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July  2015, 20(5): 1555-1572. doi: 10.3934/dcdsb.2015.20.1555

Remarks on pattern formation in a model for hair follicle spacing

Peter Rashkov 1,

1. 

Department of Mathematics and Informatics, Philipps-Universitat Marburg, Hans-Meerwein-Str., Lahnberge, 35032 Marburg, Germany

Received  June 2014 Revised  January 2015 Published  May 2015

A modified version of the Gierer-Meinhardt reaction-diffusion system (without source terms) is used in a model for hair follicle spacing in mice, proposed by Sick, Reinker, Timmer and Schlake [22]. Global existence of solutions of this model system is shown by computing uniform bounds. Analysis of conditions for emergence of spatially heterogeneous solutions is performed using a limiting form of the original reaction-diffusion system. The conditions for pattern formation given in [22] are improved by including those subregions in the parameter space where far-from-equilibrium heterogeneous solutions occur.
Keywords: pattern formation, Biological modelling, far-from-equilibrium solutions., Turing instability, dynamical systems.
Mathematics Subject Classification: Primary: 35K57, 35B36; Secondary: 35Q9.
Citation: Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555
References:
[1]

S. Abdelmalek, H. Louafi and A. Youkana, Existence of global solutions for a Gierer-Meinhardt system with three equations,, Electron. J. Differential Equations, 55 (2012), 1. Google Scholar

[2]

R. E. Baker, E. A. Gaffney and P. K. Maini, Partial differential equations for self-organization in cellular and developmental biology,, Nonlinearity, 21 (2008). doi: 10.1088/0951-7715/21/11/R05. Google Scholar

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Funct. Analysis, 8 (1971), 321. doi: 10.1016/0022-1236(71)90015-2. Google Scholar

[4]

S. Chen, J. Shi and J. Wei, Bifurcation analysis of the Gierer-Meinhardt system with a saturation in the activator production,, Appl. Anal., 93 (2014), 1115. doi: 10.1080/00036811.2013.817559. Google Scholar

[5]

M. del Pino, A priori estimates and applications to existence-nonexistence for a semilinear elliptic system,, Indiana Univ. Math. J., 43 (1994), 77. doi: 10.1512/iumj.1994.43.43030. Google Scholar

[6]

L. C. Evans, Partial Differential Equations,, American Mathematical Society, (2010). Google Scholar

[7]

F. Hecht, New development in freefem++,, J. Numer. Math., 20 (2012), 251. doi: 10.1515/jnum-2012-0013. Google Scholar

[8]

A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetik, 12 (1972), 30. doi: 10.1007/BF00289234. Google Scholar

[9]

D. S. Grebenkov and B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions,, SIAM Rev., 55 (2013), 601. doi: 10.1137/120880173. Google Scholar

[10]

H. Jiang, Global existence of solutions of an activator-inhibitor system,, Discrete Contin. Dyn. Syst., 14 (2006), 737. doi: 10.3934/dcds.2006.14.737. Google Scholar

[11]

P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations,, Texts in Applied Mathematics, (2003). Google Scholar

[12]

S. Kouachi, Global existence and boundedness of solutions for a general activator-inhibitor model,, Mat. Vesnik, 66 (2014), 274. Google Scholar

[13]

A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanisms to complex structures,, Rev. Mod Phys., 66 (1994), 1481. doi: 10.1103/RevModPhys.66.1481. Google Scholar

[14]

K. Masuda and K. Takahashi, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation,, Japan J. Appl. Math., 4 (1987), 47. doi: 10.1007/BF03167754. Google Scholar

[15]

M. Li, S. Chen and Y. Qin, Boundedness and blow up for the general activator-inhibitor model,, Acta Math. Appl. Sinica (English Ser.), 11 (1995), 59. doi: 10.1007/BF02012623. Google Scholar

[16]

J. D. Murray, Mathematical Biology,, $2^{nd}$ edition, (1993). doi: 10.1007/b98869. Google Scholar

[17]

W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system,, J. Differential Equations, 229 (2006), 426. doi: 10.1016/j.jde.2006.03.011. Google Scholar

[18]

W.-M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type,, Trans. Amer. Math. Soc., 297 (1986), 351. doi: 10.1090/S0002-9947-1986-0849484-2. Google Scholar

[19]

Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems,, SIAM J. Math. Anal., 13 (1982), 555. doi: 10.1137/0513037. Google Scholar

[20]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: a survey,, Milan J. Math., 78 (2010), 417. doi: 10.1007/s00032-010-0133-4. Google Scholar

[21]

F. Rothe, Global Solutions of Reaction-Diffusion Systems,, Springer-Verlag, (1984). Google Scholar

[22]

S. Sick, S. Reinker, J. Timmer and T. Schlake, WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism,, Science, 314 (2006), 1447. doi: 10.1126/science.1130088. Google Scholar

[23]

K. Suzuki and I. Takagi, On the role of basic production terms in an activator-inhibitor system modeling biological pattern formation,, Funkc. Ekvac., 54 (2011), 237. doi: 10.1619/fesi.54.237. Google Scholar

[24]

I. Takagi, Stability of bifurcating solutions of the Gierer-Meinhardt system,, Tôhoku Math. J., 31 (1979), 221. doi: 10.2748/tmj/1178229841. Google Scholar

