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Journal of Applied Mathematics
Volume 2013, Article ID 201936, 10 pages
Research Article

Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays

1College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China
2School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China

Received 26 April 2013; Accepted 17 September 2013

Academic Editor: Maoan Han

Copyright © 2013 Ruiqing Shi et al. 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.


We propose a three-dimensional stage-structured predatory-prey model with discrete and distributed delays. By use of a new variable, the original three-dimensional system transforms into an equivalent four-dimensional system. Firstly, we study the existence and local stability of positive equilibrium of the new system. And, by choosing the time delay as a bifurcation parameter, we show that Hopf bifurcation may occur as the time delay passes through some critical values. Secondly, by use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, some simple discussion is presented.