Abstract
We consider an almost periodic multispecies discrete Lotka-Volterra mutualism system with feedback controls. We firstly obtain the permanence of the system by utilizing the theory of difference equation. By means of constructing a suitable Lyapunov function, sufficient conditions are obtained for the existence of a unique positive almost periodic solution which is uniformly asymptotically stable. An example together with numerical simulation indicates the feasibility of the main result.
1. Introduction
The mutualism system [1] has been studied by more and more scholars. Topics such as permanence, global attractivity, and global stability of continuous differential mutualism system were extensively investigated (see [2–7] and the references cited therein). In addition, some recent attention was on the permanence and global stability of discrete mutualism system, and many excellent results have been derived (see [3, 8–13] and the references cited therein).
Recently, the multispecies descrete Lotka-Volterra ecosystem is increasingly concerned (see [12–21] and the references cited therein). Yang and Li [19] studied a discrete nonlinear N-species cooperation system with time delays and feedback controls. Sufficient conditions which ensure the permanence of the system are obtained. Li and Zhang [21] studied a discrete n-species cooperation system with time-varying delays and feedback controls. Sufficient conditions are obtained for the permanence of the system.
In real world phenomenon, the environment varies due to the factors such as seasonal effects of weather, food supplies, mating habits, and harvesting. So it is usual to assume the periodicity of parameters in the systems. However, if the various constituent components of the temporally nonuniform environment are with incommensurable (nonintegral multiples) periods, then one has to consider the environment to be almost periodic since there is no a priori reason to expect the existence of periodic solutions. For this reason, the assumption of almost periodicity is more realistic, more important, and more general when we consider the effects of the environmental factors. In fact, there have been many nice works on the positive almost periodic solutions of continuous and discrete dynamics model with almost periodic coefficients (see [7, 12, 13, 22–28] and the references cited therein).
As we all known, investigating the almost periodic solutions of discrete population dynamics model with feedback control has more extensively practical application value (see [11, 22, 23, 29–34] and the references cited therein). Wang [22] considered a nonlinear single species discrete with feedback control and obtained some sufficient conditions which assure the unique existence and global attractivity of almost positive periodic solution. Niu and Chen [30] studied a discrete Lotka-Volterra competitive system with feedback control and obtain the existence and uniqueness of the almost periodic solution which is uniformly asymptotically stable.
Motivated by above, in this paper, we are concerned with the following multispecies discrete Lotka-Volterra mutualism system with feedback controls where , , , , , and are bounded nonnegative almost periodic sequences such that , . For any bounded sequence defined on , and .
By the biological meaning, we will focus our discussion on the positive solutions of system (1). So it is assumed that the initial conditions of system (1) are the form: One can easily show that the solutions of system (1) with the initial condition (3) are defined and remain positive for all .
To the best of our knowledge, this is the first paper to investigate the uniformly asymptotical stability of positive almost periodic solution of multispecies discrete Lotka-Volterra mutualism system with feedback controls. The aim of this paper is to obtain sufficient conditions for the existence of a unique uniformly asymptotically stable almost periodic solution of system (1) with initial condition (3), by utilizing the theory of difference equation and constructing a suitable Lyapunov function and applying the analysis technique of papers [11, 22, 29, 31, 32].
The remaining part of this paper is organized as follows. In Section 2, we will introduce some definitions and several useful lemmas. In the next section, we establish the permanence of system (1). Then, in Section 4, we establish sufficient conditions to ensure the existence of a unique positive almost periodic solution, which is uniformly asymptotically stable. The main result is illustrated by an example with a numerical simulation in the last section.
2. Preliminaries
First, we give the definitions of the terminologies involved.
Definition 1 (see [35, 36]). 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 interval of length contains an integer with is called an -translation number of .
Definition 2 (see [37]). Let be an open subset of . 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 3 (see [38]). The hull of , denoted by , is defined by for some sequence , where is any compact set in .
Now, we state several lemmas which will be useful in proving our main result.
Lemma 4 (see [39]). 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.
Lemma 5 (see [9]). Assume that satisfies and for , where and are nonnegative sequences bounded above and below by positive constants. Then
Lemma 6 (see [9]). Assume that satisfies and , where and are nonnegative sequences bounded above and below by positive constants and . Then
Lemma 7 (see [40]). Assume that and , and further suppose that Then, for any integer , Specifically, if and is bounded above with respect to , then
Lemma 8 (see [40]). Assume that and , and further suppose that
Then, for any integer ,
Specifically, if and is bounded below with respect to , then
Consider the following almost periodic difference system:
where , , and is almost periodic in uniformly for and is continuous in . The product system of (18) is the following system:
and Zhang [38] obtained the following theorem.