[25]

I. Takagi, A priori estimates for stationary solutions of an activator-inhibitor model due to Gierer and Meinhardt,, Tôhoku Math. J., 34 (1982), 113. doi: 10.2748/tmj/1178229312. Google Scholar

[26]

I. Takagi, Point-condensation for a reaction-diffusion system,, J. Differential Equations, 61 (1986), 208. doi: 10.1016/0022-0396(86)90119-1. Google Scholar

[27]

T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions,, Springer-Verlag, (1975). Google Scholar

show all references

References:
[1]

S. Abdelmalek, H. Louafi and A. Youkana, Existence of global solutions for a Gierer-Meinhardt system with three equations,, Electron. J. Differential Equations, 55 (2012), 1. Google Scholar

[2]

R. E. Baker, E. A. Gaffney and P. K. Maini, Partial differential equations for self-organization in cellular and developmental biology,, Nonlinearity, 21 (2008). doi: 10.1088/0951-7715/21/11/R05. Google Scholar

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Funct. Analysis, 8 (1971), 321. doi: 10.1016/0022-1236(71)90015-2. Google Scholar

[4]

S. Chen, J. Shi and J. Wei, Bifurcation analysis of the Gierer-Meinhardt system with a saturation in the activator production,, Appl. Anal., 93 (2014), 1115. doi: 10.1080/00036811.2013.817559. Google Scholar

[5]

M. del Pino, A priori estimates and applications to existence-nonexistence for a semilinear elliptic system,, Indiana Univ. Math. J., 43 (1994), 77. doi: 10.1512/iumj.1994.43.43030. Google Scholar

[6]

L. C. Evans, Partial Differential Equations,, American Mathematical Society, (2010). Google Scholar

[7]

F. Hecht, New development in freefem++,, J. Numer. Math., 20 (2012), 251. doi: 10.1515/jnum-2012-0013. Google Scholar

[8]

A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetik, 12 (1972), 30. doi: 10.1007/BF00289234. Google Scholar

[9]

D. S. Grebenkov and B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions,, SIAM Rev., 55 (2013), 601. doi: 10.1137/120880173. Google Scholar

[10]

H. Jiang, Global existence of solutions of an activator-inhibitor system,, Discrete Contin. Dyn. Syst., 14 (2006), 737. doi: 10.3934/dcds.2006.14.737. Google Scholar

[11]

P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations,, Texts in Applied Mathematics, (2003). Google Scholar

[12]

S. Kouachi, Global existence and boundedness of solutions for a general activator-inhibitor model,, Mat. Vesnik, 66 (2014), 274. Google Scholar

[13]

A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanisms to complex structures,, Rev. Mod Phys., 66 (1994), 1481. doi: 10.1103/RevModPhys.66.1481. Google Scholar

[14]

K. Masuda and K. Takahashi, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation,, Japan J. Appl. Math., 4 (1987), 47. doi: 10.1007/BF03167754. Google Scholar

[15]

M. Li, S. Chen and Y. Qin, Boundedness and blow up for the general activator-inhibitor model,, Acta Math. Appl. Sinica (English Ser.), 11 (1995), 59. doi: 10.1007/BF02012623. Google Scholar

[16]

J. D. Murray, Mathematical Biology,, $2^{nd}$ edition, (1993). doi: 10.1007/b98869. Google Scholar

[17]

W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system,, J. Differential Equations, 229 (2006), 426. doi: 10.1016/j.jde.2006.03.011. Google Scholar

[18]

W.-M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type,, Trans. Amer. Math. Soc., 297 (1986), 351. doi: 10.1090/S0002-9947-1986-0849484-2. Google Scholar

[19]

Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems,, SIAM J. Math. Anal., 13 (1982), 555. doi: 10.1137/0513037. Google Scholar

[20]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: a survey,, Milan J. Math., 78 (2010), 417. doi: 10.1007/s00032-010-0133-4. Google Scholar

[21]

F. Rothe, Global Solutions of Reaction-Diffusion Systems,, Springer-Verlag, (1984). Google Scholar

[22]

S. Sick, S. Reinker, J. Timmer and T. Schlake, WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism,, Science, 314 (2006), 1447. doi: 10.1126/science.1130088. Google Scholar

[23]

K. Suzuki and I. Takagi, On the role of basic production terms in an activator-inhibitor system modeling biological pattern formation,, Funkc. Ekvac., 54 (2011), 237. doi: 10.1619/fesi.54.237. Google Scholar

[24]

I. Takagi, Stability of bifurcating solutions of the Gierer-Meinhardt system,, Tôhoku Math. J., 31 (1979), 221. doi: 10.2748/tmj/1178229841. Google Scholar

[25]

I. Takagi, A priori estimates for stationary solutions of an activator-inhibitor model due to Gierer and Meinhardt,, Tôhoku Math. J., 34 (1982), 113. doi: 10.2748/tmj/1178229312. Google Scholar

[26]

I. Takagi, Point-condensation for a reaction-diffusion system,, J. Differential Equations, 61 (1986), 208. doi: 10.1016/0022-0396(86)90119-1. Google Scholar

[27]

T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions,, Springer-Verlag, (1975). Google Scholar

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