Advanced Search
Article Contents
Article Contents

Remarks on pattern formation in a model for hair follicle spacing

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 35K57, 35B36; Secondary: 35Q92.


    \begin{equation} \\ \end{equation}
  • [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.


    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.


    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.


    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.


    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.


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


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


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


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


    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.


    P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Texts in Applied Mathematics, 44. Springer-Verlag, New York, 2003.


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


    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.


    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.


    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.


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


    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.


    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.


    Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593.doi: 10.1137/0513037.


    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.


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


    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.


    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.


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


    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.


    I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), 208-249.doi: 10.1016/0022-0396(86)90119-1.


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

  • 加载中

Article Metrics

HTML views() PDF downloads(115) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint