Abstract
This paper discusses a discrete multispecies Lotka-Volterra mutualism system. We first obtain the permanence of the system. Assuming that the coefficients in the system are almost periodic sequences, we obtain the sufficient conditions for the existence of a unique almost periodic solution which is globally attractive. In particular, for the discrete two-species Lotka-Volterra mutualism system, the sufficient conditions for the existence of a unique uniformly asymptotically stable almost periodic solution are obtained. An example together with numerical simulation indicates the feasibility of the main result.
1. Introduction
In this paper, we consider a discrete multispecies Lotka-Volterra mutualism system: where stand for the densities of species at the th generation, represent the natural growth rates of species at the th generation, are the intraspecific effects of the th generation of species on own population, measure the interspecific mutualism effects of the th generation of species on species , and are positive control constants.
A number of scholars have studied the difference system (see [1–8] and the references cited therein) since the discrete time models governed by the difference equation are more appropriate than the continuous ones when the populations have short life expectancy, nonoverlapping generations in the real world.
Recently, as far as the multispecies Lotka-Volterra ecosystem is concerned, Wendi and Zhengyi [1] proposed the following Lotka-Volterra model: By constructing a suitable Lyapunov function and using the finite covering theorem of mathematic analysis, they obtained a set of sufficient conditions which ensure the system to be globally asymptotically stable.
Chen [3] studied the dynamic behavior of the discrete -species Lotka-Volterra competition predator-prey systems: where and . Sufficient conditions which ensure the permanence and the global stability of the systems are obtained; for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the systems are obtained.
At the same time, a few scholars have investigated the mutualism system (see [2, 8–13] in detail). However, to the best of the authors' knowledge, still no scholar has done works on discrete multispecies mutualism system. So we propose the discrete multispecies Lotka-Volterra mutualism system (1).
Notice that the investigation of almost periodic solutions for difference equations is one of the most important topics in the qualitative theory of difference equations due to the applications in biology, ecology, neural network, and so forth (see [7, 14–21] and the references cited therein), and little work has been done previously on an almost periodic version which is corresponding to system (1). Then, we will further investigate the existence of a unique almost periodic solution of system (1) which is globally attractive.
Denote as and the set of integers and the set of nonnegative integers, respectively. For any bounded sequence defined on , define , .
Throughout this paper, we assume the following.(H1), , and are bounded nonnegative almost periodic sequences such that
From the point of view of biology, in the sequel, we assume that . Then, it is easy to see that, for given , the system (1) has a positive sequence solution passing through .
The remaining part of this paper is organized as follows. In Section 2, we will introduce some definitions and several useful lemmas. In Section 3, by applying the theory of difference inequality, we present the permanence results for system (1). In Section 4, we establish the sufficient conditions for the existence of a unique globally attractive almost periodic solution of system (1). In particular, for the discrete two-species Lotka-Volterra mutualism system, the sufficient conditions for the existence of a unique uniformly asymptotically stable almost periodic solution are obtained. The main result is illustrated by an example with a numerical simulation in the last section.
2. Preliminaries
Firstly, we give the definitions of the terminologies involved.
Definition 1 (see [22]). 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 [23]). A sequence is called an asymptotically almost periodic sequence if where is an almost periodic sequence and .
Definition 3 (see [24]). A solution of system (1) is said to be globally attractive if, for any other solution of system (1), one has
Now, we present some results which will play an important role in the proof of the main result.
Lemma 4 (see [25]). If is an almost periodic sequence, then is bounded.
Lemma 5 (see [26]). is an almost periodic sequence if and only if, for any 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 6 (see [23]). is an asymptotically almost periodic sequence if and only if, for any sequence satisfying and as , there exists a subsequence such that the sequence converges uniformly for all as .
Lemma 7 (see [25]). Suppose that and are almost periodic real sequences. Then, and are almost periodic; is also almost periodic provided that for all . Moreover, if is an arbitrary real number, then there exists a relatively dense set that is -almost periodic common to and .
Lemma 8 (see [3, 11, 27]). Assume that satisfies and for , where and are nonnegative sequences bounded above and below by positive constants. Then,
Lemma 9 (see [3, 11, 27]). Assume that satisfies
and , where and are nonnegative sequences bounded above and below by positive constants and . Then,
Consider the following almost periodic difference system:
where , , and is almost periodic in uniformly for and is continuous in . The product system of (13) is the following system:
and Zhang [28] obtained the following lemma.
Lemma 10 (see [28]). 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 (13) such that for , then there exists a unique uniformly asymptotically stable almost periodic solution of (13) which is bounded by . In particular, if is periodic of period , then there exists a unique uniformly asymptotically stable periodic solution of (13) of period .
3. Permanence
In this section, we establish a permanence result for system (1), which can be found by Lemmas 8 and 9.
Proposition 11. Assume that (H1) holds. Then, any positive solution of system (1) satisfies where
Theorem 12. Assume that (H1) holds; then system (1) is permanent.
It should be noticed that, from Proposition 11, we know that the set
is an invariant set of system (1).
The next result tells us that there exist solutions of system (1) totally in the interval of Proposition 11. To be precise, see the following.
Proposition 13. System (1) has a solution satisfying for .
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 Proposition 11 that there exists a positive integer such that Write 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 for . Since is an arbitrary small positive number, it follows that , and hence we complete the proof.
4. Main Result
The main result of this paper concerns the existence of a globally attractive almost periodic solution of system (1).
Theorem 14. Assume that (H1) and
(H2)
hold. Then, system (1) admits a unique almost periodic solution which is globally attractive.
Proof. It follows from Proposition 13 that there exists a solution of system (1) satisfying , . Let be any integer valued sequence such that as . Using the mean value theorem, for , we get
where lies between and . Then,
For convenience, we introduce through
Thus,
Let be an arbitrary positive number. By the almost periodicity of , , and and the boundedness of , it follows from Lemmas 5 and 7 that there exists a positive integer such that, for any and (if necessary, we can choose subsequences of and ),
It follows from (25) and (27)–(29) that, for and ,
Then,
And we have
for and .
Since , for arbitrary , there exists a positive integer such that, for any ,
for .
This combined with (26) gives us
It follows from Lemma 6 that the sequence is asymptotically almost periodic. Thus, we can express as
where are almost periodic in and as . In the following, we show that is an almost periodic solution of system (1).
Define
It follows from (1), (35), and the mean value theorem that
where for some . Thus,
Let
By the boundedness of the almost periodic sequences , , , and and the fact that as , we obtain
We claim that . Otherwise, there exists an integer such that . By the almost periodicity of , , , and , there exists an integer valued sequence such that as and
uniformly for all . Then, we have
as , which contradicts the fact that as . This proves the claim. Hence,
that is, is an almost periodic solution of system (1).
Assume that is the solution of system (1) satisfying (H1). Let
Then, system (1) is equivalent to
Therefore,
where , . To complete the proof, it suffices to show that
In view of (H2), we can choose such that
Let ; then . According to Proposition 11, there exists a positive integer such that
for .
Notice that implies that lies between and ; implies that lies between and . From (46), we get
for .
In view of (50), we get
This implies
Then, (47) holds, and we can obtain
Therefore, system (1) admits a unique almost periodic solution which is globally attractive. This ends the proof of Theorem 14.
In particular, if , we can obtain a discrete two-species Lotka-Volterra mutualism system:
In the following, the main result concerns the existence of a uniformly asymptomatically stable almost periodic solution of system (54).
From Proposition 11, we denote by the set of all solutions of system (54) satisfying , , for all . According to Lemma 10, we first prove that there is a bounded solution of system (54) and then structure a suitable Lyapunov function for system (54).
Proposition 15. Assume that (H1) holds. Then, .
Proof. By an inductive argument, we have from system (54) that
According to Proposition 11, for any solution of system (54) and an arbitrarily small constant , there exists sufficiently large such that
Set to be any positive integer sequence such that as ; we can show that there exists a subsequence of still denoted by , such that , , uniformly in on any finite subset of as , where , , and is a finite number.
As a matter of fact, for any finite subset , , , when is large enough. Therefore, , ; that is, are uniformly bounded for large enough.
Now, for , we can choose a subsequence of such that and uniformly converge on for large enough.
Analogously, for , we can also choose a subsequence of such that and uniformly converge on for large enough.
Repeating the above process, for , we get a subsequence of such that and uniformly converge on for large enough.
Now, we choose the sequence which is a subsequence of denoted by ; then, for all , we obtain that , , uniformly in as . Hence, the conclusion is valid by the arbitrariness of .
Recall the almost periodicity of , , and , for the above sequence , as ; there exists a subsequence denoted by such that
as uniformly on .
For any , we can assume that for large enough. Let , and, by an inductive argument of system (54) from to , we obtain
Thus, it derives that
Let ; we have
Since is arbitrary, we know that is a solution of system (54) on , and
Notice that is an arbitrarily small positive constant; it follows that
Thus, . This completes the proof.
Theorem 16. Assume that (H1) holds; furthermore, , where , and Then, there exists a unique uniformly asymptotically stable almost periodic solution of system (54) which is bounded by for all .
Proof. Denote , . It follows from system (54) that
According to Proposition 15, we can see that the system (64) has a bounded solution satisfying
Thus, , where , .
Define the norm , where . Consider the product system of system (64) as follows:
We assume that , are any two solutions of system (66) defined on ; then, , , 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 , , such that
then,
Let , , ; then, condition (i) of Lemma 10 is satisfied.
Moreover, for any , , we have
where , , and . Thus, condition (ii) of Lemma 10 is satisfied.
Finally, calculating the of along the solutions of system (64), we have
By the mean value theorem, it derives that
where lies between and . Then,
where . By the conditions of Theorem 16, we have , and hence condition (iii) of Lemma 10 is satisfied. So, it follows from Lemma 10 that there exists a unique uniformly asymptotically stable almost periodic solution of system (64) which is bounded by for all ; that is, there exists a unique uniformly asymptotically stable almost periodic solution of system (54) which is bounded by for all . This completed the proof.
5. Numerical Simulations
In this section, we give the following example to check the feasibility of our result.
Example 1. Consider the discrete multispecies Lotka-Volterra mutualism system:
A computation shows that and, moreover, we have that and that . It is easy to see that condition (H2) is satisfied. Hence, there exists a unique globally attractive almost periodic solution of system (74). Our numerical simulations support our results (see Figures 1, 2, and 3).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper, and there is no financial conflict of interests between the authors and the commercial identity.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 61132008) and the Scientific Research Program Funded by Shaanxi Provincial Education Department of China (no. 2013JK1098).