Abstract
A discrete almost periodic competitive system with delay is proposed and analyzed. The system admits a unique almost periodic solution, which is shown to be uniformly asymptotically stable by using the method of Lyapunov function. Some specific numerical examples are provided to verify our analytical results.
1. Introduction
As we know, a discrete time model governed by difference equations is more suitable than a corresponding continuous version when the populations have a short-life expectancy and nonoverlapping generations and can also provide efficient computation for numerical simulation (see [1–7]). On the other hand, the coefficients of model are changed owing to environmental variation. The assumption of almost periodicity of the coefficients is a way of incorporating the time-dependent variability of the environment if the various components of the environment are with incommensurable periods. One of the important ecological problems associated with the investigation of populations interaction in an almost periodic environment is the positive almost periodic solution which plays the significant role played by the equilibrium of the autonomous model (see [8–14]). In this contribution, we discuss the positive almost periodic solutions of a delay almost periodic competitive system in discrete time, and our motivation comes from the works of [8, 9, 15].
Firstly, let us introduce the following autonomous differential model which was proposed by Ayala et al. [15]: where and stand for the population densities of two competing species. and are the intrinsic growth rates. and represent the interspecific competitive effects, . Assume that a species needs some time to mature and the competition occurs after some time lag required for maturity of the species; a revised version was introduced by Gopalsamy [16] Furthermore, considering the biological parameters naturally being subject to almost periodic fluctuation in time and the influence of many generations on the density of species population, we establish the following model: , and the initial conditions satisfy Here and are, respectively, the densities and intrinsic growth rates of species at the th generation. and measure the interspecific influence of the th generation of species on species . The delays and are positive integers and the coefficients , , and are bounded positive almost periodic sequences, .
The rest part of this paper is organized as follows. In Section 2, some preliminaries are given. Sufficient conditions for the uniformly asymptotic stability of a unique positive almost periodic solution for the system are established in Section 3. In final section, some specific numerical examples are carried out to illustrate the feasibilities of our theoretical results.
2. Preliminaries
This section is concerned with some notations, definitions, and lemmas which will be used for our main results.
Let , , , and be, respectively, the sets of real numbers, nonnegative real numbers, integers, and nonnegative integers. Assign and which denote the cone of 2-dimensional and -dimensional real Euclidean space, respectively. Denote , , and then , where is defined in (4). For simplicity in the following discussion, we use the notations , , where is an almost periodic sequence.
Definition 1 (see [12]). A 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 discrete interval of length contains an integer such that , . is called the-translation number of .
Definition 2 (see [12]). Let , where is an open set in . is said to be almost periodic in uniformly for , if for any and any compact set in , there exists a positive integer such that any interval of length contains an integer for which and is called the -translation number of .
Lemma 3 (see [17]). is an almost periodic sequence if and only if for any sequence there exists a subsequence such that converges uniformly on as .
Zhang and Zheng [12] considered the following almost periodic delay difference system: where , , with , is almost periodic in uniformly for and is continuous in , while is defined as for all . The product system of (7) is as follows: A discrete Lyapunov function of (8) is a function which is continuous in its second and third variables. Define the difference of along the solution of system (8) by where is a solution of system (8) through .
Lemma 4 (see [12]). Suppose that there exists a Lyapunov function satisfying the following conditions.(I), where with and is continuous, increasing in .(II), where is a constant.(III), where is a constant.Moreover, if there exists a solution of system (7) such that for all , then there exists a unique uniformly asymptotically stable almost periodic solution of system (7) which satisfies for all . In particular, if is periodic of period , then system (7) has a unique uniformly asymptotically stable periodic solution of period .
Remark 5 (see [9]). Condition (III) of Lemma 4 can be replaced by
(III) , where is continuous, and for .
Lemma 6 (see [18]). Assume that and where is a bounded positive sequence and is a positive constant. Then
Lemma 7 (see [18]). Assume that satisfies and , where is a bounded positive sequence and is a positive constant such that and . Then
Denote
Lemma 8. If , then any positive solution of system (3) satisfies , .
Proof. According to the first equation of system (3), one has It follows from Lemma 6 that Analogously, from the second equation of system (3) we obtain that Assigning a positive constant arbitrarily small, it follows from (16) and (17) that there exists a large enough such that for all , From the first equation of system (3), for , Using Lemma 7, for , one has Setting , it follows that Similar to the above argument, from the second equation of system (3) we can obtain that This completes the proof.
Denote by
Lemma 9. If , then , where are defined in (14).
Proof. Since , , and , are almost periodic sequences, there exists a positive integer sequence with as such that as for . By (16), (17), (21), and (22), for any sufficiently small , there exists a positive integer such that for all , Assign For any positive integer , it is obvious that there exists a sequence such that the sequence has a subsequence, also denoted by , converging on any definite interval of as . Thus, there is a sequence satisfying which, together with (24), yields that we have from (24), (27), and (28) that It is easy to see that is a solution of system (3) and for . Since is small enough, we derive that , for . This completes the proof.
3. Main Result
In this section, we focus on the result of the uniformly asymptotic stability of positive almost periodic solutions of system (3).
Denote
Theorem 10. If and , system (3) has a unique positive almost periodic solution which is uniformly asymptotically stable, where are defined in (14).
Proof. We make the change of variables
and then system (3) can be rewritten as
It follows from Lemma 9 that there exists a bounded solution of system (32) satisfying
which implies that , , where , . Assign
are two solutions of system (32) defined on , where
Definning
where , we have
where .
Let us consider the associate product system of system (32)
Construct the following Lyapunov function defined on
Apparently,
where
Denote
and thus condition (I) in Lemma 4 is satisfied.
For any , we obtain that
Hence, for any , we obtain, by (44), that
where and are defined in (42). That is, condition (II) in Lemma 4 is also satisfied.
It follows from the mean-value theorem that we find
where lie between and and , all lie between and , respectively. Then
This one together with system (38) and (46) yields that
Combining with (48) and calculating the of along the solution of (38), one has
where . Since , . Denote , ; therefore, the condition in Remark 5 is satisfied. From Lemma 4 and Remark 5, system (32) has a unique uniformly asymptotically stable almost periodic solution denoted by , which is equivalent to saying that system (3) has a unique uniformly asymptotically stable positive almost periodic solution denoted by . This proof of Theorem 10 is completed.
If the coefficients , , and are bounded positive periodic sequences, then system (3) becomes a periodic version. Applying Lemma 4 and Theorem 10, Corollary 11 is obtained directly.
Corollary 11. Periodic system (3) shows a unique positive periodic solution which is uniformly asymptotically stable under the same assumptions of Theorem 10.
4. Numerical Simulations
In this section, we give two specific numerical examples to verify our analytical results, that is, Theorem 10 and Corollary 11.
Example 12. Consider the following delay discrete almost periodic competitive system: A computation shows that and then we have Moreover, It is easy to see that the assumptions of Theorem 10 are satisfied; that is to say, system (50) has a unique positive almost periodic solution denoted by which is uniformly asymptotically stable (see Figure 1), and the two-dimensional and three-dimensional phase portraits are displayed in Figure 2, respectively. In Figure 3, any positive solution denoted by tends to the above almost periodic solution .
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Example 13. Consider the following delay discrete periodic competitive system: Analogously, we can see that system (54) satisfies the assumptions of Theorem 10. From Corollary 11, there exists a unique positive periodic solution of system (54) which is uniformly asymptotically stable. From Figure 4, system (54) shows a positive periodic solution denoted by , and the two-dimensional and three-dimensional phase portraits are displayed in Figure 5, respectively. Figure 6 shows that any positive solution denoted by tends to the above periodic solution .
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Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the referees for their valuable comments that greatly improved the presentation of this paper. They are also grateful to Professor Zhijun Liu for his suggestions and doctoral student Qinglong Wang for his help in numerical simulations.