Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model
type: Item
instance of: scholarly article
title: Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model
zbMATH Open document ID: 1125.35009
author: Rui Peng
author: Mingxin Wang
published in: Applied Mathematics Letters
publication date: 2007-05-04T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 5149470
zbMATH Keywords: Holling-Tanner prey-predator model
zbMATH Keywords: diffusion
zbMATH Keywords: equilibrium
zbMATH Keywords: local stability
zbMATH Keywords: global stability
Wikidata QID: Q115598048
reviewed by: A. A. Martynyuk
MaRDI profile type: Publication
full work available at URL: j.aml.2006.08.020
OpenAlex ID: W2050741438
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DOI: 10.1016/J.AML.2006.08.020
description: scientific article; zbMATH DE number 5149470