Permanence and Global Attractivity of a Discrete Two-Prey One-Predator Model with Infinite Delay

Correspondence should be addressed to Zhong Li, lizhong04108@163.com Received 4 February 2009; Revised 2 July 2009; Accepted 15 August 2009 Recommended by Leonid Berezansky A discrete two-prey one-predator model with infinite delay is proposed. A set of sufficient conditions which guarantee the permanence of the system is obtained. By constructing a suitable Lyapunov functional, we also obtain sufficient conditions ensuring the global attractivity of the system. An example together with its numerical simulation shows the feasibility of the main results. Copyright q 2009 Z. Yu and Z. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Introduction
The aim of this paper is to investigate the persistence and stability property of the following discrete two-prey one-predator model with infinite delays: where x i n , i 1, 2, are the densities of the prey species i at the nth generation; x 3 n is the density of the predator at the nth generation; {a i n }, {c i n }, {d i n } i 1, 2 , {b i n } i 1, 2, 3 ; {H i n } i 1, . . ., 4 are all bounded nonnegative sequences such that Here, for any bounded sequence {g n }, set g u sup n∈N {g n } and g l inf n∈N {g n }.
From the point of view of biology, in the sequel, we assume that where s . . ., −n, −n 1, . . ., −1, 0. Then system 1.1 with the initial condition 1.3 has a unique positive solution x 1 n , x 2 n , x 3 n T .
As one of the dominant themes in mathematical biology, the predator-prey relationship has been studied in a number of ways see 1-4 and the references therein .In 1970, Parrish and Saila 5 firstly proposed the one-prey two-predator model as follows: 1.4 Gramer and May 6 studied the stability of the positive equilibrium of system 1.4 ; Takcuchi and Adachi 7 investigated the existence of the positive equilibrium and Hopf Bifurcation of the above system.Recently, Elettreby where all the parameters in system 1.5 are positive constants.By applying differential inequality theory and iterative scheme, he showed that the unique positive equilibrium of system 1.5 is globally attractive.It is well known that a suitable ecosystem should incorporate some pase of the state of system, which is represented by time delays.Li et al. 9 studied the two-prey one-predator model with delays: where x 1 t , x 2 t , y t are the densities of the prey and predator at the time t, respectively, b 1 , b 2 , b 3 , α, β, r 1 , r 2 , d are all positive constants.They investigated the Hopf bifurcation of system 1.6 .Corresponding to system 1.5 , Huang 10 proposed and studied the following system with infinite delays: where all the coefficients a, b, c, d, e are positive constants, and . By applying iterative scheme, he showed that the unique positive equilibrium of the system is globally attractive.
On the other hand, it is well known that the discrete time model governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations.Corresponding to traditional continuous Logistic model governed by differential equations, Mohamad and Gopalsamy 11 proposed the following single species discrete model: They tried to obtain a set of sufficient conditions which ensure that 1.8 admits a unique positive and globally asymptotically stable almost periodic solution.However, Zhou and Zou 12 gave an counterexample which shows that the main results of 11 are not correct.By developing some new analysis technique, Zhou and Zou 12 obtained sufficient conditions which ensure the existence of a positive and globally asymptotically stable ω-periodic solution of system 1.8 .Chen and Zhou 13 further generalized system 1.8 to the following two-species Lotka-Volterra competition system: 1.9 They obtained the sufficient conditions which guarantee the persistence of the system 1.3 .Also, for the periodic case, they obtained the sufficient conditions which guarantee the existence of a globally stable periodic solution of the system.Wang and Lu 14 proposed the following Lotka-Volterra model: where x i k is the density of population i at kth generation; r i k is the growth rate of population i at kth generation; a ij k measures the intensity of intraspecific competition or interspecific action of species.By constructing a suitable Lyapunov function and using the finite covering theorem of Mathematic Analysis, they obtained a set of sufficient conditions which ensures the system to be globally asymptotically stable.Similar to the continuous ones, some scholars also argued that the discrete model should incorporate some past state of the species and thus should consider the discrete model with time delay.Recently, Chen 15 investigated the persistent property of the following discrete two species Lotka-Volterra competition model with deviating arguments: where x i n , i 1, 2, are the densities of competition species i at nth generation.By establishing a new difference inequality, Chen 15 showed that under the same conditions as that of Chen and Zhou 13 , 1.11 is also permanent, which means that with some suitable restriction on the coefficients of the system, delay has no influence on the persistent property of the system.Chen 16 also investigated the persistent property of a discrete n-species nonautonomous Lotka-Volterra competitive systems with infinite delays and feedback controls.As we can see, both 15 and 16 considered the persistent property of the system, but they did not investigate the stability property of the system.Recently, Chen et al. 17 investigated the dynamic behaviors of the following general discrete nonautonomous system of plankton allelopathy with finite time delay: where N i k represents the densities of population i at the kth generation; r i k is the intrinsic growth rate of population i at the kth generation; a il k measures the intraspecific influence of the k − l th generation of population i on the density of own population; b il k stands for the inter-specific influence of the k − l th generation of population i on the density of own population and c il k stands for the effect of toxic inhibition of population i by population j at the k − l th generation, i, j 1, 2 and i / j. {r i k }, {a il k }, {b il k } and {c il k } are all bounded nonnegative sequences defined for k ∈ N.They obtained sufficient conditions which guarantee the permanence, global attractivity and partial extinction of the above system.
Concerned with the discrete predator-prey system, by giving the detail analysis of the right hand side of the system, Yang 18 obtained a set of sufficient conditions which ensures the uniform persistence of the system investigated.Recently, Chen and Chen 19 proposed the following discrete periodic Volterra model with mutual interference and Holling II type functional response

1.13
They also obtained sufficient conditions which ensure the permanence of the system.For more works on discrete population model, one could refer to 11-42 and the references cited therein.However, to the best of the authors knowledge, to this day, no scholars propose and study the discrete predator-prey model with infinite delays.This motivates us to propose and study 1.1 .The aim of this paper is to investigate the persistent and stability property of system 1.1 .
The rest of the paper is arranged as follows: some useful lemmas are stated in the following section.Sufficient conditions which ensure the permanence and global attractivity of system 1.1 are stated and proved in Section 3. In Section 4, an example together with its numeric simulation shows the feasibility of the main results.We end this paper by a brief discussion.

