Research Article | Open Access
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.
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 [1–9] 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  proposed the following commensalism system: where , , , , 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.
In this paper, we study the following discrete commensalism system:where and represent the sizes or the densities of species and at th generation, respectively. Parameters , , and are bounded nonnegative almost periodic sequences (for the definition, see Section 4) such that
Here, denote the sets of integers and nonnegative integers by and , respectively, and . For any bounded sequence defined on , denote and .
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 and satisfy .
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.
We first introduce some lemma.
Lemma 1 (see ). Assume that satisfies and for , , where is a positive constant. Then
Lemma 2 (see ). Assume that satisfies , , , and , where is a positive constant such that and . Then
Theorem A. For every positive solution of system (2), one has where
Proof of Theorem A. Let be an arbitrary solution of system (2). From the second equation of system (2), it follows that From the above inequality, one has Substituting (10) into the second equation of (2), it follows that From Lemma 1 and (11), one has For any positive constant small enough, it follows from (12) that there exists a such that, for all ,It follows from (13) and the second equation of (2), for , thatNoting the fact that , for , we obtain Therefore, from (14) and (15), we have By using (16), one could easily obtain that Substituting (17) into the second equation of (2), we have Noting that , Thus, according to Lemma 2 and (18), one has Setting , it follows that where .
For the above , there exists a such that, for all ,From the first equation of system (2) and (22), for all , From the above inequality, we have Substituting (24) into the first equation of (2), it follows that From Lemma 1 and (25), we have Setting , it follows that where .
There exists a such that, for all , It follows from (28) and the first equation of (2), for , thatNoting the fact that , for , we obtain such that Therefore, from the above inequality and (29), one has Using (32), we could easily obtain that Substituting (33) into the first equation of (2), for , deduces Noting the fact that , for , we obtain According to (34) and Lemma 2, one has Setting , it follows that where and . This completes the proof of Theorem A.
3. Global Attractivity
Proof of Theorem B. Let and be any positive solutions of system (2).
According to the proof of Theorem of , holds under the condition and there exists a positive constant such that . Now we just need to proof .
Condition implies that Therefore, we can choose and small enough such that For any solutions and of system (2), it follows from Theorem A that For the above , there exists an integer such that, for all , Using the mean value theorem, we get where lies between and .
Let Then, from system (2), we obtain Therefore, we have where .
Define Thenwhere .
Hence, from (43) and (48) and , we have Let Then Define Then it follows from (49) and (51) that where From and (43), we have Let Then Define Then it follows from (53), (55), and (57) that where Therefore, for all , From (59) and (61), it follows that Summating both sides of the above inequalities from to , we have which implies It follows that , , are all bounded. Hence which means that This implies that or . This completes the proof of Theorem B.
4. 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 3 (see ). Sequence is called an almost periodic sequence if the -translation set of , is a relatively dense set in for all ; that is, for any given , there exists an integer such that each interval of length contains an integer with
The integer is called an -translation number of .
Definition 4 (see ). Let be an open subset of . A function is said to be almost periodic in uniformly for , if, for any 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 ). The hull of , denoted by , is defined by for some sequence , where is any compact set in .
Lemma 7 (see ). Suppose that is an almost periodic sequence if and only if, for any integer sequence , there exists a subsequence such that the sequence converges uniformly for all as . Furthermore, the limit sequence is also an almost periodic sequence.
From Theorem in , we can easily obtain the following lemma.
Under the consumption of , , and being bounded nonnegative almost periodic sequences, we have the following theorem.
Theorem C. Assume that and hold; then the almost periodic difference system (2) admits a unique almost periodic sequence solution which is globally attractive.
Let be any integer value sequence such that as . According to Lemma 7, taking subsequence if necessary, we have , , and as for . Then the hull equation of (2) is as follows: By the almost periodic theory, we can conclude that if (2) satisfies and , then the hull equation (72) also satisfies and .
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 such that, for all , we have , . Define