Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 814069, 13 pages
http://dx.doi.org/10.1155/2012/814069
Research Article

Pattern Formation in a Cross-Diffusive Ratio-Dependent Predator-Prey Model

Xinze Lian,1 Yanhong Yue,2 and Hailing Wang3

1Chengdu Institute of Computer Application, Chinese Academy of Sciences, Chengdu 610041, China
2School of Foreign Language, Wenzhou University, Wenzhou 325000, China
3College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received 29 September 2012; Accepted 3 November 2012

Academic Editor: Yonghui Xia

Copyright © 2012 Xinze Lian 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

This paper presents a theoretical analysis of evolutionary process that involves organisms distribution and their interaction of spatial distribution of the species with self- and cross-diffusion in a Holling-III ratio-dependent predator-prey model. The diffusion instability of the positive equilibrium of the model with Neumann boundary conditions is discussed. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by self- and cross-diffusion in the model and find that the model dynamics exhibits a cross-diffusion controlled formation growth to spots, stripes, and spiral wave pattern replication, which show that reaction-diffusion model is useful to reveal the spatial predation dynamics in the real world.