American Institute of Mathematical Sciences

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October  2017, 10(5): 1051-1062. doi: 10.3934/dcdss.2017056

Pattern formation of a coupled two-cell Schnakenberg model

Guanqi Liu and  Yuwen Wang 1,2,,

1. 

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
2. 

Y. Y. Tseng Functional Analysis Research Center and School of Mathematical Sciences, Harbin Normal University, Harbin, Heilongjiang 150025, China

* Corresponding author: Yuwen Wang

Received  July 2016 Revised  January 2017 Published  June 2017

Fund Project: Partially supported by NSFC grant 11571086,11471091 and Science Research Funds for Over-seas Returned Chinese Scholars of Heilongjiang Province LC2013C01.

In this paper, we study a coupled two-cell Schnakenberg model with homogenous Neumann boundary condition, i.e.,
$\left\{ \begin{gathered} -d_1Δ u=a-u+u^2v+c(w-u),&\text{ in } Ω, \\-d_2Δ v=b-u^2v,&\text{ in } Ω , \\-d_1Δ w=a-w+w^2z+c(u-w),&\text{ in } Ω, \\-d_2Δ z=b-w^2z,&\text{ in } Ω, \\\dfrac{\partial u}{\partial ν}=\dfrac{\partial v}{\partial ν}=\dfrac{\partial w}{\partial ν}=\dfrac{\partial z}{\partial ν}=0, &\text{ on } \partialΩ.\end{gathered} \right.$
We give a priori estimate to the positive solution. Meanwhile, we obtain the non-existence and existence of positive non-constant solution as parameters
$ d_1, d_2, a$
 and b changes.
Keywords: Schnakenberg model, two-cell, pattern formation, a priori estimate, steady state, non-existence and existence.
Mathematics Subject Classification: 92C40, 92C15, 35B45, 35B35, 35A01.
Citation: Guanqi Liu, Yuwen Wang. Pattern formation of a coupled two-cell Schnakenberg model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1051-1062. doi: 10.3934/dcdss.2017056
References:
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H. I. Abdel-Gawad and A. M. El-Shrae, Symmetric patterns in the Dirichlet problem for a two-cell cubic autocatalytor reaction model, Applied Mathematics and Computation, 150 (2004), 623-645.  doi: 10.1016/S0096-3003(03)00295-9.  Google Scholar

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P. Gormley, K. Li and G. W. Irwin, Modelling molecular interaction pathways using a two-stage identification algorithm, Systems and Synthetic Biology, 1 (2007), 145-160.  doi: 10.1007/s11693-008-9012-5.  Google Scholar

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D. Iron, J. C. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model, Journal of Mathematical Biology, 49 (2004), 358-390.  doi: 10.1007/s00285-003-0258-y.  Google Scholar

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H. Kitano, Systems biology: A brief overview, Science, 295 (2002), 1662-1664.  doi: 10.1126/science.1069492.  Google Scholar

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Y. Li, Steady-state solution for a general Schnakenberg model, Nonlinear Anal. Real World Appl., 12 (2011), 1985-1990.  doi: 10.1016/j.nonrwa.2010.12.014.  Google Scholar

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Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, Journal of Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

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R. Peng and M. Wang, Positive steady-state solutions of the noyes--field model for belousov-zhabotinskii reaction, Nonlinear Analysis: Theory, Methods Applications, 56 (2004), 451-464.  doi: 10.1016/j.na.2003.09.020.  Google Scholar

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M. R. Ricard and S. Mischler, Turing instabilities at hopf bifurcation, Journal of Nonlinear Science, 19 (2009), 467-496.  doi: 10.1007/s00332-009-9041-6.  Google Scholar

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J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, Journal of Theoretical Biology, 81 (1979), 389-400.  doi: 10.1016/0022-5193(79)90042-0.  Google Scholar

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I. Schreiber and M. Marek, Strange attractors in coupled reaction-diffusion cells, Physica D: Nonlinear Phenomena, 5 (1982), 258-272.  doi: 10.1016/0167-2789(82)90021-5.  Google Scholar

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C. K. Tyson and B. Novak, Network dynamics and cell physiology, Nature Reviews Molecular Cell Biology, 2 (2001), 908-916.   Google Scholar

[15]

J. C. Wei and M. Winter, Flow-distributed spikes for Schnakenberg kinetics, J. Math. Biol., 64 (2012), 211-254.  doi: 10.1007/s00285-011-0412-x.  Google Scholar

[16]

C. Xu and J. J. Wei, Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model, Nonlinear Anal. Real World Appl., 13 (2012), 1961-1977.  doi: 10.1016/j.nonrwa.2012.01.001.  Google Scholar

[17]

Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equation, bejing: Sci, 1990.  Google Scholar

[18]

Y. You, Global attractor of a coupled two-cell brusselator model, Fields Inst. Commun., 64 (2013), 319-352, arXiv: 0906.4345. doi: 10.1007/978-1-4614-4523-4_13.  Google Scholar

[19]

J. Zhou and C. L. Mu, Pattern formation of a coupled two-cell brusselator model, Journal of Mathematical Analysis and Applications, 366 (2010), 679-693.  doi: 10.1016/j.jmaa.2009.12.021.  Google Scholar

[20]

W. J. Zuo and J. J. Wei, Multiple bifurcations and spatiotemporal patterns for a coupled two-cell Brusselator model, Dynamics of Partial Differential Equations, 8 (2011), 363-384.   Google Scholar

show all references

References:
[1]

H. I. Abdel-Gawad and A. M. El-Shrae, Symmetric patterns in the Dirichlet problem for a two-cell cubic autocatalytor reaction model, Applied Mathematics and Computation, 150 (2004), 623-645.  doi: 10.1016/S0096-3003(03)00295-9.  Google Scholar

[2]

M. Ghergu and V. D. Radulescu, Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics, Springer, 2012. doi: 10.1007/978-3-642-22664-9.  Google Scholar

[3]

P. Gormley, K. Li and G. W. Irwin, Modelling molecular interaction pathways using a two-stage identification algorithm, Systems and Synthetic Biology, 1 (2007), 145-160.  doi: 10.1007/s11693-008-9012-5.  Google Scholar

[4]

D. Iron, J. C. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model, Journal of Mathematical Biology, 49 (2004), 358-390.  doi: 10.1007/s00285-003-0258-y.  Google Scholar

[5]

H. Kitano, Systems biology: A brief overview, Science, 295 (2002), 1662-1664.  doi: 10.1126/science.1069492.  Google Scholar

[6]

Y. Li, Steady-state solution for a general Schnakenberg model, Nonlinear Anal. Real World Appl., 12 (2011), 1985-1990.  doi: 10.1016/j.nonrwa.2010.12.014.  Google Scholar

[7]

Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, Journal of Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[8]

Y. Lou and W. M. Ni, Diffusion vs cross-diffusion: An elliptic approach, Journal of Differential Equations, 154 (1999), 157-190.  doi: 10.1006/jdeq.1998.3559.  Google Scholar

[9]

J. D. Murray, Mathematical Biology: Ⅰ. An Introduction, volume 2. Springer, 2002.  Google Scholar

[10]

R. Peng and M. Wang, Positive steady-state solutions of the noyes--field model for belousov-zhabotinskii reaction, Nonlinear Analysis: Theory, Methods Applications, 56 (2004), 451-464.  doi: 10.1016/j.na.2003.09.020.  Google Scholar

[11]

M. R. Ricard and S. Mischler, Turing instabilities at hopf bifurcation, Journal of Nonlinear Science, 19 (2009), 467-496.  doi: 10.1007/s00332-009-9041-6.  Google Scholar

[12]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, Journal of Theoretical Biology, 81 (1979), 389-400.  doi: 10.1016/0022-5193(79)90042-0.  Google Scholar

[13]

I. Schreiber and M. Marek, Strange attractors in coupled reaction-diffusion cells, Physica D: Nonlinear Phenomena, 5 (1982), 258-272.  doi: 10.1016/0167-2789(82)90021-5.  Google Scholar

[14]

C. K. Tyson and B. Novak, Network dynamics and cell physiology, Nature Reviews Molecular Cell Biology, 2 (2001), 908-916.   Google Scholar

[15]

J. C. Wei and M. Winter, Flow-distributed spikes for Schnakenberg kinetics, J. Math. Biol., 64 (2012), 211-254.  doi: 10.1007/s00285-011-0412-x.  Google Scholar

[16]

C. Xu and J. J. Wei, Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model, Nonlinear Anal. Real World Appl., 13 (2012), 1961-1977.  doi: 10.1016/j.nonrwa.2012.01.001.  Google Scholar

[17]

Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equation, bejing: Sci, 1990.  Google Scholar

[18]

Y. You, Global attractor of a coupled two-cell brusselator model, Fields Inst. Commun., 64 (2013), 319-352, arXiv: 0906.4345. doi: 10.1007/978-1-4614-4523-4_13.  Google Scholar

[19]

J. Zhou and C. L. Mu, Pattern formation of a coupled two-cell brusselator model, Journal of Mathematical Analysis and Applications, 366 (2010), 679-693.  doi: 10.1016/j.jmaa.2009.12.021.  Google Scholar

[20]

W. J. Zuo and J. J. Wei, Multiple bifurcations and spatiotemporal patterns for a coupled two-cell Brusselator model, Dynamics of Partial Differential Equations, 8 (2011), 363-384.   Google Scholar

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