Discrete Dynamics in Nature and SocietyVolume 2011 (2011), Article ID 271928, 15 pagesdoi:10.1155/2011/271928

Research Article

Asymptotic Behaviors of a Delayed Nonautonomous Predator-Prey System Governed by Difference Equations

Lili Liu1,2 and Zhijun Liu1,2

¹Key Laboratory of Biological Resources Protection and Utilization of Hubei Province, Hubei University for Nationalities Enshi, Hubei 445000, China²Department of Mathematics, Hubei University for Nationalities Enshi, Hubei 445000, China

Received 3 May 2011; Accepted 18 September 2011

Academic Editor: Zhen Jin

Copyright © 2011 Lili Liu and Zhijun Liu. 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.

Based on a predator-prey differential system with continuously distributed delays, we derive a corresponding difference version by using the method of a discretization technique. A good understanding of permanence of system and global attractivity of positive solutions of system is gained. An example and its numerical simulations are presented to substantiate our theoretical results.

1. Introduction

It is well known that the classical Lotka-Volterra differential equations for predator-prey interaction can be written as where , , , and are the population densities of prey and predator, respectively, at time , and the coefficients are positive constants. is the intrinsic growth rate of prey species, and represents the death rate of predator species; and , denote their self-inhibition and interaction rates, respectively. To account for time lag effects in their interaction, Cushing [1] suggested visualizing system (1.1) as a special case of the following more general system where are nondecreasing functions on with for each , and with , respectively, equal to for .

As Hale [2] pointed out, one expects that the lag effects tend to diminish gradually in ever-moderating pace as one moves backward in time after the initial instantaneous effect described by the assumption . The effects are negligible after a certain length of time . Mathematically, for , it is reasonable that can be assumed to be of the form , , where and continuous for ; for ; and continuous for ; and both exist; and continuous for .

The assumption represents gradual diminishing effect as one moves backward in time, and the assumption stands for the moderating pace of the decrease (see [2] or [3] for the assumptions to ). Leung [4] revised system (1.2) and established the following predator-prey model with continuously distributed delays The parameters of system (1.3) are assumed to be constant; however, in the real world the parameters are not fixed constants owing to the variation of environment. The effect of a varying environment is significant for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Thus, it is realistic to assume that the parameters of system (1.3) are continuous functions with respect to , and then we obtain the following form: where the coefficients are positive continuous functions and satisfy for .

Although much progress on the predator-prey models with discrete or distributed delays has been made, such models are not sufficiently researched in the sense that most results are continuous time versions related (see [5–13]). Many authors have argued that the discrete time models governed by difference equations are more appropriate than the continuous time ones when populations have a short life expectancy, nonoverlapping generations in the real world (see [14–18]). Meanwhile, discrete time models can provide efficient computational models of continuous time models for numerical simulations. To the best of our knowledge, no such work has been done for the corresponding discrete version of system (1.4).

In the following, we employ the discretization technique to derive the discrete version of system (1.4). Throughout this paper, let , and denote the sets of all integers, nonnegative integers, and two-dimensional vector space, respectively. We begin to approximate the continuous time system by replacing the integral term with discrete sums of the form for , , , , where denotes the greatest integer function. , for , where the fixed number denotes an uniform discretization step size. We approximate system (1.4) by differential equations with piecewise constant arguments of the form for , , . Noting that , , we integrate (1.6) over , where , then (1.6) can be reformulated as Denoting , , , , , , , , , , then we have Setting in (1.8) and simplifying, we get a discrete time analogue of continuous time system (1.4) with the form Our main purpose of this paper is to derive a set of easily verifiable sufficient conditions concerning the permanence and global attractivity of system (1.9). For convenience, we introduce the following definitions and notations.

Definition 1.1. It is said that system (1.9) is permanent if there exist positive constants , and any positive solution of system (1.9) satisfies

Definition 1.2. A positive solution of system (1.9) is global attractive if each other positive solution of system (1.9) satisfies Let be a bounded sequence; we denote Meanwhile, we make a convention that for all .

This paper proceeds as follows. System (1.9) is analyzed to study the permanence and global attractivity of system (1.9) in the next two sections, respectively. In the final section, we give an example, and its numerical simulations are presented to substantiate our theoretical results.

2. Permanence of System (1.9)

In this section, we devote to investigating the permanence of system (1.9). To do so, we introduce the following lemmas.

Lemma 2.1 (see [19]). Assume that satisfies and for , where is a positive constant and . Then

Lemma 2.2 (see [19]). Assume that satisfies and for , and , where is a positive constant such that and . Then

For the simplicity of description, we denote where are, respectively, defined in (2.8) and (2.11)–(2.14) and is a sufficient small positive constant. Now, we begin to search the conditions for the permanence of (1.9).

Theorem 2.3. If the conditions hold, then (1.9) is permanent.

Proof. It is easy to verify that is a positive invariant set of (1.9) with , . Suppose that is any positive solution of (1.9), we prove Theorem 2.3 by the following two steps.Step 1. We show that is uniformly ultimately upper bounded.From the first equation of (1.9), we have Applying Lemma 2.1, we can obtain that From (2.8), there exists a sufficient large and any constant such that It follows from the second equation of (1.9) that So by Lemma 2.1 and letting in (2.10), we have Step 2. We prove that is uniformly ultimately lower bounded.For any sufficient small , it follows from (2.6) that . According to (2.11), there exists an such that for and the above constant . By the first equation of (1.9), it gives that Note that where we employ the inequality result that for . Therefore, applying Lemma 2.2 and setting , one has For the above constant , it follows from (2.14) that there exists a sufficient large such that From the second equation of (1.9), we have Clearly, Thus, applying Lemma 2.2 and letting , one has Based on the above Step 1 with Step 2, we can see that system (1.9) is permanent. This completes the proof of Theorem 2.3.

