- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 168340, 13 pages
The Hopf Bifurcation for a Predator-Prey System with -Logistic Growth and Prey Refuge
School of Mathematics and Information Sciences, Henan University, Kaifeng 475001, China
Received 12 April 2013; Accepted 6 June 2013
Academic Editor: Luca Guerrini
Copyright © 2013 Shaoli Wang and Zhihao Ge. 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.
- 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.