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Qualitative analysis of a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth. (English) Zbl 1264.34170
Summary: We investigate the dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth subject to zero-flux boundary condition. Some qualitative properties, including the dissipation, persistence, and local and global stability of positive constant solution, are discussed. Moreover, we give the refined a priori estimates of positive solutions and derive some results for the existence and nonexistence of nonconstant positive steady state.

MSC:
34K60Qualitative investigation and simulation of models
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)
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References:
[1] J. T. Tanner, “The stability and the intrinsic growth rates of prey and predator populations,” Ecology, vol. 56, pp. 855-886, 1975.
[2] 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. · Zbl 0652.92019 · doi:10.1007/BF02459707
[3] 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.
[4] M. P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, Princeton University Press, Princeton, NJ, USA, 1978. · Zbl 0429.92018
[5] 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. · Zbl 0103.12502 · doi:10.1093/biomet/47.3-4.219
[6] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA, 1973.
[7] 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. · Zbl 0832.34035 · doi:10.1137/S0036139993253201
[8] 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. · Zbl 0880.34028 · doi:10.5565/PUBLMAT_41197_09 · eudml:41283
[9] 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. · Zbl 0934.92027 · doi:10.1137/S0036139997318457
[10] 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.
[11] R. Arditi and H. Saiah, “Empirical evidence of the role of heterogeneity in ratio-dependent consumption,” Ecology, vol. 73, pp. 1544-1551, 1992.
[12] 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.
[13] 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. · Zbl 1177.34103 · doi:10.1016/j.jmaa.2009.03.072
[14] 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.
[15] 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.
[16] 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. · Zbl 1124.34030 · doi:10.1016/j.jmaa.2006.12.079
[17] 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. · Zbl 06036392 · doi:10.1016/j.mbs.2011.12.005
[18] F. E. Smith, “Population dynamics in Daphnia Magna and a new model for population growth,” Ecology, vol. 44, pp. 651-663, 1963.
[19] E. C. Pielou, An Introduction to Mathematical Ecology, Wiley, New York, NY, USA, 1969. · Zbl 0259.92001
[20] 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.
[21] 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. · Zbl 0701.92021 · doi:10.1016/0022-247X(90)90369-Q
[22] 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. · Zbl 0927.35049 · doi:10.1006/jmaa.1999.6332
[23] 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. · Zbl 1025.34070 · doi:10.1007/s102550200030
[24] 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. · Zbl 1087.35053 · doi:10.1016/j.mcm.2005.05.013
[25] D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 61 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981. · Zbl 0456.35001
[26] 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. · Zbl 0676.35030 · doi:10.1016/0022-0396(88)90147-7
[27] Y. Lou and W.-M. Ni, “Diffusion, self-diffusion and cross-diffusion,” Journal of Differential Equations, vol. 131, no. 1, pp. 79-131, 1996. · Zbl 0867.35032 · doi:10.1006/jdeq.1996.0157
[28] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute, 2nd edition, 2001. · Zbl 0992.47023
[29] 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. · Zbl 1148.35094 · doi:10.1088/0951-7715/21/7/006