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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 828219, 12 pages
http://dx.doi.org/10.1155/2012/828219
Research Article

Pattern Formation in a Cross-Diffusive Holling-Tanner Model

1College of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China
2Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand 826004, India

Received 29 November 2012; Accepted 13 December 2012

Academic Editor: Yonghui Xia

Copyright © 2012 Weiming Wang 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 present a theoretical analysis of the processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with self- as well as cross-diffusion in a Holling-Tanner predator-prey model; the sufficient conditions for the Turing instability with zero-flux boundary conditions are obtained; Hopf and Turing bifurcation in a spatial domain is presented, too. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by self- as well as cross-diffusion in the model, and find that the model dynamics exhibits a cross-diffusion controlled formation growth not only to spots, but also to strips, holes, and stripes-spots replication. And the methods and results in the present paper may be useful for the research of the pattern formation in the cross-diffusive model.