Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 260798, 17 pages
http://dx.doi.org/10.1155/2012/260798
Research Article

Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey System

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

Received 30 October 2011; Accepted 19 December 2011

Academic Editor: Junjie Wei

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. H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57, Marcel Dekker, New York, NY, USA, 1980. View at Zentralblatt MATH
  2. N.G. Hairston, F. E. Smith, and L. B. Slobodkin, “Community structure, population control and competition,” The American Naturalist, vol. 94, pp. 421–425, 1960. View at Google Scholar
  3. G. J. Butler, S. B. Hsu, and P. Waltman, “Coexistence of competing predators in a chemostat,” Journal of Mathematical Biology, vol. 17, no. 2, pp. 133–151, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. K. S. Chêng, 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
  5. 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
  6. A. Hastings and T. Powell, “Chaos in a three-species food chain,” Ecology, vol. 72, no. 3, pp. 896–903, 1991. View at Google Scholar
  7. 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
  8. Z. Luo, “Optimal control for a predator-prey system with age-dependent,” International Journal of Biomathematics, vol. 2, no. 1, pp. 45–59, 2009. View at Publisher · View at Google Scholar
  9. C.-H. Chiu and S.-B. Hsu, “Extinction of top-predator in a three-level food-chain model,” Journal of Mathematical Biology, vol. 37, no. 4, pp. 372–380, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. 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 in Applied Sciences and Engineering, vol. 15, no. 5, pp. 1689–1708, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. 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
  12. S. Tang and L. Chen, “Global qualitative analysis for a ratio-dependent predator-prey model with delay,” Journal of Mathematical Analysis and Applications, vol. 266, no. 2, pp. 401–419, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. Z. Q. Lu and G. S. K. Wolkowicz, “Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates,” SIAM Journal on Applied Mathematics, vol. 52, no. 1, pp. 222–233, 1992. View at Publisher · View at Google Scholar
  14. A. Ardito and P. Ricciardi, “Lyapunov functions for a generalized Gause-type model,” Journal of Mathematical Biology, vol. 33, no. 8, pp. 816–828, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. S.-B. Hsu, “A survey of constructing Lyapunov functions for mathematical models in population biology,” Taiwanese Journal of Mathematics, vol. 9, no. 2, pp. 151–173, 2005. View at Google Scholar · View at Zentralblatt MATH
  16. 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
  17. Y. Kuang, “Global stability of Gause-type predator-prey systems,” Journal of Mathematical Biology, vol. 28, no. 4, pp. 463–474, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. S.-B. Hsu, T.-W. Hwang, and Y. Kuang, “Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system,” Journal of Mathematical Biology, vol. 42, no. 6, pp. 489–506, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. L. Chen and F. Chen, “Global analysis of a harvested predator-prey model incorporating a constant prey refuge,” International Journal of Biomathematics, vol. 3, no. 2, pp. 205–223, 2010. View at Publisher · View at Google Scholar
  20. W. Ko and K. Ryu, “A qualitative study on general Gause-type predator-prey models with non-monotonic functional response,” Nonlinear Analysis. Real World Applications, vol. 10, no. 4, pp. 2558–2573, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. D. Xiao and S. Ruan, “Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response,” Journal of Differential Equations, vol. 176, no. 2, pp. 494–510, 2001. View at Publisher · View at Google Scholar
  22. Yu. A. Kuznetsov and S. Rinaldi, “Remarks on food chain dynamics,” Mathematical Biosciences, vol. 134, no. 1, pp. 1–33, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. 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
  24. H. Wang and W. 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
  25. J. Wei and W. 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
  26. J. J. Wei and S. G. Ruan, “Stability and global Hopf bifurcation for neutral differential equations,” Acta Mathematica Sinica, vol. 45, no. 1, pp. 93–104, 2002. View at Google Scholar · View at Zentralblatt MATH
  27. 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 Google Scholar · View at Zentralblatt MATH
  28. J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977.
  29. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41, Cambridge University Press, Cambridge, UK, 1981.
  30. B. C. Birch, Hosea, Joel and Amos, Westminster John Knox, Kentucky, Ky, USA, 1997.