About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 657286, 14 pages
http://dx.doi.org/10.1155/2013/657286
Research Article

Pattern Formation in a Semi-Ratio-Dependent Predator-Prey System with Diffusion

1Department of Mathematics Education, Catholic University of Daegu, Gyeongsan, Gyeongbuk 712-702, Republic of Korea
2Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea
3College of Mathematics and Computer, Ningxia University, Ningxia 750021, China

Received 31 October 2012; Accepted 25 February 2013

Academic Editor: Vimal Singh

Copyright © 2013 Hunki Baek 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 investigate spatiotemporal dynamics of a semi-ratio-dependent predator-prey system with reaction-diffusion and zero-flux boundary. We obtain the conditions for Hopf, Turing, and wave bifurcations of the system in a spatial domain by making use of the linear stability analysis and the bifurcation analysis. In addition, for an initial condition which is a small amplitude random perturbation around the steady state, we classify spatial pattern formations of the system by using numerical simulations. The results of numerical simulations unveil that there are various spatiotemporal patterns including typical Turing patterns such as spotted, spot-stripelike mixtures and stripelike patterns thanks to the Turing instability, that an oscillatory wave pattern can be emerged due to the Hopf and wave instability, and that cooperations of Turing and Hopf instabilities can cause occurrence of spiral patterns instead of typical Turing patterns. Finally, we discuss spatiotemporal dynamics of the system for several different asymmetric initial conditions via numerical simulations.