Permanence and Positive Periodic Solutions of a Discrete Delay Competitive System

A discrete time non-autonomous two-species competitive system with delays is proposed, which involves the influence of many generations on the density of species population. Sufficient conditions for permanence of the system are given. When the system is periodic, by using the continuous theorem of coincidence degree theory and constructing a suitable Lyapunov discrete function, sufficient conditions which guarantee the existence and global attractivity of positive periodic solutions are obtained. As an application, examples and their numerical simulations are presented to illustrate the feasibility of our main results.


Introduction
In recent years, the application of theories of functional differential equations in mathematical ecology has developed rapidly.Various delayed models have been proposed in the study of population dynamics, ecology, and epidemic.In fact, more realistic population dynamics should take into account the effect of delay.Also, delay differential equations may exhibit much more complicated dynamic behaviors than ordinary differential equations since a delay could cause a stable equilibrium to become unstable and cause the population to fluctuate see 1 .One of the famous models for dynamics of population is the delay Lotka-Volterra competitive system.Owing to its theoretical and practical significance, various delay competitive systems have been studied extensively see 2-8 .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 9-11 have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have non-overlapping generations.Discrete time models can also provide efficient computational models of continuous models for numerical simulations.Therefore, the dynamic behaviors of population models governed by difference equations have been studied by many authors, see 12-18 and the references cited therein.Noting that some studies of the dynamics of natural populations indicate that the density-dependent population regulation probably takes place over many generations 19, 20 , many authors have discussed the influence of many past generations on the density of species population and discussed the dynamic behaviors of competitive, predator-prey, and cooperative systems see 21-24 .Motivated by the above work [19][20][21][22][23][24] , in this paper we will investigate the following discrete time non-autonomous two-species competitive system with delays: 1.1 with the initial conditions x i −l ≥ 0, x i 0 > 0, l 0, 1, . . ., m, i where x i k represents the density of population x i at the kth generation, r i k is the intrinsic growth rate of population x i at the kth generation, a il k measures the intraspecific influence of the k − l th generation of population x i on the density of its own population, and c jl k stands for the interspecific influence of the k −l th generation of population x j on population x i , i, j 1, 2 and i / j.The coefficients {r i k }, {a il k }, and {c il k } i 1, 2 are bounded nonnegative sequences.The exponential form of the equations in system 1.1 ensures that any forward trajectory { x 1 k , x 2 k } of system 1.1 with initial conditions 1.2 remains positive for all k ∈ {0, 1, 2, . ..}.For the investigations of some continuous versions of 1.1 we refer to 8, 25, 26 and the references cited therein.
The principle aim of this paper is to study the dynamic behaviors of system 1.1 , such as permanence, existence, and global attractivity of positive periodic solutions.To the best of our knowledge, no work has been done for the discrete non-autonomous difference system 1.1 .The paper is organized as follows.In Section 2, we obtain sufficient conditions which guarantee the permanence of system 1.1 .In Section 3, a good understanding of the existence and global attractivity of positive periodic solutions of system 1.1 is gained by using the method of coincidence degree theory and a Lyapunov discrete function.Some illustrative examples are given to demonstrate the feasibility of the obtained results in Section 4. To do this, we need to give the following notations and Definitions 1.1 and 1.2.
For the simplicity and convenience of exposition, throughout this paper we let Z, Z , R , and R 2 denote the sets of all integers, nonnegative integers, nonnegative real numbers and two-dimensional Euclidian vector space, respectively.Meanwhile, we denote that b * sup k∈Z b k , b * inf k∈Z b k for any bounded sequence b k .Definition 1.1.System 1.1 is said to be permanent if there exist positive constants m i and M i such that

Permanence
In this section, we will establish sufficient conditions for the permanence of system 1.1 .To do this, we first give two lemmas which will be useful for establishing our main result in this section.
Lemma 2.1 see 27, Lemma 1 .Assume that {x k } satisfies x k > 0 and Before stating Theorem 2.3, for the sake of convenience, we set where i, j 1, 2, i / j, ε > 0 is a sufficiently small constant.
We are now in a position to state our main result of this section on the permanence of system 1.1 .

Theorem 2.3. If the following assumptions:
The following two steps are considered.
Step 1.According to Definition 1.1, we will prove that any positive solution of system 1.1 satisfies lim sup k → ∞ x i k ≤ M i for i 1, 2.
It follows from the first equation of system 1.1 that We will make a convention that that is, in other words, Consequently, we have

2.11
By Lemma 2.1, we can derive that lim sup

2.12
Similar to the above argument, we can verify that lim sup

2.13
Step 2. By a similar procedure to Step 1, we will prove that any positive solution of system 1.1 satisfies lim inf k → ∞ x i k ≥ m i , where

2.14
For any sufficiently small ε > 0, according to 2.5 , there exists a positive integer k 0 such that x 1 k ≤ M 1 ε for all k ≥ k 0 .Thus, for k ≥ k 0 m, it follows from the first equation of system 1.1 that Therefore, for all l 0, 1, . . ., m and k that is, where Combining 2.5 and 2.17 with the first equation of system 1.1 leads to

2.19
And hence, by applying Lemma 2.2 and letting ε → 0, it follows from 2.5 -2.6 and 2.19 that lim inf

2.20
Analogously, from the second equation of system 1.1 , we can verify that lim inf

2.21
The proof of Theorem 2.3 is completed by combining Steps 1 and 2.

Existence and Global Attractivity of Positive Periodic Solutions
In this section, we will give two main results.We first derive sufficient conditions for the existence of positive periodic solutions of system 1.1 .We further assume that r i , a il , c il : Z → R are positive ω-periodic for system 1.1 , that is, for any k ∈ Z, where ω, a fixed positive integer, denotes the prescribed common period of the parameters in system 1.1 .
In order to obtain sufficient conditions for the existence of positive periodic solutions of system 1.1 , we will use the method of coincidence degree.For convenience, we will summarize in the following a few concepts and results from 28 that will be useful in this section.
Let X and Y be two Banach spaces.Consider an operator equation where L : Dom L ∩ X → Y is a linear operator and λ is a parameter.Let P and Q denote two projectors such that Recall that a linear mapping L : Dom L ∩ X → Y with Ker L L −1 0 and Im L L Dom L will be called a Fredholm mapping if the following two conditions hold: i Ker L has a finite dimension; ii Im L is closed and has a finite codimension.
Recall also that the codimension of Im L is the dimension of Y/ Im L, that is, the dimension of the cokernel coker L of L.
When L is a Fredholm mapping, its index is the integer Ind L dim Ker L − codim Im L.
We will say that a mapping N is L-compact on Ω if the mapping QN : Ω → Y is continuous.QN Ω is bounded and K p I − Q N : Ω → X is compact, that is, it is continuous and K p I − Q N Ω is relatively compact, where K p : Im L − Dom L ∩ Ker P is an inverse of the restriction L p of L to Dom L ∩ Ker P , so that LK p I and K p I − P .Lemma 3.1 see 28, Continuation Theorem .Let X and Y be two Banach spaces and let L be a Fredholm mapping of index zero.Assume that N :

Then the equation Lx Nx has at least one solution in
In what follows, we will use the following notations: where {f k } is an ω-periodic sequence of real numbers defined for k ∈ Z.
For a a 1 , a 2 ∈ R 2 , define |a| max{a 1 , a 2 }.Let l ω ⊂ l 2 denote the subspace of all ωperiodic sequences equipped with the usual supremum norm • , that is, Then it follows that l ω is a finite dimensional Banach space.Let

3.8
Then it follows that l ω 0 and l ω c are both closed linear subspaces of l ω and 3.9 We are now in a position to state one of the main results of this section on the existence of positive periodic solutions of system 1.1 .

3.10
Then system 1.1 has at least one positive ω-periodic solution.
Proof.We first make the change of variables By substituting 3.11 into system 1.1 , we can get

3.12
It is easy to see that if system 3.12 has one ω-periodic solution, then system 1.1 has one positive ω-periodic solution.Therefore, to complete the proof, it is only to show that system 3.12 has at least one ω-periodic solution.
Set X Y l ω .Denote by L : X → X the difference operator given by Ly { Ly k } with and N : X → X as follows: It is not difficult to show that P and Q are continuous projectors such that Furthermore, the inverse to L K p : Im L → Dom L ∩ Ker P exists and is given by ω − s y s .

3.18
Then QN : X → Y and K p I − Q N : X → X are given by Ny s .

3.19
In order to apply Lemma 3.1, we need to search for an appropriately open, bounded subset Ω.

3.20
Suppose that y {y k } { y 1 k , y 2 k } ∈ X is a solution of 3.20 for a certain λ ∈ 0, 1 .Summing both sides of 3.20 from 0 to ω − 1 with respect to k, we can derive

3.21
Since y {y k } ∈ X, there exist ξ i ∈ I ω such that

3.22
It follows from 3.21 that a il exp y i ξ i , i,j 1, 2, i / j,

which implies
where A i def m l 0 a il , besides, from 3.20 and 3.21 By 3.24 , 3.25 , and Lemma 3.2, we have

3.26
On the other hand, there also exist η i ∈ I ω such that

3.27
In view of 3.21 , we can obtain c jl ω, i, j 1, 2, i / j.

3.30
That is,

3.35
If system 3.34 does not have one solution, then it is obvious that This implies that condition b in Lemma 3.1 is satisfied.Now we prove that condition c in Lemma 3.1 holds.Define Φ : Dom L × 0, 1 → X as follows: where μ is a parameter with μ ∈ 0, 1 .When y { y 1 , y 2 } ∈ ∂Ω ∩ Ker L ∂Ω ∩ R 2 , y 1 , y 2 is a constant vector in R 2 with y H.We will show that { y 1 , y 2 } ∈ ∂Ω ∩ Ker L, Φ y 1 , y 2 , μ / 0. If the conclusion is not true, then there is a constant vector y 1 , y 2 ∈ R 2 with y H satisfying Φ y 1 , y 2 , μ 0, that is, 1 exp y 1 0.

