Remarks on pattern formation in a model for hair follicle spacing

Previous Article
Convective nonlocal Cahn-Hilliard equations with reaction terms
DCDS-B Home
This Issue
Next Article
Global behaviour of a delayed viral kinetic model with general incidence rate

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

Remarks on pattern formation in a model for hair follicle spacing

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

Abstract
Related Papers

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.
[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.
[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.
[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.
[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.
[6]	L. C. Evans, Partial Differential Equations,, American Mathematical Society, (2010).
[7]	F. Hecht, New development in freefem++,, J. Numer. Math., 20 (2012), 251. doi: 10.1515/jnum-2012-0013.
[8]	A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetik, 12 (1972), 30. doi: 10.1007/BF00289234.
[9]	D. S. Grebenkov and B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions,, SIAM Rev., 55 (2013), 601. doi: 10.1137/120880173.
[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.
[11]	P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations,, Texts in Applied Mathematics, (2003).
[12]	S. Kouachi, Global existence and boundedness of solutions for a general activator-inhibitor model,, Mat. Vesnik, 66 (2014), 274.
[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.
[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.
[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.
[16]	J. D. Murray, Mathematical Biology,, $2^{nd}$ edition, (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. 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. 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. 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. doi: 10.1007/s00032-010-0133-4.
[21]	F. Rothe, Global Solutions of Reaction-Diffusion Systems,, Springer-Verlag, (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. 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. doi: 10.1619/fesi.54.237.
[24]	I. Takagi, Stability of bifurcating solutions of the Gierer-Meinhardt system,, Tôhoku Math. J., 31 (1979), 221. 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. doi: 10.2748/tmj/1178229312.
[26]	I. Takagi, Point-condensation for a reaction-diffusion system,, J. Differential Equations, 61 (1986), 208. 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, (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.
[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.
[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.
[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.
[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.
[6]	L. C. Evans, Partial Differential Equations,, American Mathematical Society, (2010).
[7]	F. Hecht, New development in freefem++,, J. Numer. Math., 20 (2012), 251. doi: 10.1515/jnum-2012-0013.
[8]	A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetik, 12 (1972), 30. doi: 10.1007/BF00289234.
[9]	D. S. Grebenkov and B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions,, SIAM Rev., 55 (2013), 601. doi: 10.1137/120880173.
[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.
[11]	P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations,, Texts in Applied Mathematics, (2003).
[12]	S. Kouachi, Global existence and boundedness of solutions for a general activator-inhibitor model,, Mat. Vesnik, 66 (2014), 274.
[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.
[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.
[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.
[16]	J. D. Murray, Mathematical Biology,, $2^{nd}$ edition, (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. 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. 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. 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. doi: 10.1007/s00032-010-0133-4.
[21]	F. Rothe, Global Solutions of Reaction-Diffusion Systems,, Springer-Verlag, (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. 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. doi: 10.1619/fesi.54.237.
[24]	I. Takagi, Stability of bifurcating solutions of the Gierer-Meinhardt system,, Tôhoku Math. J., 31 (1979), 221. 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. doi: 10.2748/tmj/1178229312.
[26]	I. Takagi, Point-condensation for a reaction-diffusion system,, J. Differential Equations, 61 (1986), 208. 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, (1975).

[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]	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
[4]	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
[5]	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
[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]	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
[8]	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
[9]	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
[10]	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
[11]	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
[12]	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
[13]	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
[14]	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
[15]	Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[16]	Rui Peng, Fengqi Yi. On spatiotemporal pattern formation in a diffusive bimolecular model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 217-230. doi: 10.3934/dcdsb.2011.15.217
[17]	Taylan Sengul, Shouhong Wang. Pattern formation and dynamic transition for magnetohydrodynamic convection. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2609-2639. doi: 10.3934/cpaa.2014.13.2609
[18]	MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777
[19]	Pietro-Luciano Buono, Daniel C. Offin. Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems. Journal of Geometric Mechanics, 2017, 9 (4) : 439-457. doi: 10.3934/jgm.2017017
[20]	Pierre Degond, Marcello Delitala. Modelling and simulation of vehicular traffic jam formation. Kinetic & Related Models, 2008, 1 (2) : 279-293. doi: 10.3934/krm.2008.1.279

2017 Impact Factor: 0.972

American Institute of Mathematical Sciences