## Abstract and Applied Analysis

### The Hopf Bifurcation for a Predator-Prey System with $\theta$-Logistic Growth and Prey Refuge

#### Abstract

The Hopf bifurcation for a predator-prey system with $\theta$-logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index-$\theta$ passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 168340, 13 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393449393

Digital Object Identifier
doi:10.1155/2013/168340

Mathematical Reviews number (MathSciNet)
MR3081591

Zentralblatt MATH identifier
1303.34067

#### Citation

Wang, Shaoli; Ge, Zhihao. The Hopf Bifurcation for a Predator-Prey System with $\theta$ -Logistic Growth and Prey Refuge. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 168340, 13 pages. doi:10.1155/2013/168340. https://projecteuclid.org/euclid.aaa/1393449393

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