w n + f (1)  w n ,z n ,u n−s 1 ,v n−s 2  ,  1+kK (2)  v n = v n−1 + k  K (2) z n + f (2)  w n ,z n ,u n−s 1 ,v n−s 2  (n = 1,2, ), u n = φ n  n ∈ I 1  , v n = ψ n  n ∈ I 2  , (2.5) where K (1) and K (2) are the Lipschitz constants in (2.1). Define a solution operator ᏼ : ᏿ → R 2 by ᏼW n ≡  P (1) W n ,P (2) W n  ≡  u n ,v n  W n ∈ ᏿  . (2.6) Then system (1.1) may be expressed as U n = ᏼU n , U n =  u n ,v n  (n = 1,2, ). (2.7) To prove the existence of a global solution to (1.1)itsuffices to show that ᏼ has a fixed point in ᏿ for every n. It is clear from hypothesis (H 1 )thatᏼ is a continuous map on ᏿ which is a closed bounded convex subset of R 2 . We show that ᏼ maps ᏿ into itself by a marching process. Given any W n ≡ (w n ,z n ) ∈ ᏿,relation(2.6) and conditions (2.1), (2.2)implythat  1+kK (1)  M (1) − P (1) W n  =  1+kK (1)  M (1) −  u n−1 + k  K (1) w n + f (1)  w n ,z n ,u n−s 1 ,v n−s 2  ≥  M (1) − u n−1  + k  K (1)  M (1) − w n  + f (1)  M (1) ,z n ,u n−s 1 ,v n−s 2  − f (1)  w n ,z n ,u n−s 1 ,v n−s 2  ≥ M (1) − u n−1 ,  1+kK (2)  M (2) − P (2) W n  =  1+kK (2)  M (2) −  v n−1 + k  K (2) z n + f (2)  w n ,z n ,u n−s 1 ,v n−s 2  ≥  M (2) − v n−1  + k  K (2)  M (2) − z n  + f (2)  w n ,M (2) ,u n−s 1 ,v n−s 2  − f (2)  w n ,z n ,u n−s 1 ,v n−s 2  ≥ M (2) − v n−1 (n = 1,2, ), (2.8) C. V. Pao 61 whenever (u n−s 1 ,v n−s 2 ) ∈ ᏿. This leads to the relation M (1) − P (1) W n ≥ M (1) − u n−1 1+kK (1) M (2) − P (2) W n ≥ M (2) − v n−1 1+kK (2) (n = 1,2, ). (2.9) A similar argument using the second inequalities in (2.2)gives P (1) W n − δ (1) ≥ u n−1 − δ (1) 1+kK (1) , P (2) W n − δ (2) ≥ v n−1 − δ (2) 1+kK (2) (n = 1,2, ), (2.10) whenever (u n−s 1 ,v n−s 2 ) ∈ ᏿. Consider the case n = 1. Since (u 1−s 1 ,v 1−s 2 ) = (φ 1−s 1 ,ψ 1−s 2 ) and (u 0 ,v 0 ) = (φ 0 ,ψ 0 )arein᏿,relations(2.9), (2.10)implythat(δ (1) ,δ (2) ) ≤ (P (1) W 1 , P (2) W 1 ) ≤ (M (1) ,M (2) ). By Brower’s fixed point theorem, ᏼ ≡ (P (1) ,P (2) )hasafixedpoint U 1 ≡ (u 1 ,v 1 )in᏿. This shows that (u 1 ,v 1 )isasolutionof(1.1)forn = 1, and (u 1 ,v 1 )and (u 2−s 1 ,v 2−s 2 )arein᏿. Using this property in (2.9), (2.10)forn = 2, the same argument shows that ᏼ has a fixed point U 2 ≡ (u 2 ,v 2 )in᏿,and(u 2 ,v 2 ) is a solution of (1.1)for n = 2and(u 3−s 1 ,v 3−s 2 ) ∈ ᏿. A continuation of the above argument shows that ᏼ has a fixed point U n ≡ (u n ,v n )in᏿ for every n,and(u n ,v n ) is a global solution of (1.1)in᏿. To show the uniqueness of the solution, we consider any two solutions (u n ,v n ), (u  n ,v  n ) in ᏿ and let (w n ,z n ) = (u n − u  n ,v n − v  n ). By (1.1), w n = w n−1 + k  f (1)  u n ,v n ,u n−s 1 ,v n−s 2  − f (1)  u  n ,v  n ,u  n−s 1 ,v  n−s 2  , z n = z n−1 + k  f (2)  u n ,v n ,u n−s 1 ,v n−s 2  − f (2)  u  n ,v  n ,u  n−s 1 ,v  n−s 2  (n = 1,2, ), w n = 0  n ∈ I 1  , z n = 0  n ∈ I 2  . (2.11) The above relation and condition (2.1)implythat   w n   ≤   w n−1   + kK (1)    w n   +   z n   +   w n−s 1   +   z n−s 2    ,   z n   ≤   z n−1   + kK (2)    w n   +   z n   +   w n−s 1   +   z n−s 2    . (2.12) Addition of the above inequalities leads to   w n   +   z n   ≤   w n−1   +   z n−1   + k  K (1) +K (2)    w n   +   z n   +   w n−s 1   +   z n−s 2    (n = 1,2, ). (2.13) Since w n = z n = 0forn = 0,−1,−2, , the above inequality for n = 1yields   w 1   +   z 1   ≤ k  K (1) + K (2)    w 1   +   z 1    . (2.14) In view of condition (2.4), this is possible only when |w 1 |=|z 1 |=0. Using w 1 = z 1 = 0 in (2.13)forn = 2yields   w 2   +   z 2   ≤ k  K (1) + K (2)    w 2   +   z 2    . (2.15) 62 Global attractor of difference equations It follows again from (2.4)that|w 2 |=|z 2 |=0. The conclusion |w n |=|z n |=0forevery n follows by an induction argument. This proves (u n ,v n ) = (u  n ,v  n ), and therefore (u n ,v n ) is the unique solution of (1.1)in᏿.  Remark 2.2. (a) Since problem (1.3) may be considered as an equivalent system of the scalar second-order differential equation u  = f  u,u  ,u τ 1 ,u  τ 2  (t>0), u(t) = φ(t)  − τ 1 ≤ t ≤ 0  , u  (t) = ψ(t)  − τ 2 ≤ t ≤ 0  , (2.16) the conclusion in Theorem 2.1 and all the results obtained in later sections are directly applicable to the difference approximation of (2.16)with(u n ,v n ) = (u n ,u  n )and(f (1) , f (2) ) = (v n , f (u n ,v n ,u n−s 1 ,v n−s 2 )). (b) System (1.1)isadifference approximation of (1.3) by the backward (or left-sided) approximation of the time derivative (du/dt,dv/dt), and this approximation preserves the nonlinear nature of the differential system. If the forward (or right-sided) approximation for (du/dt,dv/dt) is used, then the resulting difference system gives an explicit formula for (u n+1 ,v n+1 ) which can be computed by a marching process for every n = 0,1,2, and for any continuous function ( f (1) , f (2) ). From a view point of differential equations, the forward approximation may lead to misleading information about the solution of the differential system. One reason is that a global solution to the differential system may fail to exist while the difference solution (u n+1 ,v n+1 ) exists for every n. (c) The uniqueness result in Theorem 2.1 is in the set ᏿, and it does not rule out the possibility of existence of positive solutions outside of ᏿. 3. Comparison theorems To investigate the asymptotic behavior of the solution we consider a class of quasimono- tone functions which depend on the monotone property of ( f (1) , f (2) ). Specifical ly, we make the following hypothesis. (H 2 )(f (1) , f (2) )isaC 1 -function in ᏿ × ᏿ and possesses the property ∂f (1) /∂u s ≥ 0, ∂f (2) /∂v s ≥0 and one of the following quasimonotone properties for (u,v,u s ,v s )∈ ᏿ × ᏿: (a) quasimonotone nondecreasing: ∂f (1) ∂v ≥ 0, ∂f (1) ∂v s ≥ 0, ∂f (2) ∂u ≥ 0, ∂f (2) ∂u s ≥ 0; (3.1) (b) quasimonotone nonincreasing: ∂f (1) ∂v ≤ 0, ∂f (1) ∂v s ≤ 0, ∂f (2) ∂u ≤ 0, ∂f (2) ∂u s ≤ 0; (3.2) (c) mixed quasimonotone: ∂f (1) ∂v ≤ 0, ∂f (1) ∂v s ≤ 