Abstract and Applied Analysis

Volume 2013, Article ID 956703, 9 pages

http://dx.doi.org/10.1155/2013/956703

## Dynamics in a Lotka-Volterra Predator-Prey Model with Time-Varying Delays

^{1}Guizhou Key Laboratory of Economics System Simulation, School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550004, China^{2}School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China

Received 31 May 2013; Revised 18 August 2013; Accepted 18 August 2013

Academic Editor: Mark McKibben

Copyright © 2013 Changjin Xu and Yusen Wu. 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

A Lotka-Volterra predator-prey model with time-varying delays is investigated. By using the differential inequality theory, some sufficient conditions which ensure the permanence and global asymptotic stability of the system are established. The paper ends with some interesting numerical simulations that illustrate our analytical predictions.

#### 1. Introduction

In 1992, Berryman [1] pointed out that the dynamical relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Dynamical behavior of predator-prey models has been studied by a lot of papers. It is well known that the investigation on predator-prey models not only focuses on the discussion of stability, periodic oscillatory, bifurcation, and chaos [2–26], but also involves many other dynamical behaviors such as permanence. In many applications, the nature of permanence is of great interest. Recently, Chen [27] investigated the permanence of a discrete -species food-chain system with delays. Fan and Li [28] gave a theoretical study on permanence of a delayed ratio-dependent predator-prey model with Holling type functional response. Chen [29] focused on the permanence and global attractivity of Lotka-Volterra competition system with feedback control. Zhao and Jiang [30] analyzed the permanence and extinction for nonautonomous Lotka-Volterra system. Teng et al. [31] addressed the permanence criteria for delayed discrete nonautonomous-species Kolmogorov systems. For more research on the permanence behavior of predator-prey models, one can see [32–40].

In 2010, Lv et al. [41] investigated the existence and global attractivity of periodic solution to the following Lotka-Volterra predator-prey system: where denotes the density of prey species at time , and stand for the density of predator species at time , and . Using Krasnoselskii’s fixed point theorem and constructing Lyapunov function, Lv et al. obtained a set of easily verifiable sufficient conditions which guarantee the permanence and global attractivity of system (1).

For the viewpoint of biology, we shall consider (1) together with the initial conditions . The principle object of this paper is to explore the dynamics of system (1), applying the differential inequality theory to study the permanence of system (1). Using the method of Lyapunov function, we investigated the globally asymptotically stability of system (1).

The remainder of the paper is organized as follows: in Section 2, basic definitions and Lemmas are given, and some sufficient conditions for the permanence of the Lotka-Volterra predator-prey model in consideration are established. A series of sufficient conditions for the global stability of the Lotka-Volterra predator-prey model in consideration are included in Section 3. In Section 4, we give an example which shows the feasibility of the main results. Conclusions are presented in Section 5.

#### 2. Permanence

For convenience in the following discussing, we always use the notations: where is a continuous function. In order to obtain the main result of this paper, we shall first state the definition of permanence and several lemmas which will be useful in the proving the main result.

*Definition 1 (see [41]). *We say that system (1) is permanence if there are positive constants and such that for each positive solution of system (1) satisfies

Lemma 2 (see [42]). *If ,, and , when and , we have
**
If ,, and , when and , we have
*

Now we state our permanence result for system (1).

Theorem 3. *Let ,,, and be defined by (11), (18), (24), and (30), respectively. Suppose that the following conditions:*(H1)*,
*(H2)*hold, and then system (1) is permanent; that is, there exist positive constants which are independent of the solution of system (1), such that, for any positive solution of system (1) with the initial condition , one has
*

