Turing Instability of Brusselator in the Reaction-Diffusion Network

Turing instability constitutes a universal paradigm for the spontaneous generation of spatially organized patterns, especially in a chemical reaction. In this paper, we investigated the pattern dynamics of Brusselator from the view of complex networks and considered the interaction between diffusion and reaction in the random network. After a detailed theoretical analysis, we obtained the approximate instability region about the diffusion coefficient and the connection probability of the random network. In the meantime, we also obtained the critical condition of Turing instability in the network-organized system and found that how the network connection probability and diffusion coefficient affect the reaction-diffusion system of the Brusselator model. In the end, the reason for arising of Turing instability in the Brusselator with the random network was explained. Numerical simulation verified the theoretical results.

[1]  Ekaterina Ponizovskaya Devine,et al.  Stochastic resonance in the brusselator model , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  Francesco Saverio Pavone,et al.  The theory of pattern formation on directed networks. , 2014, Nature communications.

[3]  Zhonghuai Hou,et al.  Optimal system size for mesoscopic chemical oscillation. , 2004, Chemphyschem : a European journal of chemical physics and physical chemistry.

[4]  Alexander S. Mikhailov,et al.  Turing patterns in network-organized activator–inhibitor systems , 2008, 0807.1230.

[5]  G. Gambino,et al.  Turing pattern formation in the Brusselator system with nonlinear diffusion. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  L E Scriven,et al.  Instability and dynamic pattern in cellular networks. , 1971, Journal of theoretical biology.

[7]  Rui Peng,et al.  Pattern formation in the Brusselator system , 2005 .

[8]  Guang Zhang,et al.  TURING INSTABILITY AND PATTERN FORMATION IN A SEMI-DISCRETE BRUSSELATOR MODEL , 2013 .

[9]  Analysis of bifurcation patterns in reaction-diffusion systems: Effect of external noise on the Brusselator model , 1997 .

[10]  Peter K. Moore,et al.  Network topology and Turing instabilities in small arrays of diffusively coupled reactors , 2004 .

[11]  Bernard J. Matkowsky,et al.  Interaction of Turing and Hopf modes in the superdiffusive Brusselator model , 2009, Appl. Math. Lett..

[12]  Bernard J. Matkowsky,et al.  Turing Pattern Formation in the Brusselator Model with Superdiffusion , 2008, SIAM J. Appl. Math..

[13]  A. Turing.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[14]  Qianqian Zheng,et al.  Turing instability induced by random network in FitzHugh-Nagumo model , 2020, Appl. Math. Comput..

[15]  O Mason,et al.  Graph theory and networks in Biology. , 2006, IET systems biology.

[16]  Nick McCullen,et al.  Pattern Formation on Networks: from Localised Activity to Turing Patterns , 2016, Scientific Reports.

[17]  Gourab Ghoshal,et al.  Turing patterns mediated by network topology in homogeneous active systems. , 2019, Physical review. E.

[18]  Takuya Machida,et al.  Turing instability in reaction–diffusion models on complex networks , 2014, 1405.0642.

[19]  James Sharpe,et al.  Key Features of Turing Systems are Determined Purely by Network Topology , 2017, Physical Review X.

[20]  Hiroya Nakao,et al.  Localization of Laplacian eigenvectors on random networks , 2017, Scientific Reports.

[21]  Harmonic and subharmonic bifurcation in the Brussel model with periodic force , 1997 .

[22]  D. Fanelli,et al.  Stochastic Turing patterns in the Brusselator model. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.