Abstract and Applied Analysis

Volume 2014 (2014), Article ID 380612, 12 pages

http://dx.doi.org/10.1155/2014/380612

Research Article

Almost Periodic Sequence Solutions of a Discrete Predator-Prey System with Beddington-DeAngelis Functional Response

Zengji Du and Wenbin Li

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

Received 21 April 2014; Accepted 12 June 2014; Published 17 July 2014

Academic Editor: Kunquan Lan

Copyright © 2014 Zengji Du and Wenbin 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.

Abstract.

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.

1. Introduction

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 [8] who considered a Volterra model with modified Leslie-Gower Holling-type II schemes. Cai et al. [9] 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 [10] and DeAngelis et al. [11], 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 [12], Liu et al. [13], and Baek [14].

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 [15], 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 [16] 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 [18] considered the almost periodic solutions of the following discrete almost periodic logistic equation:

Niu and Chen [19] 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. [20] 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.

2. Preliminaries

In this section, we will introduce some basic definitions and several useful lemmas.

Definition 1. System (6) is said to be permanent if there exist positive constants , , which are independent of the solutions of the system, such that any positive solution of system (6) satisfies

Lemma 2 (see [21]). Assume that satisfies and

where and are nonnegative sequences bounded above and below by positive constants. Then,

Lemma 3 (see [21]). Assume that satisfies and

and , , where and are nonnegative sequences bounded above and below by positive constants and . Then,

Lemma 4 (see [22]). Let and be nonnegative sequences defined on , and is a constant. If

then

Definition 5 (see [23]). 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 [23]). 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 [23]). 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 [24], 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 [24]). 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).

3. Permanence

Theorem 9. Assume that (H₁), (H₂), and (H₃) hold, then every solution of system (6) satisfies

That is, system (6) is permanent, where

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

Notice that

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 .

In order to apply Lemma 8, we should prove that there exists a bounded solution of system (6) and then construct an adaptive Lyapunov functional for system (6).

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

and we get

where

Hence,

where . That is, there exists a positive constant such that

From , the condition [C₃] of Lemma 8 is satisfied. So, from Lemma 8, there exists a uniqueness uniformly asymptotically stable almost periodic solution of (47) which is bounded by for all . Which means that there exists a uniqueness uniformly asymptotically stable almost periodic solution of (6) which is bounded by for all . The proof is completed.

5. Application

Example 12. Consider the following discrete model:

with initial conditions ; then system (64) is persistent and has a globally attractive almost periodic solution.

By simple computation, we derive

Thus, the system (64) is persistent. Its integral curves and orbits are shown in Figure 1, respectively.

Figure 1: The orbit of predator ()-prey ().

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors express their sincere thanks to the anonymous reviewers for their valuable comments and suggestions for improving the quality of the paper. This work is supported by the Natural Science Foundation of China (Grant nos. 11071205 and 11101349), PAPD of Jiangsu Higher Education Institutions, and Postgraduate training project of Jiangsu Normal University and Jiangsu Province.

References

J. O. Alzabut, J. J. Nieto, and G. T. Stamov, “Existence and exponential stability of positive almost periodic solutions for a model of hematopoiesis,” Boundary Value Problems, vol. 2009, Article ID 127510, 10 pages, 2009.

F. Chen, “Almost periodic solution of the non-autonomous two-species competitive model with stage structure,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 685–693, 2006.

Z. Du and Y. Lv, “Permanence and almost periodic solution of a Lotka-Volterra model with mutual interference and time delays,” Applied Mathematical Modelling, vol. 37, no. 3, pp. 1054–1068, 2013.

C. H. Feng, Y. J. Liu, and W. G. Ge, “Almost periodic solutions for delay Lotka-Volterra competitive systems,” Acta Mathematicae Applicatae Sinica, vol. 28, no. 3, pp. 458–465, 2005.

X. Lin and F. D. Chen, “Almost periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response,” Applied Mathematics and Computation, vol. 214, no. 2, pp. 548–556, 2009.

C. Shen, “Permanence and global attractivity of the food-chain system with Holling IV type functional response,” Applied Mathematics and Computation, vol. 194, no. 1, pp. 179–185, 2007.

W. Qi and B. Dai, “Almost periodic solution for -species Lotka-Volterra competitive system with delay and feedback controls,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 133–146, 2008.

Y. Zhu and K. Wang, “Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Leslie-Gower Holling-type II schemes,” Journal of Mathematical Analysis and Applications, vol. 384, no. 2, pp. 400–408, 2011.

Z. Cai, L. Huang, and H. Chen, “Positive periodic solution for a multispecies competition-predator system with Holling III functional response and time delays,” Applied Mathematics and Computation, vol. 217, no. 10, pp. 4866–4878, 2011.

J. R. Beddington, “Mutual interference between parasites or predators and its effect on searching efficiency,” Journal of Animal Ecology, vol. 44, pp. 331–340, 1975.

D. L. DeAngelis, R. A. Goldstein, and R. V. O'Neill, “A model for trophic interaction,” Ecology, vol. 56, pp. 881–892, 1975.

M. Fan and Y. Kuang, “Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 15–39, 2004.

S. Liu, E. Beretta, and D. Breda, “Predator-prey model of Beddington-DeAngelis type with maturation and gestation delays,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 4072–4091, 2010.

H. K. Baek, “Qualitative analysis of Beddington-DeAngelis type impulsive predator-prey models,” Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 1312–1322, 2010.

L. Li, G. Zhang, G. Sun, and Z. Wang, “Existence of periodic positive solutions for a competitive system with two parameters,” Journal of Difference Equations and Applications, vol. 20, no. 3, pp. 341–353, 2014.

J. Zhang and J. Wang, “Periodic solutions for discrete predator-prey systems with the Beddington-DeAngelis functional response,” Applied Mathematics Letters, vol. 19, no. 12, pp. 1361–1366, 2006.

L. Li and Z. Wang, “Global stability of periodic solutions for a discrete predator-prey system with functional response,” Nonlinear Dynamics, vol. 72, no. 3, pp. 507–516, 2013.

Z. Li and F. Chen, “Almost periodic solutions of a discrete almost periodic logistic equation,” Mathematical and Computer Modelling, vol. 50, no. 1-2, pp. 254–259, 2009.

C. Y. Niu and X. X. Chen, “Almost periodic sequence solutions of a discrete Lotka-Volterra competitive system with feedback control,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 3152–3161, 2009.

Y. Li, T. Zhang, and Y. Ye, “On the existence and stability of a unique almost periodic sequence solution in discrete predator-prey models with time delays,” Applied Mathematical Modelling, vol. 35, no. 11, pp. 5448–5459, 2011.

F. Chen, “Permanence for the discrete mutualism model with time delays,” Mathematical and Computer Modelling, vol. 47, no. 3-4, pp. 431–435, 2008.

W. Yang and X. Li, “Permanence for a delayed discrete ratio-dependent predator-prey model with monotonic functional responses,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 1068–1072, 2009.

D. Cheban and C. Mammana, “Invariant manifolds, global attractors and almost periodic solutions of nonautonomous difference equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 4, pp. 465–484, 2004.

S. N. Zhang, “Existence of almost periodic solution for difference systems,” Annals of Differential Equations, vol. 16, pp. 184–206, 2000.