3.38
A similar argument to the above shows that y < H, which is a contradiction.Using the property of topological degree and taking J I :

3.39
Obviously, the following equations: Finally, we will show that N is L-compact on Ω.For any y ∈ Ω, we have
It is easy to see that

3.43
For any y ∈ Ω, k 1 , k 2 ∈ I ω , without loss of generality, let k 2 > k 1 , then we have

3.44
Thus, the set {K p I − Q Ny | y ∈ Ω} is equicontinuous and uniformly bounded.By using the Arzela-Ascoli theorem, we see that By now, we know that Ω verifies all the requirements in Lemma 3.1 and then system 3.12 has at least one ω-periodic solution.By the medium of 3.11 , we derive that system 1.1 has at least one ω-periodic solution.This completes the proof of Theorem 3.3.
Next, by constructing a suitable Lyapunov-like discrete function, we further investigate the global attractivity of positive periodic solutions of system 1.1 .Theorem 3.4.In addition to 3.10 , assume further that there exists a constant η > 0 such that where M i (i=1,2) are defined in 2.5 and

3.46
Then the positive periodic solution of system 1.1 is globally attractive.
Proof.Let { x 1 k , x 2 k } be a positive periodic solution of system 1.1 .To finish the proof of Theorem 3.4, we will consider the following two steps.
Step 1.Let V 11 k | ln x 1 k − ln x 1 k |, then it follows from the first equation of system 1.1 that

3.47
By the mean value theorem, we have where ξ 1 k lies between x 1 k and x 1 k .Then we have ln

3.50
And hence it follows from 3.47 and 3.50 that

3.52
For the sake of convenience, we will make a convention that any bounded sequence b i .By a simple calculation, it derives that

3.53
Now, we are in a position to define Therefore, it follows from 3.51 and 3.53 that

3.55
By a similar argument, we can define V 2 k by where

3.57
Then it is easy to derive that

3.58
where ξ 2 k is between x 2 k and x 2 k .Now we can define a Lyapunov-like discrete function V k by

3.59
It is easy to see that V k ≥ 0 for all k ∈ Z and V k 0 m < ∞.For the arbitrariness of ε and by 3.45 , we can choose a small enough ε > 0 such that min a i0 * , Discrete Dynamics in Nature and Society Therefore, it follows from 3.55 -3.60 that

3.61
Thus, by 3.61 we obtain

3.63
According to Definition 1.2, this result implies that the positive periodic solution { x 1 k , x 2 k } is globally attractive.This completes the proof of Theorem 3.4.

Example and Numerical Simulation
In this paper, a discrete time non-autonomous two-species competitive system with delays is investigated.By using difference inequality technique, continuous theorem of coincidence degree theory, and Lyapunov discrete function, sufficient conditions for the permanence of system 1.1 and the existence and global attractivity of positive periodic solutions of system 1.1 are obtained, respectively.To substantiate our analytical results, we construct the following example:

4.1
We first verify the sufficient conditions for permanence of system 4.In the following, we will consider the existence and global attractivity of positive periodic solutions of system 4.1 .We assume that

4.4
It is easy to verify that assumptions 3.10 of Theorem 3.3 are satisfied.Figure 2 shows that system 4.1 has a 2-periodic solution { x 1 k , x 2 k }, which implies that the two species x 1 and x 2 can coexist.Furthermore, a calculation can show that the assumptions 3.34 of Theorem 3.4 are satisfied, so { x 1 k , x 2 k } is globally attractive, that is, any
where α is a positive constant and k 1 ∈ Z .Then B i , A i , and H i in 3.33 are independent of λ, respectively.Denote H H 1 H 2 h 0 , where h 0 is taken sufficiently large such that any solution { y 1 , y 2 } of the system of algebraic equations max{| y 1 |, | y 2 |} < h 0 If system 3.34 has at least one solution .Let Ω def {y : y 1 , y 2 ∈ X | y < H}, thus condition a in Lemma 3.1 holds.When y ∈ ∂Ω ∩ Ker L ∂Ω ∩ R 2 , y { y 1 , y 2 }, y 1 , y 2 is a constant vector in R 2 with y 1 , x 1 and x * 2 , x 2 .a Time series of x * 1 , x 1 with x * 2 k } of system 4.1 tends to { x 1 k , x 2 k } see Figure 3 .From Figure 3 a , we see that x * 1 with x * 1 −1 0.03 and x * 1 0 0.0662 will tend to x 1 with x 1 −1 0.02 and x 1 0 0.0668.Similarly, from Figure 3 b , we see that x * 2 with x * positive solution { x * 1 k , x *