Theorem 9 (see [38]). Suppose that there exists a Lyapunov function defined for , , and satisfying the following conditions:(i), where with ;(ii), where is a constant;(iii), where is a constant, and Moreover, if there exists a solution of (18) such that for , then there exists a unique uniformly asymptotically stable almost periodic solution of (18) which is bounded by . In particular, if is periodic of period , then there exists a unique uniformly asymptotically stable periodic solution of (18) of period .
3. Permanence
In this section, we establish the permanence result for system (1).
Theorem 10. Assume that the conditions (2) and (3) hold; furthermore, and then system (1) is permanent; that is, there exist positive constants , , , and () which are independent of the solutions of system (1), such that, for any positive solution of system (1), one has
Proof. Let be any positive solution of system (1). From the first equation of system (1), it follows that
Thus, as a direct corollary of Lemma 5, according to (23), one has
For any small positive constant , from (24), it follows that there exists a positive constant such that, for all and ,
For , from (25) and system (1), we have
Then, as a direct corollary of Lemma 7, according to (26), one has
Letting , it follows that
Thus, there exists a positive integer , and we have, for ,
For , from (29) and system (1), we have
Assuming that , for any , there exists a positive integer such that for . Thus, as a direct corollary of Lemma 6, according to (30), one has
where
Letting , it follows that
where
From (33), for any , there exists a positive integer such that
for .
From (35) and system (1), we have
Then, as a direct corollary of Lemma 8, according to (36), one has
Letting , it follows that
Then, (24), (28), (33), and (38) show that system (1) is permanent. The proof is completed.
According to Theorem 9, we first prove that there is a bounded solution of system (1), and then construct a suitable Lyapunov function for system (1).
We denote by the set of all solutions of system (1) satisfying and for all .
Proposition 11. Assume that the conditions (2), (3), and (21) hold. Then .
Proof. By the almost periodicity of , , , , , , and , there exists an integer valued sequence with as such that
Let be an arbitrary small positive number. It follows from Theorem 10 that there exists a positive integer such that
Write and for and . For any positive integer , it is easy to see that there exists a sequence such that the sequence has a subsequence, denoted by again, converging on any finite interval of as . Thus we have a sequence such that
This, combined with
gives us
We can easily see that (, ) is a solution of system (1) and and for . Since is an arbitrary small positive number, it follows that and and hence we complete the proof.
4. Stability of Almost Periodic Solution
In this section, by constructing a nonnegative Lyapunov function, we will obtain sufficient conditions for uniform asymptotical stability of positive almost periodic solution of system (1).
Theorem 12. Assume that the conditions (2), (3), and (21) hold; moreover, , where . Then there exists a unique uniformly asymptotically stable almost periodic solution of system (1) which is bounded by for all .
Proof. Let , . From system (1), we have
From Proposition 11, we know that system (45) has a bounded solution satisfying
Hence, and , where and , .
For , we define the norm .
Consider that the product system of (45) is
We assume that and , are any two solutions of system (45) defined on ; then, and , where and , , , .
Let us construct a Lyapunov function defined on as follows:
It is obvious that the norm is equivalent to ; that is, there are two constants and , such that
and then,
Let , , ; then, condition (i) of Theorem 9 is satisfied.
Moreover, for any , , we have
where , , , and . Thus, condition (ii) of Theorem 9 is satisfied.
Finally, calculating the of along the solutions of system (47), we have
By the mean value theorem, it derives that
where lies between and . Then, we have
Then, we have
where
Hence, we have
where . That is, there exists a positive constant such that
From , the condition (iii) of Theorem 9 is satisfied. Then, according to Theorem 9, there exists a unique uniformly asymptotically stable almost periodic solution of (45) which is bounded by for all . It means that there exists a unique uniformly asymptotically stable almost periodic solution of system (1) which is bounded by for all . This completed the proof.
Remark 13. If , the conditions of Theorem 12 can be simplified. Therefore, we have the following result.
Corollary 14. Let , and assume further that , where , . Then system (1) admits a unique uniformly asymptotically stable almost periodic solution which is bounded by for all .
5. Example and Numerical Simulation
In this section, we give the following example to check the feasibility of our result.
Example 15. Consider the following almost periodic discrete Lotka-Volterra mutualism system with feedback controls:
By simple computation, we derive
Then
Also it is easy to see that the conditions of Theorem 12 are verified. Therefore, system (60) has a unique positive almost periodic solution which is uniformly asymptotically stable. Our numerical simulations support our results (see Figure 1).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper, and there are no financial conflict of interests between the authors and the commercial identity.
Acknowledgments
The authors are grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. Also, this work is supported by National Natural Science Foundation of China (nos. 61132008 and 11301415), Natural Science Basic Research Plan in Shaanxi Province of China (no. 2013JQ1020) and Scientific Research Program Funded by Shaanxi Provincial Education Department of China (no. 2013JK1098).