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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 267173, 9 pages
Qualitative Analysis of a Diffusive Ratio-Dependent Holling-Tanner Predator-Prey Model with Smith Growth
1Faculty of Science, Shaanxi University of Science and Technology, Xi’an 710021, China
2Department of Biostatistics and Computational Biology, University of Rochester Medical Center, 601 Elmwood Avenue, P.O. Box 630, Rochester, NY 14642, USA
Received 28 January 2013; Accepted 15 February 2013
Academic Editor: Yonghui Xia
Copyright © 2013 Zongmin Yue and Wenjuan Wang. 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.
- J. T. Tanner, “The stability and the intrinsic growth rates of prey and predator populations,” Ecology, vol. 56, pp. 855–886, 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.
- C. S. Holling, “The functional response of predators to prey density and its role in mimicry and population regulation,” Memoirs of the Entomological Society of Canada, vol. 45, supplement 45, pp. 5–60, 1965.
- M. P. Hassell, The Dynamics of Arthropod Predator–Prey Systems, Princeton University Press, Princeton, NJ, USA, 1978.
- P. H. Leslie and J. C. Gower, “The properties of a stochastic model for the predator-prey type of interaction between two species,” Biometrika, vol. 47, pp. 219–234, 1960.
- R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA, 1973.
- 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. 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. González-Olivares, “Dynamics of a predator-prey model,” SIAM Journal on Applied Mathematics, vol. 59, no. 5, pp. 1867–1878, 1999.
- 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, no. 5, pp. 287–296, 1991.
- R. Arditi and H. Saiah, “Empirical evidence of the role of heterogeneity in ratio-dependent consumption,” Ecology, vol. 73, pp. 1544–1551, 1992.
- A. P. Gutierrez, “The physiological basis of ratio-dependent predator-prey theory: a metabolic pool model of Nicholson’s blowflies as an example,” Ecology, vol. 73, pp. 1552–1563, 1992.
- T. Saha and C. 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.
- 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, N. Perrin, and H. Saiah, “Functional responses and heterogeneities: an experimental test with cladocerans,” Oikos, vol. 60, no. 1, pp. 69–75, 1991.
- 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.
- 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.
- F. E. Smith, “Population dynamics in Daphnia Magna and a new model for population growth,” Ecology, vol. 44, pp. 651–663, 1963.
- E. C. Pielou, An Introduction to Mathematical Ecology, Wiley, New York, NY, USA, 1969.
- T. G. Hallam and J. T. Deluna, “Effects of toxicants on populations: a qualitative approach III,” Journal of Theoretical Biology, vol. 109, no. 3, pp. 411–429, 1984.
- K. Gopalsamy, M. R. S. Kulenović, and G. Ladas, “Environmental periodicity and time delays in a “food-limited” population model,” Journal of Mathematical Analysis and Applications, vol. 147, no. 2, pp. 545–555, 1990.
- W. Feng and X. Lu, “On diffusive population models with toxicants and time delays,” Journal of Mathematical Analysis and Applications, vol. 233, no. 1, pp. 373–386, 1999.
- M. Fan and K. Wang, “Periodicity in a “food-limited” population model with toxicants and time delays,” Acta Mathematicae Applicatae Sinica, vol. 18, no. 2, pp. 309–314, 2002.
- W. Chen and M. Wang, “Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion,” Mathematical and Computer Modelling, vol. 42, no. 1-2, pp. 31–44, 2005.
- D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 61 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981.
- C.-S. Lin, W.-M. Ni, and I. Takagi, “Large amplitude stationary solutions to a chemotaxis system,” Journal of Differential Equations, vol. 72, no. 1, pp. 1–27, 1988.
- Y. Lou and W.-M. Ni, “Diffusion, self-diffusion and cross-diffusion,” Journal of Differential Equations, vol. 131, no. 1, pp. 79–131, 1996.
- L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute, 2nd edition, 2001.
- R. Peng, J. Shi, and M. Wang, “On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law,” Nonlinearity, vol. 21, no. 7, pp. 1471–1488, 2008.