About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 236484, 19 pages
http://dx.doi.org/10.1155/2012/236484
Research Article

Stability and Hopf Bifurcation in a Modified Holling-Tanner Predator-Prey System with Multiple Delays

1Key Laboratory of Advanced Process Control for Light Industry of Ministry of Education, Jiangnan University, Wuxi 214122, China
2School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China
3Department of Science, Bengbu College, Bengbu 233030, China

Received 9 August 2012; Revised 17 September 2012; Accepted 4 October 2012

Academic Editor: Kunquan Lan

Copyright © 2012 Zizhen Zhang 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. E. Beretta and Y. Kuang, “Global analyses in some delayed ratio-dependent predator-prey systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 32, no. 3, pp. 381–408, 1998. View at Publisher · View at Google Scholar
  2. 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
  3. H. J. Guo and X. X. Chen, “Existence and global attractivity of positive periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5830–5837, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. X. L. Wang, Z. J. Du, and J. Liang, “Existence and global attractivity of positive periodic solution to a Lotka-Volterra model,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 4054–4061, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. Z. D. Teng, “On the persistence and positive periodic solution for planar competing Lotka-Volterra systems,” Annals of Differential Equations, vol. 13, no. 3, pp. 275–286, 1997.
  6. P. H. Leslie, “Some further notes on the use of matrices in population mathematics,” Biometrika, vol. 35, no. 3-4, pp. 213–245, 1948. View at Zentralblatt MATH
  7. M. A. Aziz-Alaoui and M. D. Okiye, “Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes,” Applied Mathematics Letters, vol. 16, no. 7, pp. 1069–1075, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. Z. Zhao, L. Yang, and L. Chen, “Impulsive perturbations of a predator-prey system with modified Leslie-Gower and Holling type II schemes,” Journal of Applied Mathematics and Computing, vol. 35, no. 1-2, pp. 119–134, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. S. Gakkhar and A. Singh, “Complex dynamics in a prey predator system with multiple delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 914–929, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. G. O. Eduardo, M. L. Jaime, A. Rojas-Palma, and J. D. Flores, “Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey,” Applied Mathematical Modelling, vol. 35, no. 1, pp. 366–381, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. J. T. Tanner, “The stability and intrinsic growth rates of prey and predator populations,” Ecology, vol. 56, no. 4, pp. 855–867, 1975. View at Publisher · View at Google Scholar
  12. A. Gasull, R. E. Kooij, and J. Torregrosa, “Limit cycles in the Holling-Tanner model,” Publicacions Matemàtiques, vol. 41, no. 1, pp. 149–167, 1997.
  13. E. Sáez and E. G. Olivares, “Dynamics of a predator-prey model,” SIAM Journal on Applied Mathematics, vol. 59, no. 5, pp. 1867–1878, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. T. Saha and C. G. Chakrabarti, “Dynamical analysis of a delayed ratio-dependent Holling-Tanner predator-prey model,” Journal of Mathematical Analysis and Applications, vol. 358, no. 2, pp. 389–402, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. K. Q. Lan and C. R. Zhu, “Phase portraits, Hopf bifurcations and limit cycles of the Holling-Tanner models for predator-prey interactions,” Nonlinear Analysis: Real World Applications, vol. 12, no. 4, pp. 1961–1973, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. J.-F. Zhang, “Bifurcation analysis of a modified Holling-Tanner predator-prey model with time delay,” Applied Mathematical Modelling, vol. 36, no. 3, pp. 1219–1231, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. J. R. Beddington, “Mutual interference between parasites or predators and its effect on searching efficiency,” Journal of Animal Ecology, vol. 44, no. 1, pp. 331–340, 1975. View at Publisher · View at Google Scholar
  18. D. L. DeAngelis, R. A. Goldstein, and R. V. ONeill, “A model for trophic interaction,” Ecology, vol. 56, no. 4, pp. 881–892, 1975. View at Publisher · View at Google Scholar
  19. T.-W. Hwang, “Global analysis of the predator-prey system with Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 395–401, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. K. Q. Lan and C. R. Zhu, “Phase portraits of predator-prey systems with harvesting rates,” Discrete and Continuous Dynamical Systems A, vol. 32, no. 3, pp. 901–933, 2012. View at Publisher · View at Google Scholar
  22. Z. Q. Lu and X. Liu, “Analysis of a predator-prey model with modified Holling-Tanner functional response and time delay,” Nonlinear Analysis: Real World Applications, vol. 9, no. 2, pp. 641–650, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. S. G. Ruan and J. 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
  24. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191, Academic Press, New York, NY, USA, 1993.
  25. J. K. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977.
  26. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981.
  27. L. Feng and L. H. Wei, “Hopf bifurcation of a predator-prey model with time delay and stage structure for the prey,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 672–679, 2012. View at Publisher · View at Google Scholar
  28. Y. Yang, “Hopf bifurcation in a two-competitor, one-prey system with time delay,” Applied Mathematics and Computation, vol. 214, no. 1, pp. 228–235, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. Y. L. Song, S. L. Yuan, and J. M. Zhang, “Bifurcation analysis in the delayed Leslie-Gower predator-prey system,” Applied Mathematical Modelling, vol. 33, no. 11, pp. 4049–4061, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. S. Yuan and Y. Song, “Stability and Hopf bifurcations in a delayed Leslie-Gower predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 82–100, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. Y. F. Ma, “Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 370–375, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. X.-Y. Meng, H.-F. Huo, and H. Xiang, “Hopf bifurcation in a three-species system with delays,” Journal of Applied Mathematics and Computing, vol. 35, no. 1-2, pp. 635–661, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. S. Gakkhar, K. Negi, and S. K. Sahani, “Effects of seasonal growth on ratio dependent delayed prey predator system,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 850–862, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH