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Global stability in a diffusive Holling-Tanner predator-prey model. (English) Zbl 06046349
Summary: A diffusive Holling-Tanner predator-prey model with no-flux boundary condition is considered, and it is proved that the unique constant equilibrium is globally asymptotically stable under a new simpler parameter condition.

35KParabolic equations and systems
Full Text: DOI
[1] Peng, R.; Wang, M. X.: Global stability of the equilibrium of a diffusive Holling--tanner prey--predator model, Appl. math. Lett. 20, No. 6, 664-670 (2007) · Zbl 1125.35009 · doi:10.1016/j.aml.2006.08.020
[2] Tanner, J. T.: The stability and the intrinsic growth rates of prey and predator populations, Ecology 56, 855-867 (1975)
[3] May, R. M.: Stability and complexity in model ecosystems, (1974)
[4] Leslie, P. H.: Some further notes on the use of matrices in population mathematics, Biometrika 35, No. 3--4, 213 (1948) · Zbl 0034.23303
[5] Leslie, P. H.; Gower, J. C.: The properties of a stochastic model for the predator--prey type of interaction between two species, Biometrika 47, No. 3--4, 219 (1960) · Zbl 0103.12502
[6] Holling, C. S.: The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Can. entomol. 91, No. 5, 293-320 (1959)
[7] Murray, J. D.: Mathematical biology. I: an introduction, Interdisciplinary applied mathematics 17 (2002)
[8] Hsu, S. B.; Huang, T. W.: Global stability for a class of predator--prey system, SIAM J. Appl. math. 55, No. 3, 763-783 (1995) · Zbl 0832.34035 · doi:10.1137/S0036139993253201
[9] Hsu, S. B.; Hwang, T. W.: Uniqueness of limit cycles for a predator--prey system of Holling and Leslie type, Can. appl. Math. Q. 6, No. 2, 91-117 (1998) · Zbl 0981.92036
[10] Peng, R.; Wang, M. X.: Positive steady states of the Holling--tanner prey--predator model with diffusion, Proc. roy. Soc. Edinburgh sect. A 135, No. 01, 149-164 (2005) · Zbl 1144.35409 · doi:10.1017/S0308210500003814
[11] X. Li, W.H. Jiang, J.P. Shi, Hopf bifurcation and Turing instability in the reaction--diffusion Holling--Tanner predator--prey model (2010) (submitted for publication). · Zbl 1267.35236
[12] Du, Y.; Hsu, S. B.: A diffusive predator--prey model in heterogeneous environment, J. differential equations 203, No. 2, 331-364 (2004) · Zbl 02105819
[13] S.S. Chen, J.P. Shi, J.J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie--Gower predator--prey system, Internat. J. Bifur. Chaos (2011) (in press). · Zbl 1270.35376
[14] Pao, C. V.: On nonlinear reaction--diffusion systems, J. math. Anal. appl. 87, No. 1, 165-198 (1982) · Zbl 0488.35043
[15] Pao, C. V.: Nonlinear parabolic and elliptic equations, (1992) · Zbl 0777.35001
[16] S.S. Chen, J.P. Shi, Global attractivity of equilibrium in Gierer--Meinhardt system with saturation and gene expression time delays (2011) (submitted for publication). · Zbl 1274.92020
[17] Hsu, S. B.: A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwanese J. Math. 9, No. 2, 151-173 (2005) · Zbl 1087.34031