Permanence and Global Stability of Positive Periodic Solutions of a Discrete Competitive System

We consider the dynamic behaviors of a discrete 
competitive system. A good understanding of the permanence, existence, 
and global stability of positive periodic solutions is gained. Numerical 
simulations are also presented to substantiate the analytical results.


Introduction
In biomathematics, one of the most challenging aspects of mathematical biology is competition modeling.Although the mathematical idea is simple 1 , this type of modeling is so difficult to carry out in any generality since there are so many ways for a population to compete; many classical competitive models have been established to describe the relationships between species and the outer environment, and the connections between different species.Lotka-Volterra competitive model of two species communities is probably the best known model of mathematical ecology.In 1934, Gause 2 found out the competitive exclusion theory, which states that two species that compete for the exact same resources cannot stably coexist.One of the two competitors will always have an ever so slight advantage over the other that leads to extinction of the second competitor in the long run.Since then the competitive model has increasingly won attention as an important and fundament model in biomathematics.The dynamic relationship between species and their competitors has long been one of the dominant theses in both ecology and mathematical ecology.As a consequence, many excellent results concerned with permanence, extinction, stability and hopf bifurcations, and existence and global stability of positive periodic solutions of Lotka-Volterra competitive system are obtained see 3-16 .Although much progress has been seen for Lotka-Volterra competitive systems, such systems are not well studied in the sense that most results are continuous time versions related.Many authors 17-21 have argued that the discrete-time models governed by difference equations are more appropriate than the continuous ones when populations have a short life expectancy, nonoverlapping generations in the real word.Discrete-time models can provide efficient computational models of continuous models for numerical simulations.So it is reasonable to study discrete-time competitive systems governed by difference equations.
In this paper, we will consider the dynamic behavior of a discrete-time competitive system.Let us first introduce its continuous time version which is motivated in 22 where x 1 t , x 2 t are the population densities of two competing species; r 1 t , r 2 t are the intrinsic growth rates of species; a 1 t , a 2 t are the rates of intraspecific competition of the first and second species, respectively; c 1 t , c 2 t are the rates of interspecific competition of the first and second species, respectively.All the coefficients above are continuous and bounded above and below by positive constants.Following the same idea and method in 21 , one can easily derive the discrete analogue of system 1.1 , which takes the form of The exponential form of system 1.2 is more biologically reasonable than that directly derived by replacing the differential by difference in system 1.1 because this exponential form can assure Here x i n represent the densities of species x i at the nth generation, r i n are the intrinsic growth rates of species x i at the nth generation, a i n measure the intraspecific effects of the nth generation of species x i on own population, and c i n stand for the interspecific effects of the nth generation of species x i on species x j i, j 1, 2; i / j .
It is well known that, compared to the continuous time systems, the discrete-time ones are more difficult to deal with.The principle aim of this paper is to explore the permanence, existence, and global stability of positive periodic solutions of system 1.2 .To the best of our knowledge, no work has been done for system 1.2 .
For the sake of simplicity and convenience in the following discussion, the notations below will be used through this paper: where {f n } is a bounded sequence and N is the set of nonnegative integer numbers.
For biological reasons, in system 1.2 we only consider the solution {x 1 n , x 2 n } with the initial value {x 1 0 , x 2 0 } > 0.
The organization of this paper is as follows.In the next section, we establish the permanence of system 1.2 .In Section 3, we obtain sufficient conditions which ensure the existence and global stability of positive periodic solutions of system 1.2 .Numerical simulations are present to illustrate the feasibility of our main results in final section.

Permanence
In this section, we will establish sufficient conditions for the permanence of system 1.2 .Definition 2.1.System 1.2 is said to be permanent if there exist positive constants m i and M i such that each positive solution {x 1 n , x 2 n } of system 1.2 satisfies Proof.To prove Proposition 2.2, we consider Case 1 and Case 2.
Case 1. Assume that there exists an n 0 ∈ N such that x 1 n 0 1 ≥ x 1 n 0 , from the first equation of system 1.2 , it follows that which implies,

2.4
Then where we use the fact that max x∈R x exp a − bx exp a − 1 /b for a, b > 0, and R is the set of all real numbers.Hence x 1 n 0 ≤ M 1 .
We claim that x 1 n ≤ M 1 for all n ≥ n 0 .By way of contradiction, assume that there exists a p 0 > n 0 such that x 1 p 0 > M 1 , then p 0 ≥ n 0 2. Let It is easy to obtain that x 1 p 0 ≤ M 1 from the above argument, which is a contradiction.Therefore, This proves the claim.
Case 2. Suppose that x 1 n 1 < x 1 n for all n ∈ N. In particular, lim n → ∞ x 1 n exists, denoted by x 1 , we will prove x 1 ≤ r U 1 /a L 1 by way of contradiction as follows.Assume that 1 /a L 1 , taking limit in the first equation of system 1.2 , which leads to which is a contradiction.This proves the claim.By the fact that min x∈R { exp x − 1 /x} 1, we obtain that

2.10
This completes the proof of Proposition 2.2.
Proposition 2.3.Suppose that system 1.2 satisfies the following assumptions:

2.11
Then any positive solution {x 1 n , x 2 n } of system 1.2 satisfies Proof.By Proposition 2.2, since lim sup n → ∞ x 1 n ≤ M 1 for each ε > 0, then there exists an n * ∈ N such that x 1 n ≤ M 1 ε for n ≥ n * .In order to prove Proposition 2.3, there are two cases to be considered as follows.
Case 1. Assume that there exists an n 0 ≥ n * such that x 1 n 0 1 ≤ x 1 n 0 , by the first equation of system 1.2 , it derives that It follows from the inequality 2.11 that By 2.13 and 2.15 we have

2.16
Hence In the following we will prove x 1 n ≥ x ε 1 for all n ≥ n 0 .By way of contradiction, assume that there exists a p 0 > n 0 such that x 1 p 0 < x ε 1 , then p 0 ≥ n 0 2. Let that is, x 1 p 0 < x ε 1 and p 0 ≥ n 0 2, then x 1 p 0 < x ε 1 ≤ x 1 p 0 − 1 , the above argument produces that x 1 p 0 ≥ x ε 1 , which is a contradiction.Therefore, x 1 n ≥ x ε 1 for all n ≥ n 0 , since ε can be sufficiently small, it gives that This proves the claim.
Case 2. Assume that x 1 n 1 > x 1 n for a sufficiently large n ≥ n * .In this case, lim n → ∞ x 1 n exists, denoted by x 1 .For the sake of proving which is a contradiction.It implies that Since ε can be sufficiently small, we have lim inf Analogously, by the second inequality in 2.11 , we can obtain that lim inf This completes the proof of Proposition 2.3.Now, we are in a position to state Theorem 2.4 whose proof is a direct consequence of Propositions 2.2 and 2.3.Theorem 2.4.If the inequalities in 2.11 hold, then system 1.2 is permanent.

Existence and Global Stability of Positive Periodic Solutions
In this section, we suppose system 1.2 is a periodic system, and then we investigate the existence and global stability of positive periodic solutions of such system.To do this, assume that all the coefficients of system 1.2 are ω-periodic, in other words, Next, we derive sufficient conditions which guarantee that the positive periodic solution of system 1.2 is globally stable.We first give the definition of global stability.

Definition 3.2. A positive periodic solution {x *
1 n , x * 2 n } of system 1.2 is globally stable if each other solution {x 1 n , x 2 n } of system 1.2 with positive initial value defined for all n > 0 satisfies lim Now, we give the main result in this section.
Theorem 3.3.In addition to 2.11 , assume further that the following assumptions hold.Then the positive periodic solution of system 1.2 is globally stable.Proof.Let {x * 1 n , x * 2 n } be a positive periodic solution of system 1.2 .We make the change of variables x 1 n x * 1 n exp u n and x 2 n x * 2 n exp v n , then system 1.2 is rewritten as

