## Abstract and Applied Analysis

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

#### 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 956703, 9 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393449974

Digital Object Identifier
doi:10.1155/2013/956703

Mathematical Reviews number (MathSciNet)
MR3108477

#### Citation

Xu, Changjin; Wu, Yusen. Dynamics in a Lotka-Volterra Predator-Prey Model with Time-Varying Delays. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 956703, 9 pages. doi:10.1155/2013/956703. https://projecteuclid.org/euclid.aaa/1393449974

#### References

• A. A. Berryman, “The origins and evolution of predator-prey theory,” Ecology, vol. 73, no. 5, pp. 1530–1535, 1992.
• 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.
• 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.
• 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.
• 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.
• R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, vol. 568 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and its Applications, Kluwer Academic, Dordrecht, Netherlands, 1992.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• J. Dhar and K. S. Jatav, “Mathematical analysis of a delayed stage-structured predator-prey model with impulsive diffusion between čommentComment on ref. [32?]: Please update the information of this reference, if possible.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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• F. Chen and M. You, “Permanence for an integrodifferential model of mutualism,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 30–34, 2007.
• 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.
• 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.
• T. Yoshizawa, Stability Theory by Liapunov's Second Method, The Mathematical Society of Japan, Tokyo, Japan, 1966.