*Proof. * It is easy to see that system (1) with the initial value condition has positive solution passing through . Let be any positive solution of system (1) with the initial condition . It follows from the first equation of system (1) that
Integrating both sides of (7) from to , we get
which leads to
Substituting (9) into the first equation of system (1), it follows that
It follows from (10) and Lemma 2 that
For any positive constant , it follows from (11) that there exists a such that, for all ,
For , from (12) and the second equation of system (1), we have
Integrating both sides of (13) from to , we get
which leads to
Substituting (15) into the second equation of system (1), it follows that
Thus, as a direct corollary of Lemma 2, according to (16), one has
Setting , it follows that
For , from (12) and the third equation of system (1), we have
Integrating both sides of (19) from to , we get
which leads to
Substituting (21) into the third equation of system (1), it follows that
Thus, as a direct corollary of Lemma 2, according to (22), one has
Setting , it follows that
For , it follows from the first equation of system (1) that
Integrating both sides of (25) from to , one has
which leads to
Substituting (27) into the first equation of system (1), it follows that
According to Lemma 2, it follows from (28) that
Setting in (29), we can get
For , from the second equation of system (1), we have
Integrating both sides of (31) from to leads to
which leads to
Substituting (33) into the second equation of system (1), it follows that
By Lemma 2 and (34), we can get
Setting in the above inequality, it follows that
For , it follows from the third equation of system (1) that
Integrating both sides of (37) from to , we get
Hence
Substituting (39) into the third equation of system (1), we derive
In view of Lemma 2 and (40), one has
Setting in (41) leads to
Equations (11), (18), (24), (30), (36), and (42) show that system (1) is permanent. The proof of Theorem 3 is complete.

#### 3. Global Asymptotically Stability of Positive Solutions

In this section, we formulate the global asymptotically stability of positive solutions of system (1).

*Definition 4. *A bounded positive solution of system (1) is said to be globally asymptotically stable if, for any other positive bounded solution of system (1), the following equality holds:

*Definition 5 (see [24]). * Let be a real number and be a nonnegative function defined on such that is integrable on and is uniformly continuous on , then .

Theorem 6. * In addition to (H1)-(H2), assume further that *(H3)*,
**where are defined by (48), (49), and (50), respectively. Then system (1) has a unique positive solution which is global attractivity. *

*Proof. *According to the conclusion of Theorem 3, there exists and positive constants such that

Define
Calculating the upper-right derivative of along the solution of (1), it follows for that
It follows that
where is defined by Theorem 3 and
By hypothesis , there exist constants and such that
Integrating both sides of (51) on interval yields
It follows from (51) and (52) that
Since are bounded for , so are uniformly continuous on . By Barbalat’s Lemma [24], we have
By Theorems 7.4 and 8.2 in [43], we know that the positive solution of (1) is uniformly asymptotically stable. The proof of Theorem 6 is complete.

#### 4. Numerical Example

To illustrate the theoretical results, we present some numerical simulations. Let us consider the following discrete system: Here All the coefficients ,, are functions with respect to , and it is easy to see that Then ,,,. Thus it is easy to see that all the conditions of Theorem 6 are satisfied. Thus system (55) is permanent which is shown in Figures 1, 2, and 3.

#### 5. Conclusions

In this paper, we have investigated the dynamical behavior of a Lotka-Volterra predator-prey model with time-varying delays. Sufficient conditions which ensure the permanence of the system are derived. Moreover, we also deal with the global stability of the system. It is shown that delay has influence on the permanence and the global stability of system. Thus delay is an important factor to decide the permanence and global stability of the system. Numerical simulations show the feasibility of our main results.

#### Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 11261010 and no. 11101126), Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), Natural Science and Technology Foundation of Guizhou Province (J[2012]2100), Governor Foundation of Guizhou Province ([2012]53), and Doctoral Foundation of Guizhou University of Finance and Economics (2010).

#### References

- A. A. Berryman, “The origins and evolution of predator-prey theory,”
*Ecology*, vol. 73, no. 5, pp. 1530–1535, 1992. View at Publisher · View at Google Scholar · View at Scopus - B. Dai and J. Zou, “Periodic solutions of a discrete-time diffusive system governed by backward difference equations,”
*Advances in Difference Equations*, vol. 2005, no. 3, pp. 263–274, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Gyllenberg, P. Yan, and Y. Wang, “Limit cycles for competitor-competitor-mutualist Lotka-Volterra systems,”
*Physica D*, vol. 221, no. 2, pp. 135–145, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Fan and K. Wang, “Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system,”
*Mathematical and Computer Modelling*, vol. 35, no. 9-10, pp. 951–961, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Fazly and M. Hesaaraki, “Periodic solutions for a discrete time predator-prey system with monotone functional responses,”
*Comptes Rendus Mathématique*, vol. 345, no. 4, pp. 199–202, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. E. Gaines and J. L. Mawhin,
*Coincidence Degree, and Nonlinear Differential Equations*, vol. 568 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 1977. View at MathSciNet - Y. Li, “Positive periodic solutions of a discrete mutualism model with time delays,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2005, no. 4, pp. 499–506, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Sen, M. Banerjee, and A. Morozov, “Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect,”
*Ecological Complexity*, vol. 11, pp. 12–27, 2012. View at Publisher · View at Google Scholar · View at Scopus - M. Haque and E. Venturino, “An ecoepidemiological model with disease in predator: the ratio-dependent case,”
*Mathematical Methods in the Applied Sciences*, vol. 30, no. 14, pp. 1791–1809, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - L.-L. Wang, W.-T. Li, and P.-H. Zhao, “Existence and global stability of positive periodic solutions of a discrete predator-prey system with delays,”
*Advances in Difference Equations*, vol. 2004, no. 4, pp. 321–336, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Wiener, “Differential equations with piecewise constant delays,” in
*Trends in Theory and Practice of Nonlinear Differential Equations*, vol. 90 of*Lecture Notes in Pure and Applied Mathematics*, pp. 547–552, Dekker, New York, NY, USA, 1984. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Xu, L. Chen, and F. Hao, “Periodic solutions of a discrete time Lotka-Volterra type food-chain model with delays,”
*Applied Mathematics and Computation*, vol. 171, no. 1, pp. 91–103, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Zhang and H. Fang, “Multiple periodic solutions for a discrete time model of plankton allelopathy,”
*Advances in Difference Equations*, vol. 2006, Article ID 090479, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Xiong and Z. Zhang, “Periodic solutions of a discrete two-species competitive model with stage structure,”
*Mathematical and Computer Modelling*, vol. 48, no. 3-4, pp. 333–343, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Y. Zhang, Z. C. Wang, Y. Chen, and J. Wu, “Periodic solutions of a single species discrete population model with periodic harvest/stock,”
*Computers & Mathematics with Applications*, vol. 39, no. 1-2, pp. 77–90, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Zhang, D. Zhu, and P. Bi, “Multiple positive periodic solutions of a delayed discrete predator-prey system with type IV functional responses,”
*Applied Mathematics Letters*, vol. 20, no. 10, pp. 1031–1038, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zhang and J. Luo, “Multiple periodic solutions of a delayed predator-prey system with stage structure for the predator,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 5, pp. 4109–4120, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Li, K. Zhao, and Y. Ye, “Multiple positive periodic solutions of $n$ species delay competition systems with harvesting terms,”
*Nonlinear Analysis: Real World Applications*, vol. 12, no. 2, pp. 1013–1022, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. G. Sun and S. H. Saker, “Positive periodic solutions of discrete three-level food-chain model of Holling type II,”
*Applied Mathematics and Computation*, vol. 180, no. 1, pp. 353–365, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Ding and C. Lu, “Existence of positive periodic solution for ratio-dependent $N$-species difference system,”
*Applied Mathematical Modelling*, vol. 33, no. 6, pp. 2748–2756, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - K. Chakraborty, M. Chakraborty, and T. K. Kar, “Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay,”
*Nonlinear Analysis: Hybrid Systems*, vol. 5, no. 4, pp. 613–625, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.C. Li, Q.L. Zhao, and D. Ling, “Chaos in a discrete delay population model,”
