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A kind of generalized transversality theorem for $C^r$ mapping with parameter
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 |
$\left\{ \begin{gathered} -d_1Δ u=a-u+u^2v+c(w-u),&\text{ in } Ω, \\-d_2Δ v=b-u^2v,&\text{ in } Ω , \\-d_1Δ w=a-w+w^2z+c(u-w),&\text{ in } Ω, \\-d_2Δ z=b-w^2z,&\text{ in } Ω, \\\dfrac{\partial u}{\partial ν}=\dfrac{\partial v}{\partial ν}=\dfrac{\partial w}{\partial ν}=\dfrac{\partial z}{\partial ν}=0, &\text{ on } \partialΩ.\end{gathered} \right.$ |
$ d_1, d_2, a$ |
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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