Abstract and Applied Analysis

The Hopf Bifurcation for a Predator-Prey System with θ -Logistic Growth and Prey Refuge

Shaoli Wang and Zhihao Ge

Full-text: Open access

Abstract

The Hopf bifurcation for a predator-prey system with θ -logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index- θ passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 168340, 13 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393449393

Digital Object Identifier
doi:10.1155/2013/168340

Mathematical Reviews number (MathSciNet)
MR3081591

Zentralblatt MATH identifier
1303.34067

Citation

Wang, Shaoli; Ge, Zhihao. The Hopf Bifurcation for a Predator-Prey System with $\theta $ -Logistic Growth and Prey Refuge. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 168340, 13 pages. doi:10.1155/2013/168340. https://projecteuclid.org/euclid.aaa/1393449393


Export citation

References

  • C. S. Holling, “Some characteristics of simple types of predation and parasitism,” Canadian Entomologist, vol. 91, pp. 385–398, 1959.
  • M. P. Hassell and R. M. May, “Stability in insect host-parasite models,” Journal of Animal Ecology, vol. 42, pp. 693–726, 1973.
  • J. M. Smith, Models in Ecology, Cambridge University, Cambridge, UK, 1974.
  • M. P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, vol. 13, Princeton University, Princeton, NJ, USA, 1978.
  • L. Ji and C. Wu, “Qualitative analysis of a predator-prey model with constant-rate prey harvesting incorporating a constant prey refuge,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2285–2295, 2010.
  • A. Sih, “Prey refuges and predator-prey stability,” Theoretical Population Biology, vol. 31, no. 1, pp. 1–12, 1987.
  • R. J. Taylor, Predation, Chapman and Hall, New York, NY, USA, 1984.
  • E. González-Olivares and R. Ramos-Jiliberto, “Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability,” Ecological Modelling, vol. 166, pp. 135–146, 2003.
  • V. Krivan, “Effects of optimal antipredator behavior of prey on predator-prey dynamics: the role of refuges,” Theoretical Population Biology, vol. 53, pp. 131–142, 1998.
  • Z. Ma, W. Li, Y. Zhao, W. Wang, H. Zhang, and Z. Li, “Effects of prey refuges on a predator-prey model with a class of functional responses: the role of refuges,” Mathematical Biosciences, vol. 218, no. 2, pp. 73–79, 2009.
  • L. Chen, F. Chen, and L. Chen, “Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 246–252, 2010.
  • Y. D. Tao, X. Wang, and X. Y. Song, “Effect of prey refuge on a harvested predator prey model with generalized functional response,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, pp. 1052–1059, 2010.
  • W. Ko and K. Ryu, “A qualitative study on general Gause-type predator-prey models with constant diffusion rates,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 217–230, 2008.
  • A. Tsoularis and J. Wallace, “Analysis of logistic growth models,” Mathematical Biosciences, vol. 179, no. 1, pp. 21–55, 2002.
  • J. Hale, Theory of Functional Differential Equations, Springer, Heidelberg, Germany, 1977.
  • E. Beretta and Y. Kuang, “Geometric stability switch criteria in delay differential systems with delay dependent parameters,” SIAM Journal on Mathematical Analysis, vol. 33, no. 5, pp. 1144–1165, 2002.
  • B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41, Cambridge University Press, Cambridge, UK, 1981.
  • 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.
  • 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.