About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 679602, 17 pages
http://dx.doi.org/10.1155/2013/679602
Research Article

On the Nature of Bifurcation in a Ratio-Dependent Predator-Prey Model with Delays

1Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 550004, China
2Department of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, China

Received 1 April 2013; Revised 29 May 2013; Accepted 2 June 2013

Academic Editor: Jinde Cao

Copyright © 2013 Changjin Xu and Yusen Wu. 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. Bhattacharyya and B. Mukhopadhyay, “On an eco-epidemiological model with prey harvesting and predator switching: local and global perspectives,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3824–3833, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. T. K. Kar and A. Ghorai, “Dynamic behaviour of a delayed predator-prey model with harvesting,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 9085–9104, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. K. Chakraborty, M. Chakraborty, and T. K. Kar, “Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay,” Nonlinear Analysis: Hybrid Systems, vol. 5, no. 4, pp. 613–625, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. Bhattacharyya and B. Mukhopadhyay, “Spatial dynamics of nonlinear prey-predator models with prey migration and predator switching,” Ecological Complexity, vol. 3, no. 2, pp. 160–169, 2006. View at Publisher · View at Google Scholar · View at Scopus
  5. X. Chang and J. Wei, “Hopf bifurcation and optimal control in a diffusive predator-prey system with time delay and prey harvesting,” Lithuanian Association of Nonlinear Analysts (LANA), vol. 17, no. 4, pp. 379–409, 2012. View at MathSciNet
  6. W. Ko and K. Ryu, “Coexistence states of a nonlinear Lotka-Volterra type predator-prey model with cross-diffusion,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e1109–e1115, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. Gao, L. Chen, and Z. Teng, “Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 721–729, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. T. K. Kar and U. K. Pahari, “Modelling and analysis of a prey-predator system with stage-structure and harvesting,” Nonlinear Analysis: Real World Applications, vol. 8, no. 2, pp. 601–609, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Y. Kuang and Y. Takeuchi, “Predator-prey dynamics in models of prey dispersal in two-patch environments,” Mathematical Biosciences, vol. 120, no. 1, pp. 77–98, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. K. Li and J. Wei, “Stability and Hopf bifurcation analysis of a prey-predator system with two delays,” Chaos, Solitons & Fractals, vol. 42, no. 5, pp. 2606–2613, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. M. May, “Time delay versus stability in population models with two and three trophic levels,” Ecology, vol. 54, no. 2, pp. 315–325, 1973.
  12. Prajneshu and P. Holgate, “A prey-predator model with switching effect,” Journal of Theoretical Biology, vol. 125, no. 1, pp. 61–66, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  13. 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
  14. Y. Song and J. Wei, “Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 301, no. 1, pp. 1–21, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. E. Teramoto, K. Kawasaki, and N. Shigesada, “Switching effect of predation on competitive prey species,” Journal of Theoretical Biology, vol. 79, no. 3, pp. 303–315, 1979. View at Publisher · View at Google Scholar · View at MathSciNet
  16. R. Xu, M. A. J. Chaplain, and F. A. Davidson, “Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments,” Nonlinear Analysis: Real World Applications, vol. 5, no. 1, pp. 183–206, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  17. R. Xu and Z. Ma, “Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure,” Chaos, Solitons & Fractals, vol. 38, no. 3, pp. 669–684, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. 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, vol. 28, no. 8, pp. 1373–1394, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. X. Zhou, X. Shi, and X. Song, “Analysis of nonautonomous predator-prey model with nonlinear diffusion and time delay,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 129–136, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. N. Bairagi and D. Jana, “On the stability and Hopf bifurcation of a delay-induced predator-prey system with habitat complexity,” Applied Mathematical Modelling, vol. 35, no. 7, pp. 3255–3267, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. L. Zhang and C. Lu, “Periodic solutions for a semi-ratio-dependent predator-prey system with Holling IV functional response,” Journal of Applied Mathematics and Computing, vol. 32, no. 2, pp. 465–477, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. X. Tian and R. Xu, “Global dynamics of a predator-prey system with Holling type II functional response,” Lithuanian Association of Nonlinear Analysts (LANA), vol. 16, no. 2, pp. 242–253, 2011. View at MathSciNet
  23. 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
  24. Y. Xia, J. Cao, and M. Lin, “Discrete-time analogues of predator-prey models with monotonic or nonmonotonic functional responses,” Nonlinear Analysis: Real World Applications, vol. 8, no. 4, pp. 1079–1095, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Z. Cheng, Y. Lin, and J. Cao, “Dynamical behaviors of a partial-dependent predator-prey system,” Chaos, Solitons and Fractals, vol. 28, no. 1, pp. 67–75, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. M. Xiao and J. Cao, “Genetic oscillation deduced from Hopf bifurcation in a genetic regulatory network with delays,” Mathematical Biosciences, vol. 215, no. 1, pp. 55–63, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. W.-Y. Wang and L.-J. Pei, “Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system,” Acta Mechanica Sinica, vol. 27, no. 2, pp. 285–296, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  28. 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
  29. J. Cao and M. Xiao, “Stability and Hopf bifurcation in a simplified BAM neural network with two time delays,” IEEE Transactions on Neural Networks, vol. 18, no. 2, pp. 416–430, 2007. View at Publisher · View at Google Scholar · View at Scopus
  30. Z. Ge and J. Yan, “Hopf bifurcation of a predator-prey system with stage structure and harvesting,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 2, pp. 652–660, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. J. Hale, Theory of Functional Differential Equations, vol. 3, Springer, Berlin, Germany, 2nd edition, 1977. View at MathSciNet
  32. H. Hu and L. Huang, “Stability and Hopf bifurcation analysis on a ring of four neurons with delays,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 587–599, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press; INC, Boston, Mass, USA, 1993. View at MathSciNet