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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 363051, 17 pages
http://dx.doi.org/10.1155/2012/363051
Research Article

Hopf Bifurcation Analysis on General Gause-Type Predator-Prey Models with Delay

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2School of Mathematical Sciences, Daqing Normal University, Daqing 163712, China

Received 27 November 2011; Accepted 26 December 2011

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Shuang Guo and Weihua Jiang. 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. J. F. Andrews, “A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates,” Biotechnology and Bioengineering, vol. 10, no. 6, pp. 707–723, 1968. View at Publisher · View at Google Scholar
  2. M. S. Bartlett, “On theoretical models for competitive and predatory biological systems,” Biometrika, vol. 44, pp. 27–42, 1957. View at Zentralblatt MATH
  3. F. Brauer and A. C. Soudack, “Stability regions and transition phenomena for harvested predator-prey systems,” Journal of Mathematical Biology, vol. 7, no. 4, pp. 319–337, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. H. I. Freedman, Deterministic Mathematical Models in Population Ecology, HIFR Consulting, Edmonton, Canada, 1987.
  5. K. S. Cheng, S. B. Hsu, and S. S. Lin, “Some results on global stability of a predator-prey system,” Journal of Mathematical Biology, vol. 12, no. 1, pp. 115–126, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. A. Klebanoff and A. Hastings, “Chaos in three-species food chains,” Journal of Mathematical Biology, vol. 32, no. 5, pp. 427–451, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. S. B. Hsu and T. W. Huang, “Global stability for a class of predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 55, no. 3, pp. 763–783, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. M. C. Varriale and A. A. Gomes, “A study of a three species food chain,” Ecological Modelling, vol. 110, no. 2, pp. 119–133, 1998. View at Publisher · View at Google Scholar
  9. S. W. Zhang and L. S. Chen, “Chaos in three species food chain system with impulsive perturbations,” Chaos, Solitons and Fractals, vol. 24, no. 1, pp. 73–83, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. K. McCann and P. Yodzis, “Bifurcation structure of a 3-species food chain model,” Theoretical Population Biology, vol. 48, no. 2, pp. 93–125, 1995. View at Publisher · View at Google Scholar
  11. S. B. Hsu and T. W. Hwang, “Hopf bifurcation analysis for a predator-prey system of Holling and Leslie type,” Taiwanese Journal of Mathematics, vol. 3, no. 1, pp. 35–53, 1999. View at Zentralblatt MATH
  12. H. I. Freedman and P. Waltman, “Mathematical analysis of some three-species food-chain models,” Mathematical Biosciences, vol. 33, no. 3-4, pp. 257–276, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. J. M. Ginoux, B. Rossetto, and J. L. Jamet, “Chaos in a three-dimensional Volterra-Gause model of predator-prey type,” International Journal of Bifurcation and Chaos, vol. 15, no. 5, pp. 1689–1708, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. A. Hastings and T. Powell, “Chaos in three-species food chain,” Ecology, vol. 72, no. 3, pp. 896–903, 1991. View at Publisher · View at Google Scholar
  15. W. Jiang and J. Wei, “Bifurcation analysis in a limit cycle oscillator with delayed feedback,” Chaos, Solitons and Fractals, vol. 23, no. 3, pp. 817–831, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. H. B. Wang and W. H. Jiang, “Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback,” Journal of Mathematical Analysis and Applications, vol. 368, no. 1, pp. 9–18, 2010. View at Publisher · View at Google Scholar
  17. J. J. Wei and W. H. Jiang, “Bifurcation analysis in van der Pol's oscillator with delayed feedback,” Journal of Computational and Applied Mathematics, vol. 213, no. 2, pp. 604–615, 2008. View at Publisher · View at Google Scholar
  18. R. Xu and M. A. J. Chaplain, “Persistence and global stability in a delayed Gause-type predator-prey system without dominating instantaneous negative feedbacks,” Journal of Mathematical Analysis and Applications, vol. 265, no. 1, pp. 148–162, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. G. Liu, W. Yan, and J. Yan, “Positive periodic solutions for a class of neutral delay Gause-type predator-prey system,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 10, pp. 4438–4447, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. Y. K. Li and Y. Kuang, “Periodic solutions in periodic delayed Gause-type predator-prey systems,” Preceedings of Dynamical Systems and Applications, vol. 3, pp. 375–382, 2001.
  21. T. Zhao, Y. Kuang, and H. L. Smith, “Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems,” Nonlinear Analysis. Theory, Methods & Applications. Series A, vol. 28, no. 8, pp. 1373–1394, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. X. Ding and J. Jiang, “Positive periodic solutions in delayed Gause-type predator-prey systems,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1220–1230, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. 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
  24. S. G. Ruan and J. J. Wei, “On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion,” Mathematical Medicine and Biology, vol. 18, no. 1, pp. 41–52, 2001.
  25. X. Li and J. Wei, “On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays,” Chaos, Solitons and Fractals, vol. 26, no. 2, pp. 519–526, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridgem, UK, 1981.