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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 681641, 9 pages
http://dx.doi.org/10.1155/2013/681641
Research Article

Spatiotemporal Pattern in a Self- and Cross-Diffusive Predation Model with the Allee Effect

College of Sciences, Nanjing University of Technology, Nanjing, Jiangsu 211816, China

Received 16 April 2013; Accepted 12 June 2013

Academic Editor: Qiru Wang

Copyright © 2013 Feng Rao. 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 proposes and analyzes a mathematical model for a predator-prey interaction with the Allee effect on prey species and with self- and cross-diffusion. The effect of diffusion which can drive the model with zero-flux boundary conditions to Turing instability is investigated. We present 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 spotted and striped-like coexisting and spotted pattern replication. Moreover, we discuss the effect of cross-diffusivity on the stability of the nontrivial equilibrium of the model, which depends upon the magnitudes of the self- and cross-diffusion coefficients. The obtained results show that cross-diffusion plays an important role in the pattern formation of the predator-prey model. It is also useful to apply the reaction-diffusion model to reveal the spatial predation in the real world.