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 and 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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[16]

J. D. Murray, Mathematical Biology, $2^{nd}$ edition, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.

[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.

[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.

[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.

[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.

[21]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Springer-Verlag, Berlin, 1984.

[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.

[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.

[24]

I. Takagi, Stability of bifurcating solutions of the Gierer-Meinhardt system, Tôhoku Math. J., 31 (1979), 221-246. doi: 10.2748/tmj/1178229841.

[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.

[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.

[27]

T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Springer-Verlag, New York-Heidelberg, 1975.

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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[16]

J. D. Murray, Mathematical Biology, $2^{nd}$ edition, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.

[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.

[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.

[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.

[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.

[21]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Springer-Verlag, Berlin, 1984.

[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.

[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.

[24]

I. Takagi, Stability of bifurcating solutions of the Gierer-Meinhardt system, Tôhoku Math. J., 31 (1979), 221-246. doi: 10.2748/tmj/1178229841.

[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.

[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.

[27]

T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Springer-Verlag, New York-Heidelberg, 1975.

[1]

Weihua Jiang, Xun Cao, Chuncheng Wang. Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1163-1178. doi: 10.3934/dcdsb.2021085
[2]

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
[3]

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 and Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031
[4]

Julien Barré, Pierre Degond, Diane Peurichard, Ewelina Zatorska. Modelling pattern formation through differential repulsion. Networks and Heterogeneous Media, 2020, 15 (3) : 307-352. doi: 10.3934/nhm.2020021
[5]

Olha Ivanyshyn. Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Problems and Imaging, 2007, 1 (4) : 609-622. doi: 10.3934/ipi.2007.1.609
[6]

Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations and Control Theory, 2014, 3 (1) : 1-14. doi: 10.3934/eect.2014.3.1
[7]

Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260
[8]

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

Steffen Härting, Anna Marciniak-Czochra, Izumi Takagi. Stable patterns with jump discontinuity in systems with Turing instability and hysteresis. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 757-800. doi: 10.3934/dcds.2017032
[10]

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 and Applied Analysis, 2004, 3 (3) : 527-543. doi: 10.3934/cpaa.2004.3.527
[11]

Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170
[12]

Kolade M. Owolabi. Numerical analysis and pattern formation process for space-fractional superdiffusive systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 543-566. doi: 10.3934/dcdss.2019036
[13]

Abed Bounemoura, Edouard Pennamen. Instability for a priori unstable Hamiltonian systems: A dynamical approach. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 753-793. doi: 10.3934/dcds.2012.32.753
[14]

Hongyu He, Naohiro Kato. Equilibrium submanifold for a biological system. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1429-1441. doi: 10.3934/dcdss.2011.4.1429
[15]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285
[16]

Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162
[17]

Joseph G. Yan, Dong-Ming Hwang. Pattern formation in reaction-diffusion systems with $D_2$-symmetric kinetics. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 255-270. doi: 10.3934/dcds.1996.2.255
[18]

Qi Wang, Ling Jin, Zengyan Zhang. Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2105-2134. doi: 10.3934/dcds.2020108
[19]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395
[20]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (94)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy