Journal of Applied Mathematics

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

Shuang Guo and Weihua Jiang

Full-text: Open access

Abstract

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.

Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 260798, 17 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.jam/1355495084

Digital Object Identifier
doi:10.1155/2012/260798

Mathematical Reviews number (MathSciNet)
MR2904519

Zentralblatt MATH identifier
1248.34125

Citation

Guo, Shuang; Jiang, Weihua. Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey System. J. Appl. Math. 2012 (2012), Article ID 260798, 17 pages. doi:10.1155/2012/260798. https://projecteuclid.org/euclid.jam/1355495084


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