# American Institute of Mathematical Sciences

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

## Pattern formation of a coupled two-cell Schnakenberg model

 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.
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.

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##### 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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