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Journal of Applied Mathematics
Volume 2012, Article ID 260798, 17 pages
Research Article

Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey System

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2School of Mathematical Sciences, Daqing Normal University, Daqing 163712, China

Received 30 October 2011; Accepted 19 December 2011

Academic Editor: Junjie Wei

Copyright © 2012 Shuang Guo and Weihua Jiang. 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.


A class of three-dimensional Gause-type predator-prey model is considered. Firstly, local stability of equilibrium indicating the extinction of top-predator is obtained. Meanwhile, we construct a Lyapunov function, which is an extension of the Lyapunov functions constructed by Hsu for predator-prey system (2005), to give the global stability of the equilibrium. Secondly, we analyze the stability of coexisting equilibrium of predator-prey system with time delay when the predator catches the prey of pregnancy or with growth time. The delay can lead to periodic solutions, which is consistent with the law of growth for birds and some mammals. Further, an explicit formula is given which determines the stability of the bifurcating periodic solutions theoretically and the existence of periodic solutions is displayed by numerical simulations.