Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2013, Article ID 321930, 15 pages
http://dx.doi.org/10.1155/2013/321930
Research Article

Stability and Global Hopf Bifurcation Analysis on a Ratio-Dependent Predator-Prey Model with Two Time Delays

1Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, China
2Department of Mathematics and Information Science, Zhoukou Normal University, Zhoukou, Henan 466001, China

Received 4 September 2013; Revised 3 November 2013; Accepted 5 November 2013

Academic Editor: István Györi

Copyright © 2013 Huitao Zhao 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. R. Arditi and L. R. Ginzburg, “Coupling in predator-prey dynamics: ratio-dependence,” Journal of Theoretical Biology, vol. 139, no. 3, pp. 311–326, 1989. View at Google Scholar
  2. D. Xiao, W. Li, and M. Han, “Dynamics in a ratio-dependent predator-prey model with predator harvesting,” Journal of Mathematical Analysis and Applications, vol. 324, no. 1, pp. 14–29, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. Xiao and J. Cao, “Hopf bifurcation and non-hyperbolic equilibrium in a ratio-dependent predator-prey model with linear harvesting rate: analysis and computation,” Mathematical and Computer Modelling, vol. 50, no. 3-4, pp. 360–379, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. Xu, Q. Gan, and Z. Ma, “Stability and bifurcation analysis on a ratio-dependent predator-prey model with time delay,” Journal of Computational and Applied Mathematics, vol. 230, no. 1, pp. 187–203, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Y. Kuang and E. Beretta, “Global qualitative analysis of a ratio-dependent predator-prey system,” Journal of Mathematical Biology, vol. 36, no. 4, pp. 389–406, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S. Ruan and J. Wei, “Periodic solutions of planar systems with two delays,” Proceedings of the Royal Society of Edinburgh A, vol. 129, no. 5, pp. 1017–1032, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. 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 · View at MathSciNet
  8. B. Zheng, Y. Zhang, and C. Zhang, “Global existence of periodic solutions on a simplified BAM neural network model with delays,” Chaos, Solitons and Fractals, vol. 37, no. 5, pp. 1397–1408, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. C. Sun and M. Han, “Global Hopf bifurcation analysis on a BAM neural network with delays,” Mathematical and Computer Modelling, vol. 45, no. 1-2, pp. 61–67, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  10. F. Botelho and V. A. Gaiko, “Global analysis of planar neural networks,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 5, pp. 1002–1011, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. X.-Y. Meng, H.-F. Huo, and X.-B. Zhang, “Stability and global Hopf bifurcation in a delayed food web consisting of a prey and two predators,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4335–4348, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Y. Song, Y. Peng, and J. Wei, “Bifurcations for a predator-prey system with two delays,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 466–479, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. Wu, “Symmetric functional-differential equations and neural networks with memory,” Transactions of the American Mathematical Society, vol. 350, no. 12, pp. 4799–4838, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993. View at MathSciNet
  15. 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, Cambridge, Mass, USA, 1981. View at MathSciNet
  16. T. Faria and L. T. Magalhães, “Normal forms for retarded functional-differential equations and applications to Bogdanov-Takens singularity,” Journal of Differential Equations, vol. 122, no. 2, pp. 201–224, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. A. Chaplygin, Complete Works III, Akademia Nauk SSSR, Leningrad, Russia, 1935.
  18. N. S. Kurpel and V. I. Grechko, “On some modifications of Chaplygin's method for equations in partially orderedspaces,” Ukrainian Mathematical Journal, vol. 25, no. 1, pp. 30–36, 1973. View at Publisher · View at Google Scholar
  19. A. Dzieliński, “Stability of a class of adaptive nonlinear systems,” International Journal of Applied Mathematics and Computer Science, vol. 15, no. 4, pp. 455–462, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. X. Song and L. Chen, “Optimal harvesting and stability for a two-species competitive system with stage structure,” Mathematical Biosciences, vol. 170, no. 2, pp. 173–186, 2001. View at Publisher · View at Google Scholar · View at MathSciNet