Preliminaries
Now let us state several lemmas which will be useful in proving main results.Lemma 2.1 see 29 .Assume that {x n } satisfies x n > 0 and for n ∈ N, where {r n } and {a n } are all positive sequences bounded above and below by positive constants.Then x n . 2.5

Main Results
Now, we investigate the persistence property and stability property of system 1.1 .

Theorem 3.1. Assume that
where

3.2
Proof.It follows from the first two equations of system 1.1 that So, as a consequence of Lemma 2.1, for any solution x 1 n , x 2 n , x 3 n ∞ n 0 of system 1.1 with initial condition 1.3 , one has lim sup

3.4
According to Lemma 2.3, from 3.4 , we have

3.5
For any small positive constant ε > 0, it follows from 3.5 that there exists a positive integer n 1 such that for all n ≥ n 1 , Thus, for all n ≥ n 1 , from the last equation of system 1.1 , if follows that By applying Lemma 2.1 to 3.7 , we have lim sup 3.9 Next, we show that under the assumption of Theorem 3.1,

3.10
According to Lemma 2.3, from 3.9 we have lim sup

3.11
Conditions H 1 and H 2 imply that for enough small positive constant ε 1 , we have

3.12
For ε 1 , it follows from 3.11 that there exists an positive integer n 2 ≥ n 1 such that for all

3.13
For n > n 2 , from 3.13 and the first two equations of system 1.1 , we have Thus, according to Lemma 2.2, one has lim inf

3.16
According to Lemma 2.3, from 3.16 we have 3.17 For any ε 2 > 0 small enough, without loss of generality, we may assume that ε 2 < 1/2 min{m 1 , m 2 }.From 3.17 , it follows that there exists a n 3 , such that for all

3.18
For n ≥ n 3 , from 3.18 and the last equation of 1.1 , we have

3.19
By applying Lemma 2.2 to 3.19 , it follows that lim inf

3.21
This ends the proof of Theorem 3.1.
Theorem 3.2.Assume that H 1 and H 2 hold.Assume further that there exist positive constants α, β, γ and δ such that hold.Then for any two positive solutions x 1 n , x 2 n , x 3 n T and y 1 n , y 2 n , y 3 n T of system 1.1 , one has Proof.From conditions H 3 -H 5 , there exits an enough small positive constant ε 3 such that

3.25
Since H 1 and H 2 hold, for any solutions x 1 n , x 2 n , x 3 n T and y 1 n , y 2 n , y 3 n T of system 1.1 with the initial conditions 1.3 , it follows from Theorem 3.1 that lim sup For the above ε 3 and 3.26 , there exists an n * > 0 such that for all n > n * , Firstly, let

3.28
Then from the first equation of the system 1.1 , we have

3.29
Using the Mean Value Theorem, we get where ξ 1 n lies between x 1 n and y 1 n , then it follows that and so

3.32
Secondly, let then, similar to the aforementioned analysis, we have

3.34
Now, set then from 3.32 and 3.34 , we have

3.36
Let where

3.38
Similar to the aforementioned analysis, we have where ξ 2 n lies between x 2 n and y 2 n .Let where

3.41
Similar to the aforementioned analysis, we have where ξ 3 n lies between x 3 n and y 3 n .Now, we define a Lyapunou functional as follows: V n αV 1 n βV 2 n γV 3 n .

3.43
Calculating the difference of V along the solution of system 1.1 , for n > n * , it follows from 3.23 , 3.24 , 3.25 , 3.27 , 3.36 , 3.39 and 3.42 that x i n − y i n .

3.44
Summating both sides of the above inequalities from n * to n, we have

3.49
This completes the proof of Theorem 3.2.

Example
The following example shows the feasibility of the main results.

4.2
Clearly, conditions H 1 -H 5 are satisfied.From Theorems 3.1 and 3.2, 1.1 is permanent and globally attractive.Numeric simulation Figure 1 strongly supports our results.

Discussion
In this paper, we propose a discrete two-prey one-predator model with infinite delay.Theorem 3.1 shows that to ensure the permanence of the system, one should ensure a l i , i 1, 2 and b l 3 enough large, that is, the net birth rate of prey species and the density restriction of predator species should be large enough.We also obtain a set of sufficient conditions which ensures the global attractivity of the system.
and x N 0 > 0, where {r n } and {a n } are all positive sequences bounded above and below by positive constants and N 0 ∈ N. Then 2 Lemma 2.2 see 29 .Assume that {x n } satisfies x n 1 ≥ x n exp r n − a n x n , k ≥ N 0 , 2.3 lim sup n → ∞ x n ≤ x * Lemma 2.3 see 43 .Let x : Z → R be a nonnegative bounded sequences, and let