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Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 286031, 18 pages
Almost Periodic Solutions of a Discrete Mutualism Model with Feedback Controls
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
Received 31 October 2009; Accepted 16 March 2010
Academic Editor: Leonid Berezansky
Copyright © 2010 Zheng Wang and Yongkun Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a discrete mutualism model with feedback controls. Assuming that the coefficients in the system are almost periodic sequences, we obtain the existence and uniqueness of the almost periodic solution which is uniformly asymptotically stable.
Two species cohabit a common habitat and each species enhances the average growth rate of the other; this type of ecological interaction is known as facultative mutualism . A two species mutualism model can be described in the following form:
where are continuously differentiable such that
One of the simplest models which satisfies the above assumption is the traditional Lotka-Volterra two species mutualism model, which takes the form
Since the above system could exhibit unbounded solutions [2, 3] and it is well known that in nature, with the restriction of resources, it is impossible for one species to survive if its density is too high. Thus, the above model is not so good in describing the mutualism of two species. Gopalsamy  has proposed the following model to describe the mutualism mechanism:
where denotes the intrinsic growth rate of species and The carrying capacity of species is in the absence of other species, while with the help of the other species, the carrying capacity becomes The above mutualism can be classified as facultative, obligate, or a combination of both. For more details of mutualistic interactions, we refer to [5–9]. Realistic models require the inclusion of the effect of changing environment. This motivate us to consider the following nonautonomous model:
Since many authors [10, 11] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations, then, discrete-timemodels can provide efficient computational types of continuous models for numerical simulations. It is reasonable to study the discrete-time mutualism model governed by difference equations. One of the ways of deriving difference equations modeling the dynamics of populations with nonoverlapping generations is based on appropriate modifications of the corresponding models with overlapping generations [4, 12]. In this approach, differential equations with piecewise constant arguments have been proved to be useful. Following the same idea and the same method in [4, 12], one can easily derive the following discrete analog of (1.5), which takes the form of
The exponential form of (1.6) is more biologically reasonable than that directly derived by replacing the differential by difference in (1.5). Feedback control is the basic mechanism by which systems, whether mechanical, electrical, or biological, maintain their equilibrium or homeostasis. During the last decade, a series of mathematical models have been established to describe the dynamics of feedback control systems [13–17].
In this paper, we are concerned with the following discrete mutualism model with feedback controls:
To the best of our knowledge, though many works have been done for the population dynamic system with feedback controls, most of the works dealt with the continuous time model. For more results about the existence of almost periodic solutions of a continuous time system, we can refer to [18–22] and the references cited therein. There are few works that consider the existence of almost periodic solutions for discrete time population dynamic model with feedback controls. So, our main purpose of this paper is to study the existence and uniqueness of almost periodic solutions for the model (1.7).
Throughout this paper, we assume that() and for are bounded non-negative almost periodic sequences such that and for .
Here, for any bounded sequence , and . By the biological meaning, we focus our discussion on the positive solution of the system (1.7). So it is assumed that the initial conditions of (1.7) are of the form
In this section, we will introduce two definitions and a useful lemma.
Definition 2.1 (see ). A sequence is called an almost periodic sequence if the - translation set of : for all 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 for all , is called the -translation number of .
Definition 2.2 (see ). Let , where is an open set in , is said to be almost periodic in uniformly for , or uniformly almost periodic for short, if for any and any compact set in , there exists a positive integer such that any interval of length contains an integer for which for all and . is called the -translation number of .
Lemma 2.3 (see ). is an almost periodic sequence if and only if for any sequence there exists a subsequence such that converges uniformly on as . Furthermore, the limit sequence is also an almost periodic sequence.
In this section, we establish a persistence result for model (1.7).
Proposition 3.1. Assume that holds. For every solution of (1.7) where
Proof. We first present two cases to prove that
Case 1. By the first equation of model (1.7), and (1.9), we have
Then there exists an such that . So, . Hence, , and
Here we used for . We claim that for .
In fact, if there exists an integer such that , and letting be the least integer between and such that , then and which implies . This is impossible. The claim is proved.Case 2 ( for ). In particular, exists, denoted by . We claim that . By way of contradiction, assume that . Taking . Noting that , hence for , which is a contradiction. This proves the claim.We can prove that in the similar way. Therefore, for each , there exists a large enough integer such that , whenever . The proof of is very similar to that of Proposition 1 in . Here we omit the details here.
Proof. We also present two cases to prove that .
For any which satisfies , according to Proposition 3.1, there exists such that
for . Case 1. There exists a positive integer such that . Note that for , In particular, with , we get which implies that . Then
Let . We claim that for .
By a way of contradiction, assume that there exists a such that . Then , let be the smallest integer such that . Then . The above argument produces that , a contradiction. This proves the claim.Case 2. We assume that for all large . Then exists, denoted by We claim that . By way of contradiction, assume that . Taking − , which is a contradiction, since
Noting that we see that , and . We can easily see that holds. Similarly, we can prove that Thus for any small enough, there exists a positive integer , such that for
The proof of is very similar to that of Proposition 2 in . Here we omit the details.
4. Main Results
For our purpose, we first introduce the following results which are given in Persistence.
Lemma 4.1. Assume that (1.9), , and hold, then where .
In , Zhang considered the following almost periodic difference system
where , and is almost periodic in uniformly for and is continuous in . Related to system (4.2), he also considered the following product system:
and obtained the following theorem.
Theorem 4.2 (see ). Suppose that there exists a Lyapunov functional defined for 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 (4.2) such that for , then there exists a unique uniformly asymptotically stable almost periodic solution of system (4.2) which is bounded by . In particular, if is periodic of period , then there exists a unique uniformly asymptotically stable periodic solution of (4.2) of period .
We denote by the set of all solutions of system (1.7) satisfying for all .
Lemma 4.3. Assume that () and the conditions of Lemma 4.1 hold, then .
Proof. It is now possible to show by an inductive argument that system (1.7) leads to
for From Lemma 4.1, for any solution of (1.7) with initial condition (1.9) satisfies (4.2). Hence, for any , there exist , if is sufficiently large, we have
Let be any integer-valued sequence such that as , we claim that there exists a subsequence of , we still denote by , such that
uniformly in on any finite subset of as , where and is a finite number.
In fact, for any finite subset , when is large enough, So
That is, are uniformly bounded for large enough .
Now, for , we can choose a subsequence of such that , uniformly converges on for large enough.
Similarly, for , we can choose a subsequence of such that , uniformly converges on for large enough.
Repeating this procedure, for , we obtain a subsequence of such that , uniformly converges on for large enough.
Now pick the sequence which is a subsequence of , we still denote it as , then for all , we have , uniformly in as .
By the arbitrariness of , the conclusion is valid.
Since and are almost periodic sequences, for above sequence , as , there exists a subsequence still denote by (if necessary, we take subsequence), such that
as uniformly on .
For any , we can assume that for large enough. Let and , by an inductive argument of (1.7) from to leads to
Then, for , we have Let , for any , By the arbitrariness of , is a solution of model (1.7) on . It is clear that , for all . So . Lemma 4.3 is valid.
Theorem 4.4. Suppose that the conditions of Lemma 4.3 are satisfied, moreover, , where , then there exists a unique uniformly asymptotically stable almost periodic solution of (1.7) which is bounded by for all .
Proof. Let . From (1.7), we have
where . From Lemma 4.3, we know that system (4.13) has bounded solution satisfying
Hence, , where
For , we define the norm .
Consider the product system of system (4.13)
Suppose , are any two solutions of system (4.15) defined on , then , where
Consider a Lyapunov function defined on as follows:
It is easy to see that the norm and the norm are equivalent that is, there exist two constants , such that then Let , thus condition (i) in Theorem 4.2 is satisfied.
where . Hence the condition (ii) of Theorem 4.2 is satisfied. Finally, calculate the of along the solutions of (4.15), we can obtain Using the mean value theorem, we get where lies between and , . From (4.21), (4.22), we have we get where