- 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
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 679602, 17 pages
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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Prajneshu and P. Holgate, “A prey-predator model with switching effect,” Journal of Theoretical Biology, vol. 125, no. 1, pp. 61–66, 1987.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- J. Hale, Theory of Functional Differential Equations, vol. 3, Springer, Berlin, Germany, 2nd edition, 1977.
- 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.
- 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.