Pattern formation of a coupled two-cell Schnakenberg model

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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 and Continuous Dynamical Systems - S, 2017, 10 (5) : 1051-1062. doi: 10.3934/dcdss.2017056

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.
[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.
[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.
[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.
[5]	H. Kitano, Systems biology: A brief overview, Science, 295 (2002), 1662-1664. doi: 10.1126/science.1069492.
[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.
[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.
[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.
[9]	J. D. Murray, Mathematical Biology: Ⅰ. An Introduction, volume 2. Springer, 2002.
[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.
[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.
[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.
[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.
[14]	C. K. Tyson and B. Novak, Network dynamics and cell physiology, Nature Reviews Molecular Cell Biology, 2 (2001), 908-916.
[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.
[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.
[17]	Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equation, bejing: Sci, 1990.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[5]	H. Kitano, Systems biology: A brief overview, Science, 295 (2002), 1662-1664. doi: 10.1126/science.1069492.
[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.
[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.
[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.
[9]	J. D. Murray, Mathematical Biology: Ⅰ. An Introduction, volume 2. Springer, 2002.
[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.
[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.
[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.
[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.
[14]	C. K. Tyson and B. Novak, Network dynamics and cell physiology, Nature Reviews Molecular Cell Biology, 2 (2001), 908-916.
[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.
[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.
[17]	Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equation, bejing: Sci, 1990.
[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.
[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.
[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.

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