0, ∂f (2) ∂u ≥ 0, ∂f (2) ∂u s ≥ 0. (3.3) C. V. Pao 63 Notice that if ( f (1) , f (2) ) ≡ ( f (1) (u,v), f (2) (u,v)) is independent of (u s ,v s ), then the above conditions are reduced to those required for the standard three types of quasimonotone functions (cf. [15, 18]). It is easy to see from (H 2 ) that for quasimonotone functions the conditions on (M (1) , M (2) ), (δ (1) ,δ (2) )in(2.2) are reduced to the following. (a) For quasimonotone nondecreasing functions: f (1)  M (1) ,M (2) ,M (1) ,M (2)  ≤ 0 ≤ f (1)  δ (1) ,δ (2) ,δ (1) ,δ (2)  , f (2)  M (1) ,M (2) ,M (1) ,M (2)  ≤ 0 ≤ f (2)  δ (1) ,δ (2) ,δ (1) ,δ (2)  . (3.4) (b) For quasimonotone nonincreasing functions: f (1)  M (1) ,δ (2) ,M (1) ,δ (2)  ≤ 0 ≤ f (1)  δ (1) ,M (2) ,δ (1) ,M (2)  , f (2)  δ (1) ,M (2) ,δ (1) ,M (2)  ≤ 0 ≤ f (2)  M (1) ,δ (2) ,M (1) ,δ (2)  . (3.5) (c) For mixed quasimonotone functions: f (1)  M (1) ,δ (2) ,M (1) ,δ (2)  ≤ 0 ≤ f (1)  δ (1) ,M (2) ,δ (1) ,M (2)  , f (2)  M (1) ,M (2) ,M (1) ,M (2)  ≤ 0 ≤ f (2)  δ (1) ,δ (2) ,δ (1) ,δ (2)  . (3.6) In this section, we show some comparison results among solutions with different initial functions for each of the above three types of quasimonotone functions. The comparison results for the first two types of quasimonotone functions are based on the following positivity lemma for a function (w n ,z n ) satisfying the relation γ (1) n w n ≥ w n−1 + a (1) n z n + b (1) n w n−s 1 + c (1) n z n−s 2 , γ (2) n z n ≥ z n−1 + a (2) n w n + b (2) n w n−s 1 + c (2) n z n−s 2 (n = 1,2, ), w n ≥ 0  n ∈ I 1  , z n ≥ 0(n ∈ I), (3.7) where for each l = 1,2, and n = 1,2, ,γ (l) n is positive, and a (l) n , b (l) n ,andc (l) n are nonnega- tive. Lemma 3.1. Let (w n ,z n ) satisfy (3.7), and let a (1) n a (2) n <γ (1) n γ (2) n (n = 1,2, ). (3.8) Then (w n ,z n ) ≥ (0,0) for every n = 1, 2, Proof. Consider the case n = 1. Since w n ≥ 0forn ∈ I 1 and z n ≥ 0forn ∈ I 2 , the inequal- ities in (3.7)yield γ (1) 1 w 1 ≥ a (1) 1 z 1 , γ (2) 1 z 1 ≥ a (2) 1 w 1 . (3.9) 64 Global attractor of difference equations The positivity of γ (1) 1 , γ (2) 1 implies that w 1 ≥  a (1) 1 γ (1) 1  z 1 ≥  a (1) 1 a (2) 1 γ (1) 1 γ (2) 1  w 1 , z 1 ≥  a (2) 1 γ (2) 1  w 1 ≥  a (1) 1 a (2) 1 γ (1) 1 γ (2) 1  z 1 . (3.10) In view o f (3.8), the above inequalities can hold only if (w 1 ,z 1 ) ≥ (0,0). Assume, by in- duction, that (w n ,z n ) ≥ (0,0) for n = 1,2, ,m − 1forsomem>1. Then by (3.7), γ (1) m w m ≥ a (1) m z m , γ (2) m z m ≥ a (2) m w m . (3.11) This leads to w m ≥  a (1) m a (2) m γ (1) m γ (2) m  , z m ≥  a (1) m a (2) m γ (1) m γ (2) m  z m . (3.12) It follows again from (3.8)that(w m ,z m ) ≥ (0,0). The conclusion of the lemma follows by the principle of induction.  