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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 592547, 10 pages
http://dx.doi.org/10.1155/2013/592547
Research Article

Global Stability and Bifurcations of a Diffusive Ratio-Dependent Holling-Tanner System

College of Science, China University of Petroleum (East China), Qingdao, Shandong 266580, China

Received 8 May 2013; Accepted 17 July 2013

Academic Editor: Chun-Lei Tang

Copyright © 2013 Wenjie Zuo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey system subject to Neumann boundary conditions are considered. By choosing the ratio of intrinsic growth rates of predators to preys as a bifurcation parameter, the existence and stability of spatially homogeneous and nonhomogeneous Hopf bifurcations and steady state bifurcation are investigated in detail. Meanwhile, we show that Turing instability takes place at a certain critical value; that is, the stationary solution becomes unstable induced by diffusion. Particularly, the sufficient conditions of the global stability of the positive constant coexistence are given by the upper-lower solutions method.