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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 856725, 21 pages
http://dx.doi.org/10.1155/2012/856725
Research Article

Spatially Nonhomogeneous Periodic Solutions in a Delayed Predator-Prey Model with Diffusion Effects

School of Mathematics and Information Sciences, Henan University, Henan, Kaifeng 475001, China

Received 5 April 2012; Accepted 14 May 2012

Academic Editor: Toka Diagana

Copyright © 2012 Jia-Fang Zhang. 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

This paper is concerned with a delayed predator-prey diffusion model with Neumann boundary conditions. We study the asymptotic stability of the positive constant steady state and the conditions for the existence of Hopf bifurcation. In particular, we show that large diffusivity has no effect on the Hopf bifurcation, while small diffusivity can lead to the fact that spatially nonhomogeneous periodic solutions bifurcate from the positive constant steady-state solution when the system parameters are all spatially homogeneous. Meanwhile, we study the properties of the spatially nonhomogeneous periodic solutions applying normal form theory of partial functional differential equations (PFDEs).