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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 306467, 9 pages
http://dx.doi.org/10.1155/2013/306467
Research Article

Cross-Diffusion-Driven Instability in a Reaction-Diffusion Harrison Predator-Prey Model

1Faculty of Science, Shaanxi University of Science and Technology, Xi’an 710021, China
2School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China

Received 4 August 2012; Accepted 14 December 2012

Academic Editor: Xiaodi Li

Copyright © 2013 Xiaoqin Wang and Yongli Cai. 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 processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with cross-diffusion in a Harrison-type predator-prey model. We analyze the global behaviour of the model by establishing a Lyapunov function. We carry out the analytical study in detail and find out the certain conditions for Turing’s instability induced by cross-diffusion. And the numerical results reveal that, on increasing the value of the half capturing saturation constant, the sequences “spots → spot-stripe mixtures → stripes → hole-stripe mixtures → holes” are observed. The results show that the model dynamics exhibits complex pattern replication controlled by the cross-diffusion.