American Institute of Mathematical Sciences

Advanced
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

[1]

Martin Baurmann, Wolfgang Ebenhöh, Ulrike Feudel. Turing instabilities and pattern formation in a benthic nutrient-microorganism system. Mathematical Biosciences & Engineering, 2004, 1 (1) : 111-130. doi: 10.3934/mbe.2004.1.111
[2]

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
[3]

Julien Barré, Pierre Degond, Diane Peurichard, Ewelina Zatorska. Modelling pattern formation through differential repulsion. Networks & Heterogeneous Media, 2020, 15 (3) : 307-352. doi: 10.3934/nhm.2020021
[4]

Olha Ivanyshyn. Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Problems & Imaging, 2007, 1 (4) : 609-622. doi: 10.3934/ipi.2007.1.609
[5]

Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations & Control Theory, 2014, 3 (1) : 1-14. doi: 10.3934/eect.2014.3.1
[6]

Steffen Härting, Anna Marciniak-Czochra, Izumi Takagi. Stable patterns with jump discontinuity in systems with Turing instability and hysteresis. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 757-800. doi: 10.3934/dcds.2017032
[7]

Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020260
[8]

Tian Ma, Shouhong Wang. Dynamic transition and pattern formation for chemotactic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2809-2835. doi: 10.3934/dcdsb.2014.19.2809
[9]

Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (II): interfacial instability and pattern formation at early stage of growth. Communications on Pure & Applied Analysis, 2004, 3 (3) : 527-543. doi: 10.3934/cpaa.2004.3.527
[10]

Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170
[11]

Kolade M. Owolabi. Numerical analysis and pattern formation process for space-fractional superdiffusive systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 543-566. doi: 10.3934/dcdss.2019036
[12]

Abed Bounemoura, Edouard Pennamen. Instability for a priori unstable Hamiltonian systems: A dynamical approach. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 753-793. doi: 10.3934/dcds.2012.32.753
[13]

Hongyu He, Naohiro Kato. Equilibrium submanifold for a biological system. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1429-1441. doi: 10.3934/dcdss.2011.4.1429
[14]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285
[15]

Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020162
[16]

Joseph G. Yan, Dong-Ming Hwang. Pattern formation in reaction-diffusion systems with $D_2$-symmetric kinetics. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 255-270. doi: 10.3934/dcds.1996.2.255
[17]

Qi Wang, Ling Jin, Zengyan Zhang. Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2105-2134. doi: 10.3934/dcds.2020108
[18]

Georg Hetzer, Anotida Madzvamuse, Wenxian Shen. Characterization of turing diffusion-driven instability on evolving domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3975-4000. doi: 10.3934/dcds.2012.32.3975
[19]

Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160
[20]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences