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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), 1-8.  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), R251-R290. doi: 10.1088/0951-7715/21/11/R05.  Google Scholar

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M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

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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), 1115-1134. doi: 10.1080/00036811.2013.817559.  Google Scholar

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[6]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 2010.  Google Scholar

[7]

F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265. doi: 10.1515/jnum-2012-0013.  Google Scholar

[8]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234.  Google Scholar

[9]

D. S. Grebenkov and B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions, SIAM Rev., 55 (2013), 601-667. doi: 10.1137/120880173.  Google Scholar

[10]

H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst., 14 (2006), 737-751. 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, 44. Springer-Verlag, New York, 2003.  Google Scholar

[12]

S. Kouachi, Global existence and boundedness of solutions for a general activator-inhibitor model, Mat. Vesnik, 66 (2014), 274-282.  Google Scholar

[13]

A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanisms to complex structures, Rev. Mod Phys., 66 (1994), 1481-1507. 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), 47-58. 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), 59-68. doi: 10.1007/BF02012623.  Google Scholar

[16]

J. D. Murray, Mathematical Biology, $2^{nd}$ edition, Springer-Verlag, Berlin, 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), 426-465. 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), 351-368. 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), 555-593. 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), 417-455. doi: 10.1007/s00032-010-0133-4.  Google Scholar

[21]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Springer-Verlag, Berlin, 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), 1447-1450. 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), 237-274. 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), 221-246. 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), 113-132. doi: 10.2748/tmj/1178229312.  Google Scholar

[26]

I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), 208-249. 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, New York-Heidelberg, 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), 1-8.  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), R251-R290. 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), 321-340. 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), 1115-1134. 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), 77-129. doi: 10.1512/iumj.1994.43.43030.  Google Scholar

[6]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 2010.  Google Scholar

[7]

F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265. doi: 10.1515/jnum-2012-0013.  Google Scholar

[8]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234.  Google Scholar

[9]

D. S. Grebenkov and B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions, SIAM Rev., 55 (2013), 601-667. doi: 10.1137/120880173.  Google Scholar

[10]

H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst., 14 (2006), 737-751. 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, 44. Springer-Verlag, New York, 2003.  Google Scholar

[12]

S. Kouachi, Global existence and boundedness of solutions for a general activator-inhibitor model, Mat. Vesnik, 66 (2014), 274-282.  Google Scholar

[13]

A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanisms to complex structures, Rev. Mod Phys., 66 (1994), 1481-1507. 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), 47-58. 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), 59-68. doi: 10.1007/BF02012623.  Google Scholar

[16]

J. D. Murray, Mathematical Biology, $2^{nd}$ edition, Springer-Verlag, Berlin, 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), 426-465. 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), 351-368. 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), 555-593. 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), 417-455. doi: 10.1007/s00032-010-0133-4.  Google Scholar

[21]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Springer-Verlag, Berlin, 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), 1447-1450. 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), 237-274. 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), 221-246. 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), 113-132. doi: 10.2748/tmj/1178229312.  Google Scholar

[26]

I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), 208-249. 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, New York-Heidelberg, 1975.  Google Scholar

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