-
Previous Article
Global behaviour of a delayed viral kinetic model with general incidence rate
- DCDS-B Home
- This Issue
-
Next Article
Convective nonlocal Cahn-Hilliard equations with reaction terms
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 |
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
Other articles
by authors
[Back to Top]