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Abstract and Applied Analysis
Volume 2014, Article ID 167856, 12 pages
http://dx.doi.org/10.1155/2014/167856
Research Article

Effect of Diffusion and Cross-Diffusion in a Predator-Prey Model with a Transmissible Disease in the Predator Species

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received 16 September 2013; Accepted 27 February 2014; Published 3 June 2014

Academic Editor: Francisco Solis

Copyright © 2014 Guohong Zhang and Xiaoli Wang. 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 study a Lotka-Volterra type predator-prey model with a transmissible disease in the predator population. We concentrate on the effect of diffusion and cross-diffusion on the emergence of stationary patterns. We first show that both self-diffusion and cross-diffusion can not cause Turing instability from the disease-free equilibria. Then we find that the endemic equilibrium remains linearly stable for the reaction diffusion system without cross-diffusion, while it becomes linearly unstable when cross-diffusion also plays a role in the reaction-diffusion system; hence, the instability is driven solely from the effect of cross-diffusion. Furthermore, we derive some results for the existence and nonexistence of nonconstant stationary solutions when the diffusion rate of a certain species is small or large.