The above positivity lemma can be extended to a function (w n ,z n ,w n ,z n ) satisfying the relation γ (1) n w n ≥ w n−1 + a (1) n z n + b (1) n w n−s 1 + c (1) n z n−s 2 , γ (2) n z n ≥ z n−1 + a (2) n w n + b (2) n w n−s 1 + c (2) n z n−s 2 , ˆ γ (1) n w n ≥ w n−1 + ˆ a (1) n z n + ˆ b (1) n w n−s 1 + ˆ c (1) n z n−s 2 , ˆ γ (2) n z n ≥ z n−1 + ˆ a (2) n w n + ˆ b (2) n w n−s 1 + ˆ c (2) n z n−s 2 (n = 1,2, ), w n ≥ 0, w n ≥ 0  n ∈ I 1  , z n ≥ 0, z n ≥ 0  n ∈ I 2  , (3.13) where γ (l) n , a (l) n , b (l) n ,andc (l) n , l = 1,2, are the same as that in (3.7)and ˆ γ (l) n , ˆ a (l) n , ˆ b (l) n ,and ˆ c (l) n are nonnegative with ˆ γ (l) n > 0, n = 1,2, Lemma 3.2. Let (w n ,z n ,w n ,z n ) satisfy (3.13), and let  a (1) n a (2) n  ˆ a (1) n ˆ a (2) n  <  γ (1) n γ (2) n  ˆ γ (1) n ˆ γ (2) n  (n = 1,2, ). (3.14) Then (w n ,z n ,w n ,z n ) ≥ (0,0,0,0) for every n. Proof. By (3.13)withn = 1, we have γ (1) 1 w 1 ≥ a (1) 1 z 1 , γ (2) 1 z 1 ≥ a (2) 1 w 1 , ˆ γ (1) 1 w 1 ≥ ˆ a (1) 1 z 1 , ˆ γ (2) 1 z 1 ≥ ˆ a (2) 1 w 1 . (3.15) C. V. Pao 65 This implies that w 1 ≥  a (1) 1 γ (1) 1  z 1 ≥  a (1) 1 γ (1) 1  ˆ a (2) 1 ˆ γ (2) 1  w 1 ≥  a (1) 1 γ (1) 1  ˆ a (2) 1 ˆ γ (2) 1  ˆ a (1) 1 ˆ γ (1) 1  z 1 ≥  a (1) 1 a (2) 1 γ (1) 1 γ (2) 2  ˆ a (1) 1 ˆ a (2) 2 ˆ γ (1) 1 ˆ γ (2) 1  w 1 . (3.16) In view of condition (3.14), we have w 1 ≥ 0. This implies that z 1 ≥ 0, w 1 ≥ 0andz 1 ≥ 0 which proves the case for n = 1. Assume, by induction, that (w n ,z n ,w n ,z n ) ≥ (0,0,0,0) for n = 1,2, ,m − 1forsomem>1. Then by (3.13), γ (1) m w m ≥ a (1) m z m , γ (2) m z m ≥ a (2) m w m , ˆ γ (1) m w m ≥ ˆ a (1) m z m , ˆ γ (2) m z m ≥ ˆ a (2) m w m . (3.17) This leads to w m ≥  a (1) m a (2) m γ (1) m γ (2) m  ˆ a (1) m ˆ a (2) m ˆ γ (1) m ˆ γ (2) m  w m . (3.18) It follows again from (3.14)thatw m ≥ 0fromwhichweobtainz m ≥ 0, w m ≥ 0andz m ≥ 0. The conclusion of the lemma follows from the principle of induction.  To obtain comparison results among solutions, we need to impose a condition on the time increment k.Define σ (1) 1 ≡ max  ∂f (1) ∂u  u,v,u s ,v s  ;(u,v),  u s ,v s  ∈ ᏿  , σ (1) 2 ≡ max       ∂f (1) ∂v  u,v,u s ,v s       ;(u,v),  u s ,v s  ∈ ᏿  , σ (2) 1 ≡ max       ∂f (2) ∂u  u,v,u s ,v s       ;(u,v),  u s ,v s  ∈ ᏿  , σ (2) 2 ≡ max  ∂f (2) ∂v  u,v,u s ,v s  ;(u,v),  u s ,v s  ∈ ᏿  . (3.19) Our condition on k is given by k  K (1) + K (2)  < 1,  kσ (1) 2  kσ (2) 1  <  1 − kσ (1) 1  1 − kσ (2) 2  , (3.20) where K (l) , l = 1, 2, are the Lipschitz constants in (2.1). Since σ (1) 1 ≤ K (1) , σ (2) 2 ≤ K (2) , it follows that kσ (1) 1 < 1andkσ (2) 2 < 1. Notice that σ (1) 2 and σ (2) 1 are nonnegative while σ (1) 1 and σ (2) 2 are not necessarily nonnegative. The following comparison theorem is for quasimonotone nondecreasing functions. 