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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.

MSC:
35B35 Stability in context of PDEs
92D25 Population dynamics (general)
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