Abstract and Applied Analysis

Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay

Jianming Zhang, Lijun Zhang, and Chaudry Masood Khalique

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Abstract

The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 539684, 7 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412279783

Digital Object Identifier
doi:10.1155/2014/539684

Mathematical Reviews number (MathSciNet)
MR3191051

Zentralblatt MATH identifier
07022582

Citation

Zhang, Jianming; Zhang, Lijun; Khalique, Chaudry Masood. Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 539684, 7 pages. doi:10.1155/2014/539684. https://projecteuclid.org/euclid.aaa/1412279783


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References

  • A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific, Singapore, 1998.
  • Y. A. Kuznetsov, “Practical computation of normal forms on center manifolds at degenerate Bogdanov-Takens bifurcations,” International Journal of Bifurcation and Chaos, vol. 15, no. 11, pp. 3535–3536, 2005.
  • J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer, Heidelberg, Germany, 1977.
  • K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, The Netherlands, 1992.
  • Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, NY, USA, 1993.
  • E. Beretta and Y. Kuang, “Convergence results in a well-known delayed predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 204, no. 3, pp. 840–853, 1996.
  • 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.
  • T. Faria, “Stability and bifurcation for a delayed predator-prey model and the effect of diffusion,” Journal of Mathematical Analysis and Applications, vol. 254, no. 2, pp. 433–463, 2001.
  • K. Gopalsamy, “Harmless delays in model systems,” Bulletin of Mathematical Biology, vol. 45, no. 3, pp. 295–309, 1983.
  • K. Gopalsamy, “Delayed responses and stability in two-species systems,” Australian Mathematical Society Journal B, vol. 25, no. 4, pp. 473–500, 1984.
  • R. M. May, “Time delay versus stability in population models with two and three trophic levels,” Ecology, vol. 4, no. 2, pp. 315–325, 1973.
  • 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.
  • 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.
  • Y. Song, S. Yuan, and J. Zhang, “Bifurcation analysis in the delayed Leslie-Gower predator-prey system,” Applied Mathematical Modelling, vol. 33, no. 11, pp. 4049–4061, 2009.
  • 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.
  • Z. Liu and R. Yuan, “Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 521–537, 2004.
  • 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: Mathematical Analysis, vol. 10, no. 6, pp. 863–874, 2003.
  • B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981.
  • Y. Song, M. Han, and J. Wei, “Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays,” Physica D, vol. 200, pp. 185–204, 2005. \endinput