Research Article | Open Access
Almost Periodic Sequence Solutions of a Discrete Predator-Prey System with Beddington-DeAngelis Functional Response
This paper considers a discrete predator-prey system with Beddington-DeAngelis functional response. Sufficient conditions are obtained for the existence of the almost periodic solution which is uniformly asymptotically stable by constructing a Lyapunov function.
In the past decades, the predator-prey competition models have been extensively studied by many authors (see [1–7]). Many excellent works have been done for the predator-prey model with functional response, such as Zhu and Wang  who considered a Volterra model with modified Leslie-Gower Holling-type II schemes. Cai et al.  studied the positive periodic solution for a multispecies competition-predator system with the Holling III functional and time delays. A well-known model of such systems is the predator-prey model with a Beddington-DeAngelis functional response which was originally proposed by Beddington  and DeAngelis et al. , independently. The dynamics of this model is described by the following differential equations: where and represent prey and predator densities, respectively. Usually, the constant is called intrinsic growth rate of the prey; is the carrying capacity of the prey; and are positive constants that describe the effects of capture rate and handling time, respectively, on the feeding rate; is the death rate of the predator; is the birth rate of the predator; (units: 1/predator) is a constant describing the magnitude of interference among predators. Many excellent works have been done for the predator-prey system with a Beddington-DeAngelis functional response, such as Fan and Kuang , Liu et al. , and Baek .
It has been found that discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations. There are some existing results on discrete predator-prey systems [15–17]. For example, in , by using the method of upper and lower solutions and the degree theory the authors have studied the existence of periodic positive solutions for a competitive system with two parameters. Zhang and Wang  studied the following discrete model: The authors used the Mawhin’s coincidence degree theory to obtain some sufficient conditions for the existence of positive solutions.
But, nowadays, models with almost periodic coefficient have drawn more attention. Li and Chen  considered the almost periodic solutions of the following discrete almost periodic logistic equation:
Niu and Chen  proposed the following model: By constructing a suitable Lyapunov function, they obtained the existence and uniqueness of the almost periodic solution which is uniformly asymptotically stable.
Li et al.  studied the following system: With the method of the theory of difference inequality and constructing a suitable Lyapunov function, they obtained the permanence and the almost periodic solution of the system.
Inspired by the above papers, in this paper, we consider the following discrete predator-prey system: where and represent prey and predator densities at time .
From the point of view of biology, we assumed that the initial conditions of (6) are of the form Then, it is easy to see that the solutions of (6) with the initial condition (7) are defined and remain positive for all . Throughout this paper, for any bounded sequence , we denote and , and we assume that (H1), , , , , , , and are bounded nonnegative almost periodic sequences such that (H2), (H3).
The organization of this paper is as follows. In Section 2, we will introduce some definitions and several useful definitions and lemmas. In Section 3, by applying the theory of difference inequality, we get the permanence of system (6). In Section 4, by constructing a suitable Lyapunov function, we obtain the existence of the almost periodic solution for system (6) which is uniformly asymptotically stable. Finally, we give some examples and numerical simulations to verify our results.
In this section, we will introduce some basic definitions and several useful lemmas.
Lemma 2 (see ). Assume that satisfies and where and are nonnegative sequences bounded above and below by positive constants. Then,
Lemma 3 (see ). Assume that satisfies and and , , where and are nonnegative sequences bounded above and below by positive constants and . Then,
Lemma 4 (see ). Let and be nonnegative sequences defined on , and is a constant. If then
Definition 5 (see ). A sequence is called an almost periodic sequence if the -translation number 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 such that is called the -translation number or almost period.
Definition 6 (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 a integer for which for all and . is called the -translation number of .
Lemma 7 (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 , Zhang considered the following almost periodic difference system: where , , and is almost periodic in uniformly for and is continuous in . The product system of (19) is the following system: and the following lemma is obtained.
Lemma 8 (see ). Suppose that there exists a Lyapunov functional defined for , , satisfying the following conditions: [C1], where with [C2], where is a constant,[C3], where is a constant and
Moreover, if there exists a solution of system (20) such that for , then there exists a unique uniformly asymptotically stable almost periodic solution of system (20) which is bounded by . In particular, if is -periodic function, then there exists a unique uniformly asymptotically stable -periodic solution of (20).
Proof. Let be any positive solution of system (6); from the first equation of system (6), it follows that
Thus, as a direct corollary of Lemma 2, it follows that
On the other hand, from the first equation of system (6), we obtain
According to Lemma 3 and assumption (H2), we have
In the above inequality, using the Bernoulli inequality , for , one has
From the second equation of system (6), we can obtain Let ; then where Since by applying Lemma 4, we have Thus,
By the second equation of system (6), we can get that According to (H3), applying Lemma 3 to inequality (37), one has The proof of the theorem is completed.
4. Existence of Globally Attractive Almost Periodic Solutions
In this section, we study the existence of a globally attractive almost periodic sequence solution of system (6).
First, we denote by the set of all solutions of system (6) satisfying , for all .
Theorem 10. Assume that (H1), (H2), and (H3) hold. Then .
Proof. By the almost periodicity of , , , , , , , and , there exists an integer valued sequence with as such that
as uniformly on .
Let be an arbitrary small positive number. It follows from Theorem 9 that there exists a positive integer such that Write , , for , . For any positive integer , it is easy to see that there exist sequence such that the sequence have subsequences, denoted by again, converging on any finite interval of as , respectively. Thus we have sequence such that Combined with it gives We can easily see that is a solution of system (6); furthermore, Since is an arbitrary small positive number, it follows that The proof of theorem is completed.
In the following, we denote
Theorem 11. Suppose the conditions (H1), (H2), and (H3) are satisfied, and let ; then there exists a uniqueness uniformly asymptotically stable almost periodic solution of system (6) which is bounded by for all .
Proof. Let , . From system (6),we have
From Theorem 11, we know that the system (47) have bounded solution satisfying
Hence, , where , . For , we define the norm .
Consider the product system of system (47) Suppose , are any two solutions of system (49) 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 , , , and ; thus the condition [C1] in Lemma 8 is satisfied.
In addition, where . Hence, the condition [C2] of Lemma 8 is satisfied.
Finally, calculate the of along the solutions of (49); we can obtain In the view of (49), we get Substituting (56) into (55), we have Using the mean value theorem, we get where lies between and , . From (57) and (58), we have