Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model. (English) Zbl 1125.35009

In this paper a Holling-Tanner prey-predator model is considered in the form \[ \begin{cases} \frac{\partial u}{\partial t} - d_1\Delta u = au-u^2 - \frac{uv}{m+u} &\quad\text{in}\quad \Omega\times (0,\infty),\\ \frac{\partial v}{\partial t} - d_2\Delta v = bv - \frac{v^2}{\gamma u} &\quad\text{in}\quad \Omega\times (0,\infty),\\ \partial_\eta u = \partial_\eta v = 0&\quad\text{on}\quad \partial\Omega\times (0,\infty),\\ u(x,0)=u_0(x)>0,\quad v(x,0)=v_0(x)\geq 0,\not\equiv0 &\quad\text{on}\quad \overline\Omega, \end{cases} \tag{1} \] where \(u(x,t)\) and \(v(x,t)\) respectively represent the species densities of the prey and predator. \(\eta\) is the outward unit normal vector on the smooth boundary \(\partial\Omega\) and \(\partial_\eta = \partial/\partial\eta\). The constants \(d_i\) \((i=1,2)\) are the diffusion coefficients corresponding to \(u\) and \(v\) respectively, and all the parameters appearing in (1) are assumed to be positive. The authors obtain some results for the global stability of the unique positive equilibrium of this model, and thus improve some previous results.


35B35 Stability in context of PDEs
92D25 Population dynamics (general)
Full Text: DOI


[1] Braza, P. A., The bifurcation structure of the Holling-Tanner model for predator-prey interactions using two-timing, SIAM J. Appl. Math., 63, 889-904 (2003) · Zbl 1035.34043
[2] Collings, J. B., Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge, Bull. Math. Biol., 57, 63-76 (1995) · Zbl 0810.92024
[3] Henry, D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840 (1993), Springer-Verlag: Springer-Verlag Berlin, New York, 3rd printing
[4] Hassell, M. P., The Dynamics of Arthropod Predator-Prey Systems (1978), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0429.92018
[5] Holling, C. S., The functional response of invertebrate predators to prey density, Mem. Entomol. Soc. Can., 45, 3-60 (1965)
[6] Hsu, S. B.; Huang, T. W., Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55, 763-783 (1995) · Zbl 0832.34035
[7] May, R. M., Stability and Complexity in Model Ecosystems (1973), Princeton University Press: Princeton University Press Princeton, NJ
[8] May, R. M., Limit cycles in predator-prey communities, Science, 177, 900-902 (1972)
[9] Peng, R.; Wang, M. X., Positive steady-states of the Holling-Tanner prey-predator model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 135, 149-164 (2005) · Zbl 1144.35409
[10] Saez, E.; Gonzalez-Olivares, E., Dynamics of a predator-prey model, SIAM J. Appl. Math., 59, 1867-1878 (1999) · Zbl 0934.92027
[11] Tanner, J. T., The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56, 855-867 (1975)
[12] Wang, M. X., Non-constant positive steady states of the Sel’kov model, J. Differential Equations, 190, 2, 600-620 (2003) · Zbl 1163.35362
[13] Wollkind, D. J.; Collings, J. B.; Logan, J. A., Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit flies, Bull. Math. Biol., 50, 379-409 (1988) · Zbl 0652.92019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.