66 Global attractor of difference equations Theorem 3.3. Let hypotheses (H 1 ), (H 2 )(a), and condition (3.20)besatisfied.Denoteby (u n ,v n ), (u n ,v n ),and(u n ,v n ) the solutions of (1.1)with(φ n ,ψ n ) = (M (1) ,M (2) ), (φ n ,ψ n ) = (δ (1) ,δ (2) ), and arbitrary (φ n ,ψ n ) ∈ ᏿,respectively.Then  u n ,v n  ≤  u n ,v n  ≤  u n ,v n  , n = 1,2, (3.21) Proof. Let (w n ,z n ) = (u n − u n ,v n − v n ). By (1.1) and the mean value theorem, w n = w n−1 + k  f (1)  u n ,v n ,u n−s 1 ,v n−s 2  − f (1)  u n ,v n ,u n−s 1 ,v n−s 2  = w n−1 + k  ∂f (1) ∂u  ξ n   w n +  ∂f (1) ∂v  ξ n   z n +  ∂f (1) ∂u s  ξ n   w n−s 1 +  ∂f (1) ∂v s  ξ n   z n−s 2  , z n = z n−1 + k  f (2)  u n ,v n ,u n−s 1 ,v n−s 2  − f (2)  u n ,v n ,u n−s 1 ,v n−s 2  = z n−1 + k  ∂f (2) ∂u  ξ  n   w n +  ∂f (2) ∂v  ξ  n   z n +  ∂f (2) ∂u s  ξ  n   w n−s 1 +  ∂f (2) ∂v s  ξ  n   z n−s 2  (n = 1,2, ), w n = φ n − δ (1) ≥ 0  n ∈ I 1  , z n = ψ n − δ (2) ≥ 0  n ∈ I 2  , (3.22) where ξ n ≡ (ξ n ,η n ,ξ n−s 1 ,η n−s 2 )andξ  n ≡ (ξ  n ,η  n ,ξ  n−s 1 ,η  n−s 2 ) are some intermediate values between (u n ,v n ,u n−s 1 ,v n−s 2 )and(u n ,v n ,u n−s 1 ,v n−s 2 ) and therefore are in ᏿ × ᏿.Define α (1) n = k ∂f (1) ∂u  ξ n  , a (1) n = k ∂f (1) ∂v  ξ n  , b (1) n = k ∂f (1) ∂u s  ξ n  , c (1) n = k ∂f (1) ∂v s  ξ n  , α (2) n = k ∂f (2) ∂v  ξ  n  , a (2) n = k ∂f (2) ∂u  ξ  n  , b (2) n = k ∂f (2) ∂u s  ξ  n  , c (2) n = k ∂f (2) ∂v s  ξ  n  . (3.23) Then (3.22)maybewrittenas  1 − α (1) n  w n = w n−1 + a (1) n z n + b (1) n w n−s 1 + c (1) n z n−s 2 ,  1 − α (2) n  z n = z n−1 + a (2) n w n + b (2) n w n−s 1 + c (2) n z n−s 2 (n = 1,2, ), w n ≥ 0  n ∈ I 1  , z n ≥ 0  n ∈ I 2  . (3.24) Since by hypothesis (H 2 )(a), a (l) n , b (l) n ,andc (l) n are nonnegative, and by conditions (3.19) and (3.20), γ (1) n ≡ 1 − α (1) n ≥ 1 − kσ (1) 1 > 0, γ (2) n ≡ 1 − α (2) n ≥ 1 − kσ (2) 2 > 0, a (1) n a (2) n ≤  kσ (1) 2  kσ (2) 1  <  1 − kσ (1) 1  1 − kσ (2) 2  ≤ γ (1) n γ (2) n (n = 1,2, ), (3.25) [...]... converge to (u∗ ,v∗ ) as n → ∞ The uniqueness of (u∗ ,v∗ ) in ᏿ and the convergence of (un ,vn ) to (u∗ ,v∗ ) follow from Theorem 3.5 The previous theorems imply that for each type of quasimonotone functions if (u,v) = (u,v) ≡ (u∗ ,v∗ ), then (u∗ ,v∗ ) is a global attractor relative to the set ᏿ In the following theorem, we give a sufficient condition for the global attraction of (u∗ ,v∗ ) relative to the... ,v∗ ) as n → ∞ 78 Global attractor of difference equations Remark 5.4 The convergence of the solution (un ,vn ) to (u∗ ,v∗ ) for arbitrary nonnegative initial functions (φn ,ψn ) (with (φo ,ψ0 ) > (0,0)) in Theorems 5.1, 5.2, and 5.3 shows that the positive equilibrium (u∗ ,v∗ ) in the corresponding model problems (1.4), (1.5), and (1.6) is a global attractor of all positive solutions of (1.1) This implies... Dynamics of a finite difference system of reaction diffusion equations with time delay, J Differ Equations Appl 4 (1998), no 1, 1–11 , Numerical analysis of coupled systems of nonlinear parabolic equations, SIAM J Numer Anal 36 (1999), no 2, 393–416 S Sugiyama, On the asymptotic behaviors of solutions of difference equations I, Proc Japan Acad 47 (1971), 477–480 C V Pao [20] [21] 79 , On the asymptotic behaviors... Ladas, J Popenda, and J Rakowski, Communications in difference equations, Pro´ ceedings of the 4th International Conference on