Almost Periodic Solution of a Discrete Commensalism System

A nonautonomous discrete two-species Lotka-Volterra commensalism system with delays is considered in this paper. Based on the discrete comparison theorem, the permanence of the system is obtained.Then, by constructing a new discrete Lyapunov functional, a set of sufficient conditions which guarantee the system global attractivity are obtained. If the coefficients are almost periodic, there exists an almost periodic solution and the almost periodic solution is globally attractive.


Introduction
The study of the dynamic behaviors of discrete time systems governed by difference equation has become one of the most important topics in mathematic biology during the last decade.There are three main types of interaction between two species: (i) predator-prey, (ii) competition, and (iii) mutualism or symbiosis.Topics such as permanence and global attracitvity of these types were extensively investigated by scholars; see [1][2][3][4][5][6][7][8][9] and the references cited therein.However, commensalism, a typical relationship like epiphyte and plants with epiphyte, has few people to study it.
Sun and Wei [10] proposed the following commensalism system: where  1 ,  2 ,  1 ,  2 , and  are all positive constants.By linearization of the system at positive equilibrium, they obtain that its corresponding linearization matrix has two negative eigenvalues; that is, the unique positive equilibrium is a stable node (Type I).Therefore, from First Degree Approximation Theory, we know that the unique positive equilibrium is asymptotically stable without any conditions.
Though much progress has been seen in the traditional Lotka-Volterra model, such models are not well studied in the sense that most results are continuous time cases related.Many authors have argued that the discrete time systems governed by difference equations are more approximate than the continuous ones when the populations have nonoverlapping generations.Discrete time models can also provide efficient computational models for numerical simulations.In a traditional continuous Lotka-Volterra cooperative system, if there is a positive equilibrium, the positive equilibrium is globally stable.But discrete Lotka-Volterra cooperative system cannot be permanent under any conditions.Therefore, the discrete systems are more difficult and complex to deal with compared to the continuous ones.
For biological reasons, we consider system (2) with the following initial conditions: It is not difficult to see that solutions of (2) are well defined for all  ≥ 0 and satisfy () > 0.
The organization of this paper is as follows.In the next section, we show that (2) is permanent.In Section 3, a set of conditions which ensure that ( 2) is globally attractive are obtained.In Section 4, we will show that the almost periodic solutions are globally attractive.In Section 5, an example together with its numeric simulation is given to illustrate the feasibility of the main result.

Permanence
We first introduce some lemma.

Almost Periodic Solution
The main purpose of this paper is to study the existence of a globally attractive almost periodic sequence solution of system (2).First, we give some definitions and lemmas.
Definition 4 (see [14]).Let  be an open subset of   .A function  :  ×  →   is said to be almost periodic in  uniformly for  ∈ , if, for any  > 0 and any compact set  ⊂ , there exists a positive integer  = (, ), such that any interval of length  contains an integer , for which is called an -translation number of (, ).
Definition 5 (see [15]).The hull of , denoted by (), is defined by for some sequence {  }, where  is any compact set in .
From Theorem 3.4 in [17], we can easily obtain the following lemma.

Lemma 8. If each hull equation of system (2) has a unique strictly positive solution, then the almost periodic difference system (2) has a unique strictly positive almost periodic solution.
Under the consumption of {  ()}, {  ()} ( = 1, 2), and {()} being bounded nonnegative almost periodic sequences, we have the following Theorem.
Theorem C. Assume that ( 1 ) and ( 2 ) hold; then the almost periodic difference system (2) admits a unique almost periodic sequence solution which is globally attractive.
Proof of Theorem C. By Lemma 7, we only need to prove that each hull equation of (2) has a unique strictly positive solution.Suppose ((), ()) is any positive solution of hull equation of (72).First, we prove that the hull equation of ( 2) has at least one strictly positive solution.By Theorem A, we have which gives Let  be an arbitrary small positive number.It follows from (73) that there exists a positive integer  0 such that, for all  >  0 , we have  1 −  ≤ () ≤ gives us We can easily see that (( 1 ()), ( 1 ())) is a solution of (72) and  1 −  ≤  1 () ≤  1 + ,  2 −  ≤  1 () ≤  2 +  for  ∈ .Since  is an arbitrary small positive number, it follows that  1 ≤  1 () ≤  1 ,  2 ≤  1 () ≤  2 for  ∈ ; that is, Hence each hull equation of almost periodic difference system (2) has at least one strictly positive solution.Now we prove the uniqueness of the strictly positive solution of each hull equation (72).Suppose that the hull equation has two arbitrary strictly positive solutions ( * (),  * ()) and ( * 1 (),  * 1 ()).From Theorem 1.3 in [12], we easily obtain that the hull equation of the second equation of (2) has a unique almost periodic solution which is globally attractive under condition ( 1 ).
Similar to the proof of Theorem B, we define a Lyapunov functional where (80) Calculating the difference of  * () along the solution of the hull equation (72) and the uniqueness of the second equation of (72), we have From (81), we can see that  * () is a nonincreasing function on .Summating both sides of the above inequality from  to 0, we obtain for  < 0. ( Denote  = (1/ (84) It follows from (79) and the above inequalities that  * () ≤ (/) = ,  < − 1 , so lim  → −∞  * () = 0. Note that  * () is a positive and nonincreasing function on .Then  * () ≡ 0; that is,  * () =  * 1 (), for all  ∈ .Therefore, each hull equation of system (2) has a unique strictly positive solution.
In view of the above discussion, any hull equation of system (2) has a unique strictly positive solution.By Lemma 8, the almost periodic difference system (2) has a unique strictly positive almost periodic solution (( * ()), ( * ())).Similar to the proof of Theorem B, we can obtain lim  → ∞ |() −  * ()| = 0, where () is any positive solution of system (2).Therefore, the almost periodic system (2) has a unique strictly positive almost periodic solution which is globally attractive.This completes the proof of Theorem C.

Numerical Simulations
In this section, we give an example to check the feasibility of our result.
It is easy to calculate that  2 ≈ 5.4498,  2 ≈ 0.6249 and  1 ≈ 5.2540,  1 ≈ 0.8608; then (86) It follows from Theorem C that system (85) admits a unique almost periodic sequence solution which is globally attractive (see Figures 1 and 2).

Discussion
In this paper, we investigate an almost periodic discrete commensalism system with discrete delays.By constructing suitable Lyapunov functional, we study the global attractivity