*Discrete Dynamics in Nature and Society*, vol. 2012, Article ID 482459, 14 pages, 2012. View at Publisher · View at Google Scholar - H. Xiang, K.-M. Yan, and B.-Y. Wang, “Existence and global stability of periodic solution for delayed discrete high-order hopfield-type neural networks,”
*Discrete Dynamics in Nature and Society*, vol. 2005, no. 3, pp. 281–297, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Gopalsamy,
*Stability and Oscillations in Delay Differential Equations of Population Dynamics*, vol. 74 of*Mathematics and its Applications*, Kluwer Academic, Dordrecht, Netherlands, 1992. View at MathSciNet - Y. Kuang,
*Delay Differential Equations with Applications in Population Dynamics*, vol. 191 of*Mathematics in Science and Engineering*, Academic Press, New York, NY, USA, 1993. View at MathSciNet - L. Fan, Z. Shi, and S. Tang, “Critical values of stability and Hopf bifurcations for a delayed population model with delay-dependent parameters,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 1, pp. 341–355, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen, “Permanence of a discrete $n$-species food-chain system with time delays,”
*Applied Mathematics and Computation*, vol. 185, no. 1, pp. 719–726, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - Y.-H. Fan and W.-T. Li, “Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response,”
*Journal of Mathematical Analysis and Applications*, vol. 299, no. 2, pp. 357–374, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen, “The permanence and global attractivity of Lotka-Volterra competition system with feedback controls,”
*Nonlinear Analysis: Real World Applications*, vol. 7, no. 1, pp. 133–143, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Zhao and J. Jiang, “Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system,”
*Journal of Mathematical Analysis and Applications*, vol. 299, no. 2, pp. 663–675, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Teng, Y. Zhang, and S. Gao, “Permanence criteria for general delayed discrete nonautonomous $n$-species Kolmogorov systems and its applications,”
*Computers & Mathematics with Applications*, vol. 59, no. 2, pp. 812–828, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - J. Dhar and K. S. Jatav, “Mathematical analysis of a delayed stage-structured predator-prey model with impulsive diffusion between two predators territories,”
*Ecological Complexity*. In press. - S. Liu and L. Chen, “Necessary-sufficient conditions for permanence and extinction in Lotka-Volterra system with distributed delays,”
*Applied Mathematics Letters*, vol. 16, no. 6, pp. 911–917, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Liao, S. Zhou, and Y. Chen, “Permanence and global stability in a discrete $n$-species competition system with feedback controls,”
*Nonlinear Analysis: Real World Applications*, vol. 9, no. 4, pp. 1661–1671, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - H. Hu, Z. Teng, and H. Jiang, “On the permanence in non-autonomous Lotka-Volterra competitive system with pure-delays and feedback controls,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 3, pp. 1803–1815, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Muroya, “Permanence and global stability in a Lotka-Volterra predator-prey system with delays,”
*Applied Mathematics Letters*, vol. 16, no. 8, pp. 1245–1250, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kuniya and Y. Nakata, “Permanence and extinction for a nonautonomous SEIRS epidemic model,”
*Applied Mathematics and Computation*, vol. 218, no. 18, pp. 9321–9331, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Hou, “On permanence of Lotka-Volterra systems with delays and variable intrinsic growth rates,”
*Nonlinear Analysis: Real World Applications*, vol. 14, no. 2, pp. 960–975, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - C.-H. Li, C.-C. Tsai, and S.-Y. Yang, “Analysis of the permanence of an SIR epidemic model with logistic process and distributed time delay,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 9, pp. 3696–3707, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen and M. You, “Permanence for an integrodifferential model of mutualism,”
*Applied Mathematics and Computation*, vol. 186, no. 1, pp. 30–34, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Lv, S. Lu, and P. Yan, “Existence and global attractivity of positive periodic solutions of Lotka-Volterra predator-prey systems with deviating arguments,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 1, pp. 574–583, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Montes de Oca and M. Vivas, “Extinction in two dimensional Lotka-Volterra system with infinite delay,”
*Nonlinear Analysis. Real World Applications*, vol. 7, no. 5, pp. 1042–1047, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Yoshizawa,
*Stability Theory by Liapunov's Second Method*, The Mathematical Society of Japan, Tokyo, Japan, 1966. View at MathSciNet