Difference Equations, Poznan, 1988 M E Fisher, Stability of a class of delay-difference equations, Nonlinear Anal 8 (1984), no 6, 645–654 J E Franke and A A Yakubu, Global attractors in competitive systems, Nonlinear Anal 16 (1991), no 2, 111–129 K Gopalsamy, Stability and Oscillations... Agarwal, Difference Equations and Inequalities Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics, vol 155, Marcel Dekker, New York, 1992 R D Driver, G Ladas, and P N Vlahos, Asymptotic behavior of a linear delay difference equation, Proc Amer Math Soc 115 (1992), no 1, 105–112 A Drozdowicz and J Popenda, Asymptotic behavior of the solutions of the second order... (2) ≥ 0 n ∈ I2 , (l) (l) where α(l) , a(l) , bn , and cn are given by (3.23) with possibly some different intermen n diate values ξ n , ξ n Since (un ,vn ) ∈ ᏿ for all n, the values of ξ n and ξ n remain in ᏿, (l) (l) and therefore the coefficients a(l) , bn , and cn are nonnegative It follows from the proof n 70 Global attractor of difference equations of Theorem 3.3 that (wn ,zn ) ≥ (0,0) for n = 1,2,... I2 The existence and uniqueness of a solution to (3.28) can be treated by the same argument as that for (1.1) The following theorem gives an analogous result as that in Theorem 3.3 68 Global attractor of difference equations Theorem 3.5 Let hypotheses (H1 ), (H2 )(c) and condition (3.20) be satisfied Let also ((un , vn ),(un ,vn )) be the solution of (3.28) and (un ,vn ) the solution of (1.1) with arbitrary... ᏿, and for any (φ,ψ) ∈ ᏿, the corresponding solution (un ,vn ) of (4.16) converges to (u∗ ,v∗ ) as n → ∞ Moreover, this convergence property holds true for the solution (un ,vn ) corresponding to an arbitrary (φ,ψ) if condition (4.13) holds for some n0 > 0 5 Applications to Lotka-Volterra systems In this section, we give some applications of the global stability results in the previous section to the... (2.2), or the corresponding conditions (3.4), (3.5), and (3.6); 74 Global attractor of difference equations (ii) to show that the limits (u,v), (u,v) in (4.5) or in (4.10) coincide; (iii) to verify that the solution (un ,vn ) corresponding to an arbitrary nonnegative (φn ,ψn ) with (φ0 ,ψ0 ) > (0,0) converges to (u∗ ,v∗ ) as n → ∞ To do this for each of the three models, we set a(2) = a(2) + c(2) , b... attractivity in nonlinear delay difference equations, Proc Amer c Math Soc 115 (1992), no 4, 1083–1088 , Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and Its Applications, vol 256, Kluwer Academic Publishers, Dordrecht, 1993 Y Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, vol 191, Academic . 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. (u ,v) = (u ,v) (≡ (u ∗ ,v ∗ )), then (u ∗ ,v ∗ ) is a solution of (4.1), and both (u n ,v n )and( u n ,v n )convergeto(u ∗ ,v ∗ )asn →∞. The uniqueness of (u ∗ ,v ∗ )in᏿ and the convergence of (u n ,v n )to( u ∗ ,v ∗ )followfromTheorem. 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