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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 454097, 12 pages
http://dx.doi.org/10.1155/2013/454097
Research Article

Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay

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 3 March 2013; Accepted 27 May 2013

Academic Editor: Peixuan Weng

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.

Abstract

We propose a two-dimensional predatory-prey model with discrete and distributed delay. By the use of a new variable, the original two-dimensional system transforms into an equivalent three-dimensional system. Firstly, we study the existence and local stability of equilibria of the new system. And, by choosing the time delay as a bifurcation parameter, we show that Hopf bifurcation can occur as the time delay passes through some critical values. Secondly, by the use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, an example with numerical simulations is provided to verify the theoretical results. In addition, some simple discussion is also presented.