3. Global Attractivity of Positive Solutions of System (1.9)

In this section, we investigate the global attractivity of positive solutions of system (1.9).

Theorem 3.1. Assume that there exists a positive constant such that hold. Then any positive solution of system (1.9) is global attractive.

Proof. Denote be any other positive solution of system (1.9). Let , then it follows from the first equation of (1.9) that By the Mean Value theorem, we get where lies between and . Then we have Combining (3.2) with (3.4), we have Next, we let Then we have We set then it follows from (3.5) and (3.7) that Similarly, we define where Then we have where is between and . Consequently, one has Now, we consider a Lyapunov-like discrete function defined by According to (2.8) and (2.11), there exists an such that , for and any positive constant . Obviously, for all and . In view of (3.1), we choose a sufficient small such that Therefore, combining (3.9) with (3.13) for all , we obtain Summing both sides of (3.16) from to , it derives that which implies that from which we conclude that that is, According to Definition 1.2, this result implies that is global attractive. The proof is complete.

4. An Example and Its Numerical Simulations

In this paper, we have investigated the asymptotic behaviors of a delayed nonautonomous predator-prey difference system. Sufficient conditions which guarantee the permanence of system and global attractivity of positive solutions of system are obtained, respectively. The theoretical results are substantiated by the following example and numerical results.

Example 4.1. Consider the following system: Obviously, It is easy to verify that the assumptions of Theorem 2.3 are satisfied, so system (4.1) is permanent (see Figure 1). Furthermore, a calculation shows that According to Theorem 3.1, any positive solution of system (4.1) is global attractive (see Figure 2). From Figures 2(a1) and 2(a2), we see that with tends to with . Similarly, from Figures 2(b1) and 2(b2), we see that with tends to with .

(a)

(b)

Figure 1: Permanence of (1.9) with and . (a) Time series of for . (b) Time series of for .

Figure 2: Dynamic behaviors of (1.9) with different initial values. (a1) and (a2) show with , , , for and , respectively. (b1) and (b2) show with , , , for and , respectively.

Acknowledgments

This work is supported by the Key Project of Chinese Ministry of Education (no. 210134), the Innovation Term of Educational Department of Hubei Province of China (no. T200804), and the Project of Key Laboratory of Biological Resources Protection and Utilization of Hubei Province. The authors would like to thank the anonymous referees for their helpful comments which improved the presentation of this work.

References

J. M. Cushing, “Periodic solutions of two species interaction models with lags,” Mathematical Biosciences, vol. 31, no. 1-2, pp. 143–156, 1976.

J. K. Hale, Functional Differential Equations, Springer, New York, NY, USA, 1971.

J. J. Levin and J. A. Nohel, “On a nonlinear delay equation,” Journal of Mathematical Analysis and Applications, vol. 8, pp. 31–44, 1964.

A. Leung, “Conditions for global stability concerning a prey-predator model with delay effects,” SIAM Journal on Applied Mathematics, vol. 36, no. 2, pp. 281–286, 1979.

S. Q. 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.

W. Zuo and J. Wei, “Stability and Hopf bifurcation in a diffusive predatory-prey system with delay effect,” Nonlinear Analysis. Real World Applications, vol. 12, no. 4, pp. 1998–2011, 2011.

Z. J. Liu, R. H. Tan, and L. S. Chen, “Global stability in a periodic delayed predator-prey system,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 389–403, 2007.

C.-H. Zhang, X.-P. Yan, and G.-H. Cui, “Hopf bifurcations in a predator-prey system with a discrete delay and a distributed delay,” Nonlinear Analysis. Real World Applications, vol. 11, no. 5, pp. 4141–4153, 2010.

Z. Luo, B. Dai, and Q. Zhang, “Existence of positive periodic solutions for an impulsive semi-ratio-dependent predator-prey model with dispersion and time delays,” Applied Mathematics and Computation, vol. 215, no. 9, pp. 3390–3398, 2010.

Z.-P. Ma, H.-F. Huo, and C.-Y. Liu, “Stability and Hopf bifurcation analysis on a predator-prey model with discrete and distributed delays,” Nonlinear Analysis. Real World Applications, vol. 10, no. 2, pp. 1160–1172, 2009.

L. Chen and F. Chen, “Dynamic behaviors of the periodic predator-prey system with distributed time delays and impulsive effect,” Nonlinear Analysis. Real World Applications, vol. 12, no. 4, pp. 2467–2473, 2011.

W. Yan and J. R. Yan, “Periodicity and asymptotic stability of a predator-prey system with infinite delays,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1465–1472, 2010.

X. X. Liu, “A note on the existence of periodic solutions in discrete predator-prey models,” Applied Mathematical Modelling, vol. 34, no. 9, pp. 2477–2483, 2010.

R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Application, Marcel Dekker, New York, NY, USA, 2000.

H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, NY, USA, 1980.

W. Qin and Z. Liu, “Asymptotic behaviors of a delay difference system of plankton allelopathy,” Journal of Mathematical Chemistry, vol. 48, no. 3, pp. 653–675, 2010.

W. J. Qin, Z. J. Liu, and Y. P. Chen, “Permanence and global stability of positive solutions of a discrete competitive system,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 830537, 13 pages, 2009.

J. D. Murray, Mathematical Biology, vol. 19, Springer, Berlin, Germany, 1989.

X. T. Yang, “Uniform persistence and periodic solutions for a discrete predator-prey system with delays,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 161–177, 2006.