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Abstract and Applied Analysis
Volume 2013, Article ID 891738, 10 pages
Research Article

Turing Patterns in a Predator-Prey System with Self-Diffusion

School of Science, Nanchang University, Nanchang 330031, China

Received 24 June 2013; Revised 17 September 2013; Accepted 2 October 2013

Academic Editor: Francisco Solís Lozano

Copyright © 2013 Hongwei Yin 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.


For a predator-prey system, cross-diffusion has been confirmed to emerge Turing patterns. However, in the real world, the tendency for prey and predators moving along the direction of lower density of their own species, called self-diffusion, should be considered. For this, we investigate Turing instability for a predator-prey system with nonlinear diffusion terms including the normal diffusion, cross-diffusion, and self-diffusion. A sufficient condition of Turing instability for this system is obtained by analyzing the linear stability of spatial homogeneous equilibrium state of this model. A series of numerical simulations reveal Turing parameter regions of the interaction of diffusion parameters. According to these regions, we further demonstrate dispersion relations and spatial patterns. Our results indicate that self-diffusion plays an important role in the spatial patterns.