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Advances in Mathematical Physics
Volume 2017, Article ID 9648538, 9 pages
Research Article

Turing Bifurcation and Pattern Formation of Stochastic Reaction-Diffusion System

1College of Information Science and Technology, Donghua University, Shanghai 201620, China
2Institute of Applied Mathematics, Xuchang University, Xuchang, Henan 461000, China

Correspondence should be addressed to Zhijie Wang; nc.ude.uhd@jzgnaw and Jianwei Shen; moc.liamg@nehswjcx

Received 28 August 2016; Revised 18 November 2016; Accepted 15 December 2016; Published 12 February 2017

Academic Editor: Zhi-Yuan Sun

Copyright © 2017 Qianiqian Zheng 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.


Noise is ubiquitous in a system and can induce some spontaneous pattern formations on a spatially homogeneous domain. In comparison to the Reaction-Diffusion System (RDS), Stochastic Reaction-Diffusion System (SRDS) is more complex and it is very difficult to deal with the noise function. In this paper, we have presented a method to solve it and obtained the conditions of how the Turing bifurcation and Hopf bifurcation arise through linear stability analysis of local equilibrium. In addition, we have developed the amplitude equation with a pair of wave vector by using Taylor series expansion, multiscaling, and further expansion in powers of small parameter. Our analysis facilitates finding regions of bifurcations and understanding the pattern formation mechanism of SRDS. Finally, the simulation shows that the analytical results agree with numerical simulation.