- About this Journal ·
- Abstracting and Indexing ·
- 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 2014 (2014), Article ID 539684, 7 pages
Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay
1Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China
2International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa
Received 10 January 2014; Accepted 30 January 2014; Published 27 March 2014
Academic Editor: Hossein Jafari
Copyright © 2014 Jianming Zhang 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.
- 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.