- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 236484, 19 pages
Stability and Hopf Bifurcation in a Modified Holling-Tanner Predator-Prey System with Multiple Delays
1Key Laboratory of Advanced Process Control for Light Industry of Ministry of Education, Jiangnan University, Wuxi 214122, China
2School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China
3Department of Science, Bengbu College, Bengbu 233030, China
Received 9 August 2012; Revised 17 September 2012; Accepted 4 October 2012
Academic Editor: Kunquan Lan
Copyright © 2012 Zizhen 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.
- 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.
- A. A. Berryman, “The origins and evolution of predator-prey theory,” Ecology, vol. 73, no. 5, pp. 1530–1535, 1992.
- H. J. Guo and X. X. Chen, “Existence and global attractivity of positive periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5830–5837, 2011.
- X. L. Wang, Z. J. Du, and J. Liang, “Existence and global attractivity of positive periodic solution to a Lotka-Volterra model,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 4054–4061, 2010.
- Z. D. Teng, “On the persistence and positive periodic solution for planar competing Lotka-Volterra systems,” Annals of Differential Equations, vol. 13, no. 3, pp. 275–286, 1997.
- P. H. Leslie, “Some further notes on the use of matrices in population mathematics,” Biometrika, vol. 35, no. 3-4, pp. 213–245, 1948.
- M. A. Aziz-Alaoui and M. D. Okiye, “Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes,” Applied Mathematics Letters, vol. 16, no. 7, pp. 1069–1075, 2003.
- Z. Zhao, L. Yang, and L. Chen, “Impulsive perturbations of a predator-prey system with modified Leslie-Gower and Holling type II schemes,” Journal of Applied Mathematics and Computing, vol. 35, no. 1-2, pp. 119–134, 2011.
- S. Gakkhar and A. Singh, “Complex dynamics in a prey predator system with multiple delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 914–929, 2012.
- G. O. Eduardo, M. L. Jaime, A. Rojas-Palma, and J. D. Flores, “Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey,” Applied Mathematical Modelling, vol. 35, no. 1, pp. 366–381, 2011.
- J. T. Tanner, “The stability and intrinsic growth rates of prey and predator populations,” Ecology, vol. 56, no. 4, pp. 855–867, 1975.
- A. Gasull, R. E. Kooij, and J. Torregrosa, “Limit cycles in the Holling-Tanner model,” Publicacions Matemàtiques, vol. 41, no. 1, pp. 149–167, 1997.
- E. Sáez and E. G. Olivares, “Dynamics of a predator-prey model,” SIAM Journal on Applied Mathematics, vol. 59, no. 5, pp. 1867–1878, 1999.
- T. Saha and C. G. Chakrabarti, “Dynamical analysis of a delayed ratio-dependent Holling-Tanner predator-prey model,” Journal of Mathematical Analysis and Applications, vol. 358, no. 2, pp. 389–402, 2009.
- K. Q. Lan and C. R. Zhu, “Phase portraits, Hopf bifurcations and limit cycles of the Holling-Tanner models for predator-prey interactions,” Nonlinear Analysis: Real World Applications, vol. 12, no. 4, pp. 1961–1973, 2011.
- J.-F. Zhang, “Bifurcation analysis of a modified Holling-Tanner predator-prey model with time delay,” Applied Mathematical Modelling, vol. 36, no. 3, pp. 1219–1231, 2012.
- J. R. Beddington, “Mutual interference between parasites or predators and its effect on searching efficiency,” Journal of Animal Ecology, vol. 44, no. 1, pp. 331–340, 1975.
- D. L. DeAngelis, R. A. Goldstein, and R. V. ONeill, “A model for trophic interaction,” Ecology, vol. 56, no. 4, pp. 881–892, 1975.
- T.-W. Hwang, “Global analysis of the predator-prey system with Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 395–401, 2003.
- M. Fan and Y. Kuang, “Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 15–39, 2004.
- K. Q. Lan and C. R. Zhu, “Phase portraits of predator-prey systems with harvesting rates,” Discrete and Continuous Dynamical Systems A, vol. 32, no. 3, pp. 901–933, 2012.
- Z. Q. Lu and X. Liu, “Analysis of a predator-prey model with modified Holling-Tanner functional response and time delay,” Nonlinear Analysis: Real World Applications, vol. 9, no. 2, pp. 641–650, 2008.
- S. G. Ruan and J. 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.
- Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191, Academic Press, New York, NY, USA, 1993.
- J. K. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977.
- B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981.
- L. Feng and L. H. Wei, “Hopf bifurcation of a predator-prey model with time delay and stage structure for the prey,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 672–679, 2012.
- Y. Yang, “Hopf bifurcation in a two-competitor, one-prey system with time delay,” Applied Mathematics and Computation, vol. 214, no. 1, pp. 228–235, 2009.
- Y. L. Song, S. L. Yuan, and J. M. Zhang, “Bifurcation analysis in the delayed Leslie-Gower predator-prey system,” Applied Mathematical Modelling, vol. 33, no. 11, pp. 4049–4061, 2009.
- S. Yuan and Y. Song, “Stability and Hopf bifurcations in a delayed Leslie-Gower predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 82–100, 2009.
- Y. F. Ma, “Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 370–375, 2012.
- X.-Y. Meng, H.-F. Huo, and H. Xiang, “Hopf bifurcation in a three-species system with delays,” Journal of Applied Mathematics and Computing, vol. 35, no. 1-2, pp. 635–661, 2011.
- S. Gakkhar, K. Negi, and S. K. Sahani, “Effects of seasonal growth on ratio dependent delayed prey predator system,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 850–862, 2009.