American Institute of Mathematical Sciences

Advanced
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:
[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

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

[1]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3785-3801. doi: 10.3934/dcdss.2020433
[2]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041
[3]

Delia Schiera. Existence and non-existence results for variational higher order elliptic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5145-5161. doi: 10.3934/dcds.2018227
[4]

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209
[5]

Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete & Continuous Dynamical Systems, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237
[6]

Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure & Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549
[7]

Brock C. Price, Xiangsheng Xu. Global existence theorem for a model governing the motion of two cell populations. Kinetic & Related Models, 2020, 13 (6) : 1175-1191. doi: 10.3934/krm.2020042
[8]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031
[9]

Yuta Ishii, Kazuhiro Kurata. Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1633-1679. doi: 10.3934/cpaa.2021035
[10]

Alberto Bressan, Truyen Nguyen. Non-existence and non-uniqueness for multidimensional sticky particle systems. Kinetic & Related Models, 2014, 7 (2) : 205-218. doi: 10.3934/krm.2014.7.205
[11]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
[12]

Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059
[13]

Shu-Yu Hsu. Non-existence and behaviour at infinity of solutions of some elliptic equations. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769
[14]

Ivan Landjev, Assia Rousseva. The non-existence of $(104,22;3,5)$-arcs. Advances in Mathematics of Communications, 2016, 10 (3) : 601-611. doi: 10.3934/amc.2016029
[15]

J. F. Toland. Non-existence of global energy minimisers in Stokes waves problems. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3211-3217. doi: 10.3934/dcds.2014.34.3211
[16]

Fuqin Sun, Mingxin Wang. Non-existence of global solutions for nonlinear strongly damped hyperbolic systems. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 949-958. doi: 10.3934/dcds.2005.12.949
[17]

Xueying Sun, Renhao Cui. Existence and asymptotic profiles of the steady state for a diffusive epidemic model with saturated incidence and spontaneous infection mechanism. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4503-4520. doi: 10.3934/dcdss.2021120
[18]

Qi Wang, Lu Zhang, Jingyue Yang, Jia Hu. Global existence and steady states of a two competing species Keller--Segel chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 777-807. doi: 10.3934/krm.2015.8.777
[19]

Lorena Bociu, Jean-Paul Zolésio. Existence for the linearization of a steady state fluid/nonlinear elasticity interaction. Conference Publications, 2011, 2011 (Special) : 184-197. doi: 10.3934/proc.2011.2011.184
[20]

Yuta Ishii, Kazuhiro Kurata. Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2807-2875. doi: 10.3934/dcds.2019118

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (356)
  • HTML views (73)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy