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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 454097, 12 pages
http://dx.doi.org/10.1155/2013/454097
Research Article

Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay

1College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China
2School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China

Received 3 March 2013; Accepted 27 May 2013

Academic Editor: Peixuan Weng

Copyright © 2013 Ruiqing Shi et al. 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.

Linked References

  1. A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, Md, USA, 1925.
  2. V. Volterra, “Variazioni e fluttuazioni del numero d'individui in specie animali conviventi,” Memorie Dell'Accademia Nazionale dei Lincei, vol. 2, pp. 31–113, 1926.
  3. A. Leung, “Periodic solutions for a prey-predator differential delay equation,” Journal of Differential Equations, vol. 26, no. 3, pp. 391–403, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. K. Gopalsamy, “Time lags and global stability in two-species competition,” Bulletin of Mathematical Biology, vol. 42, no. 5, pp. 729–737, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. X. Wen and Z. Wang, “The existence of periodic solutions for some models with delay,” Nonlinear Analysis. Real World Applications, vol. 3, no. 4, pp. 567–581, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. X. Chen, “Periodicity in a nonlinear discrete predator-prey system with state dependent delays,” Nonlinear Analysis. Real World Applications, vol. 8, no. 2, pp. 435–446, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. X.-P. Yan and C.-H. Zhang, “Hopf bifurcation in a delayed Lokta-Volterra predator-prey system,” Nonlinear Analysis. Real World Applications, vol. 9, no. 1, pp. 114–127, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  8. X. He, “Stability and delays in a predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 198, no. 2, pp. 355–370, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. S. Ruan, “Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays,” Quarterly of Applied Mathematics, vol. 59, no. 1, pp. 159–173, 2001. View at Zentralblatt MATH · View at MathSciNet
  10. L. Wang, W. Li, and P. Zhao, “Existence and global stability of positive periodic solutions of discrete predator-prey system with delays,” Advances in Difference Equations, vol. 4, pp. 321–336, 2004.
  11. Y. Song and Y. Peng, “Stability and bifurcation analysis on a logistic model with discrete and distributed delays,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1745–1757, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. W. Ma and Y. Takeuchi, “Stability analysis on a predator-prey system with distributed delays,” Journal of Computational and Applied Mathematics, vol. 88, no. 1, pp. 79–94, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. B. Liu, Z. Teng, and L. Chen, “Analysis of a predator-prey model with Holling II functional response concerning impulsive control strategy,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 347–362, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. R. Xu and Z. Wang, “Periodic solutions of a nonautonomous predator-prey system with stage structure and time delays,” Journal of Computational and Applied Mathematics, vol. 196, no. 1, pp. 70–86, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. X.-P. Yan and Y.-D. Chu, “Stability and bifurcation analysis for a delayed Lotka-Volterra predator-prey system,” Journal of Computational and Applied Mathematics, vol. 196, no. 1, pp. 198–210, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Z. Teng and M. Rehim, “Persistence in nonautonomous predator-prey systems with infinite delays,” Journal of Computational and Applied Mathematics, vol. 197, no. 2, pp. 302–321, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. G. Jiang and Q. Lu, “Impulsive state feedback control of a predator-prey model,” Journal of Computational and Applied Mathematics, vol. 200, no. 1, pp. 193–207, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191, Academic Press, Boston, Mass, USA, 1993. View at MathSciNet
  19. L. Zhou, Y. Tang, and S. Hussein, “Stability and Hopf bifurcation for a delay competition diffusion system,” Chaos, Solitons & Fractals, vol. 14, no. 8, pp. 1201–1225, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. Krise and S. Roy Choudhury, “Bifurcations and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations,” Chaos, Solitons and Fractals, vol. 16, no. 1, pp. 59–77, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. X. Liao and G. Chen, “Hopf bifurcation and chaos analysis of Chen's system with distributed delays,” Chaos, Solitons & Fractals, vol. 25, no. 1, pp. 197–220, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. Z. Liu and R. Yuan, “Stability and bifurcation in a harvested one-predator–two-prey model with delays,” Chaos, Solitons and Fractals, vol. 27, no. 5, pp. 1395–1407, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. S. Zhang, D. Tan, and L. Chen, “Chaotic behavior of a chemostat model with Beddington-DeAngelis functional response and periodically impulsive invasion,” Chaos, Solitons and Fractals, vol. 29, no. 2, pp. 474–482, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. X. Liu and D. Xiao, “Complex dynamic behaviors of a discrete-time predator-prey system,” Chaos, Solitons & Fractals, vol. 32, no. 1, pp. 80–94, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. F. Wang and G. Zeng, “Chaos in a Lotka-Volterra predator-prey system with periodically impulsive ratio-harvesting the prey and time delays,” Chaos, Solitons and Fractals, vol. 32, no. 4, pp. 1499–1512, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. C. Sun, M. Han, Y. Lin, and Y. Chen, “Global qualitative analysis for a predator-prey system with delay,” Chaos, Solitons & Fractals, vol. 32, no. 4, pp. 1582–1596, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. C. Çelik, “The stability and Hopf bifurcation for a predator-prey system with time delay,” Chaos, Solitons and Fractals, vol. 37, no. 1, pp. 87–99, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. C. Çelik, “Hopf bifurcation of a ratio-dependent predator-prey system with time delay,” Chaos, Solitons and Fractals, vol. 42, no. 3, pp. 1474–1484, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. C. Çelik, “Dynamical behavior of a ratio dependent predator-prey system with distributed delay,” Discrete and Continuous Dynamical Systems B, vol. 16, no. 3, pp. 719–738, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. J. M. Cushing, Integro-Differential Equations and Delay Models in Population Dynamics, Springer, Heidelberg, Germany, 1977.
  31. J. Wei and W. Jiang, “Stability and bifurcation analysis in Van der Pol's oscillator with delayed feedback,” Journal of Sound and Vibration, vol. 283, no. 3-5, pp. 801–819, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. S. Ruan and J. Wei, “On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 10, no. 6, pp. 863–874, 2003. View at Zentralblatt MATH · View at MathSciNet
  33. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981. View at MathSciNet