- 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 2013 (2013), Article ID 592547, 10 pages
Global Stability and Bifurcations of a Diffusive Ratio-Dependent Holling-Tanner System
College of Science, China University of Petroleum (East China), Qingdao, Shandong 266580, China
Received 8 May 2013; Accepted 17 July 2013
Academic Editor: Chun-Lei Tang
Copyright © 2013 Wenjie Zuo. 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.
- G. Guo and J. Wu, “The effect of mutual interference between predators on a predator-prey model with diffusion,” Journal of Mathematical Analysis and Applications, vol. 389, no. 1, pp. 179–194, 2012.
- J. Tanner, “The stability and the intrinstic growth rates of prey and predator populations,” Ecology, vol. 56, pp. 855–867, 1975.
- D. J. Wollkind, J. B. Collings, and J. A. Logan, “Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees,” Bulletin of Mathematical Biology, vol. 50, no. 4, pp. 379–409, 1988.
- E. Sáez and E. González-Olivares, “Dynamics of a predator-prey model,” SIAM Journal on Applied Mathematics, vol. 59, no. 5, pp. 1867–1878, 1999.
- C. S. Holling, “The functional response of predator to prey density and its role in mimicry and population regulation,” Memoirs of the Entomological Society of Canada, vol. 45, pp. 1–69, 1965.
- S. B. Hsu and T. W. Huang, “Global stability for a class of predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 55, no. 3, pp. 763–783, 1995.
- A. Gasull, R. Kooij, and J. Torregrosa, “Limit cycles in the Holling-Tanner model,” Publicacions Matemàtiques, vol. 41, pp. 149–167, 1997.
- J. Shi and X. Wang, “On global bifurcation for quasilinear elliptic systems on bounded domains,” Journal of Differential Equations, vol. 246, no. 7, pp. 2788–2812, 2009.
- F. Yi, J. Wei, and J. Shi, “Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system,” Journal of Differential Equations, vol. 246, no. 5, pp. 1944–1977, 2009.
- W. Zuo and J. Wei, “Multiple bifurcations and spatiotemporal patterns for a coupled two-cell Brusselator model,” Dynamics of Partial Differential Equations, vol. 8, no. 4, pp. 363–384, 2011.
- Z.-P. Ma and W.-T. Li, “Bifurcation analysis on a diffusive Holling-Tanner predator-prey model,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 37, no. 6, pp. 4371–4384, 2013.
- R. Peng and M. Wang, “Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model,” Applied Mathematics Letters, vol. 20, no. 6, pp. 664–670, 2007.
- S. Chen and J. Shi, “Global stability in a diffusive Holling-Tanner predator-prey model,” Applied Mathematics Letters, vol. 25, no. 3, pp. 614–618, 2012.
- R. Arditi and L. R. Ginzburg, “Coupling in predator-prey dynamics: ratio-dependence,” Journal of Theoretical Biology, vol. 139, no. 3, pp. 311–326, 1989.
- R. Arditi, L. R. Ginzburg, and H. R. Akcakaya, “Variation in plankton densities among lakes: a case for ratio-dependent models,” The American Naturalist, vol. 138, pp. 1287–1296, 1991.
- L. R. Ginzburg and H. R. Akcakaya, “Consequences of ratio-dependent predation for steady-state properties of ecosystems,” Ecology, vol. 73, no. 5, pp. 1536–1543, 1992.
- Q. Wang, Y. Zhang, Z. Wang, M. Ding, and H. Zhang, “Periodicity and attractivity of a ratio-dependent Leslie system with impulses,” Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 212–220, 2011.
- Q. Wang, Jizhou, Z. Wang, M. Ding, and H. Zhang, “Existence and attractivity of a periodic solution for a ratio-dependent Leslie system with feedback controls,” Nonlinear Analysis. Real World Applications, vol. 12, no. 1, pp. 24–33, 2011.
- M. Banerjee and S. Banerjee, “Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model,” Mathematical Biosciences, vol. 236, no. 1, pp. 64–76, 2012.
- Z. Liang and H. Pan, “Qualitative analysis of a ratio-dependent Holling-Tanner model,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 954–964, 2007.
- B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41, Cambridge University Press, Cambridge, UK, 1981.
- 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.
- C. V. Pao, “On nonlinear reaction-diffusion systems,” Journal of Mathematical Analysis and Applications, vol. 87, no. 1, pp. 165–198, 1982.
- C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992.