3.6
By the mean-value theorem, it derives that

3.8
Discrete Dynamics in Nature and Society 9 In view of Propositions 2.2 and 2.3, there exists an n 0 ∈ N such that n ≥ n 0 , we have

3.9
Since θ 1 , θ 2 , θ 3 , θ 4 ∈ 0, 1 , both x * 1 n exp θ 1 u n and x * 1 n exp θ 4 u n are between x 1 n and x * 1 n .Meanwhile, both x * 2 n exp θ 2 v n and x * 2 n exp θ 3 v n are between x 2 n and x * 2 n .From the first equation of system 3.7 , it follows that

3.10
Similarly, it follows from 3.5 that

Example and Numerical Simulation
In this paper, we have investigated the permanence and global stability of positive periodic solutions of a discrete competitive system.Each species is not isolated from its living environment, but competes with the other for the same resource.Sufficient conditions which guarantee the permanence, existence and global stability of positive periodic solutions are established, respectively.The theoretical results are confirmed by the following examples and their numerical results.To verify the sufficient conditions for permanence of system 1.2 , we assume that r 1 n 0.46 0.01 sin πn, r 2 n 0.88 − 0.02 sin πn, a 1 n 0.75 0.01 cos πn, a 2 n 0.52 0.01 sin πn, c 1 n 0.28 0.02 sin πn, c 2 n 0.13 − 0.01 sin πn, and the initial condition x 1 0 , x 2 0 0.25, 0.01 .Clearly, 2.11 in Theorem 2.4 are satisfied, and hence system 1.2 is permanent see Figure 1 .Now, we further verify the sufficient conditions for the existence and global stability of positive periodic solutions of periodic system 1.2 .Let us assume that all the coefficients of system 1.2 are periodic and listed in Table 1.
Besides, we choose the positive periodic solution with initial values 0.09, 0.01 , denoted by x * 1 , x * 2 , and the positive solution with initial value 0.08, 0.02 denoted by x 1 , x 2 .By Theorem 3.3, a simple calculation shows that the assumptions in 3.4 and 3.5 hold.   1 , x * 2 and x 1 , x 2 for n ∈ 30, 50 in the same coordinate system.and x 1 0 , x 2 0 0.08, 0.02 for n ∈ 0, 50 , respectively.From Figure 4 c , we can see that periodic system 1.2 has a positive 2-periodic solution which is globally stable.

Acknowledgement
The work is supported by the Innovation Term of Educational Department of Hubei Province in China T200804 , the National Science Foundation of Hubei Province in China 2008CDB068 , and the Innovation Project of Hubei Institute for Nationalities for postgraduate students.We would like to thank the Editor Professor A. Vecchio and the referee for careful reading of the original manuscript and valuable comments and suggestions that greatly improved the presentation of this work.

Figure 2 :
Figure 2: a Time series of x * 1 with x * 1 0 0.09 for n ∈ 0, 50 .b Time series of x 1 with x 1 0 0.08 for n ∈ 0, 50 .c Time series of x * 1 and x 1 for n ∈ 0, 50 in the same coordinate system.

Figure 3 :
Figure 3: a Time series of x * 2 with x * 2 0 0.01 for n ∈ 0, 50 .b Time series of x 2 with x 2 0 0.02 for n ∈ 0, 50 .c Time series of x * 2 and x 2 for n ∈ 0, 50 in the same coordinate system.

Figure 4 :
Figure 4: a Phase portrait of x * 1 and x * 2 with initial value 0.09, 0.01 for n ∈ 0, 50 .b Phase portrait of x 1 and x 2 with initial value 0.08, 0.02 for n ∈ 0, 50 .c Phase portraits of x * 1 , x * 2 and x 1 , x 2 for n ∈ 30, 50 in the same coordinate system.
By using Definition 3.2, it follows that the positive periodic solution {x * 1 n , x * 2 n } of system 1.2 is globally stable.This completes the proof.

Table 1 :
The coefficient values when n is under different conditions.