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Congestion-driven dendritic growth
1. | Laboratoire de Mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay Cedex |
2. | Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, 91405 Orsay cedex |
References:
[1] |
L. Ambrosio, N. Gigli and G. Savare, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Lectures in Mathematics, ETH Zürich . Birkhäuser Verlag, Basel, 2005. |
[2] |
L. Ambrosio and G. Savare, Gradient flows of probability measures, Handbook of differential equations, Evolutionary equations, III (2007), 1–136. Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam.
doi: 10.1016/S1874-5717(07)80004-1. |
[3] |
M. Badoual, P. Berbez, M. Aubert and B. Grammaticos, Simulating the migration and growth patterns of Bacillus subtilis, Physica A, 388 (2009), 549-559.
doi: 10.1016/j.physa.2008.10.046. |
[4] |
E. Ben-Jacob, From snowflake formation to growth of bacterial colonies II: Cooperative formation of complex colonial patterns, Contemporary Physics, 38 (1997), 205-241.
doi: 10.1080/001075197182405. |
[5] |
E. Ben-Jacob, O. Shochet, A. Tenenbaum, I. Cohen, A. Czirok and T. Vicsek, Generic modeling of cooperative growth patterns in bacterial colonies, Nature, 368 (1994), 46-49.
doi: 10.1038/368046a0. |
[6] |
G. Buttazzo and F. Santambrogio, A mass transportation model for the optimal planning of an urban region SIAM Review, 51 (2009), 593-610.
doi: 10.1137/090759197. |
[7] |
J. Dambrine, B. Maury, N. Meunier and A. Roudneff-Chupin, A congestion model for cell migration, Comm. Pure Appl. Anal., 11 (2012), 243-260.
doi: 10.3934/cpaa.2012.11.243. |
[8] |
E. Feireisl, D. Hilhorst, M. Mimura and R. Weidenfeld, On a nonlinear diffusion system with resource-consumer interaction, Hiroshima Math. J., 33 (2003), 253-295. |
[9] |
H. Fujikawa and M. Matsushita, Fractal growth of Bacillus subtilis on agar plates, J. Phys. Soc. Japan, 58 (1989), 3875-3878.
doi: 10.1143/JPSJ.58.3875. |
[10] |
H. Fujikawa and M. Matsushita, Bacterial fractal growth in the concentration field of a nutrient, J. Phys. Soc. Japan, 60 (1991), 88-94.
doi: 10.1143/JPSJ.60.88. |
[11] |
I. Golding, Y. Kozlovsky, I. Cohen and E. Ben-Jacob, Studies of bacterial branching growth using reaction-diffusion models for colonial development, Physica A, 260 (1998), 510-554.
doi: 10.1016/S0378-4371(98)00345-8. |
[12] |
K. Kawasaki, A. Mochizuki, M. Matsushita, T. Umeda and N. Shigesada, Modeling spatio-temporal patterns generated by bacillus subtilis, Journal of Theoretical Biology, 188 (1997), 177-185. |
[13] |
S. Kitsunezaki, Interface dynamics for bacterial colony formation, J. Phys. Soc. Japan, 66 (1997), 1544-1550.
doi: 10.1143/JPSJ.66.1544. |
[14] |
A. M. Lacasta, I. R. Cantalapiedra, C. E. Auguet, A. Peñaranda and L. Ramìrez-Piscina, Modelling of spatio-temporal patterns in bacterial colonies, Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics, 59 (1999), 7036. |
[15] |
A. Marrocco, H. Henry, I. B. Holland, M. Plapp, S. J. Séror and B. Perthame, Models of self-organizing bacterial communities and comparisons with experimental observations, Math. Model. Nat. Phenom., 5 (2010), 148-162.
doi: 10.1051/mmnp/20105107. |
[16] |
M. Matsushita and H. Fujikawa, Diffusion-limited growth in bacterial colony formation, Physica A, 168 (1990), 498-506.
doi: 10.1016/0378-4371(90)90402-E. |
[17] |
M. Matsushita, F. Hiramatsu, N. Kobayashi, T. Ozawa, Y. Yamasaki and T. Matsuyama, Colony formation in bacteria, experiments and modeling, Biofilms, 1 (2004), 305-317.
doi: 10.1017/S1479050505001626. |
[18] |
M. Matsushita, J. Wakita, H. Itoh, I. Rafols, T. Matsuyama, H. Sakaguchi and M. Mimura, Interface growth and pattern formation in bacterial colonies, Physica A, 249 (1998), 517-524.
doi: 10.1016/S0378-4371(97)00511-6. |
[19] |
M. Matsushita, J. Wakita, H. Itoh, K. Watanabe, T. Arai, T. Matsuyama, H. Sakaguchi and M. Mimura, Formation of colony patterns by a bacterial cell population, Physica A, 274 (1999), 190-199.
doi: 10.1016/S0378-4371(99)00328-3. |
[20] |
M. Mimura, H. Sakaguchi and M. Matsushita, Reaction-diffusion modelling of bacterial colony patterns, Physica A, 282 (2000), 283-303.
doi: 10.1016/S0378-4371(00)00085-6. |
[21] |
B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Math. Mod. Meth. Appl. Sci., 20 (2010), 1787-1821.
doi: 10.1142/S0218202510004799. |
[22] |
B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling, Netw. Heterog. Media, 6 (2011), 485-519.
doi: 10.3934/nhm.2011.6.485. |
[23] |
M. Mimura, Pattern formation in consumer-finite resource reaction-diffusion systems, Publ. RIMS, Kyoto Univ., 40 (2004), 1413-1431.
doi: 10.2977/prims/1145475451. |
[24] |
A. Roudneff-Chupin, "Modélisation Macroscopique De Mouvements De Foule,'' Ph. D thesis, Université Paris-Sud (2011), available at http://tel.archives-ouvertes.fr/tel-00678596. |
[25] |
F. Santambrogio, Gradient flows in Wasserstein spaces and applications to crowd movement, to appear in the proceedings of the Seminar X-EDP, École Polytechnique, Palaiseau, 2010. |
[26] |
N. Sukumar, Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids, Int. J. Numer. Meth. Engng, 57 (2003), 1-34.
doi: 10.1002/nme.664. |
[27] |
C. Villani, "Topics in Optimal Transportation," Grad. Stud. Math., 58 (AMS, Providence 2003).
doi: 10.1007/b12016. |
[28] |
C. Villani, "Optimal Transport, Old and New," Grundlehren der mathematischen Wissenschaften, 338 (2009).
doi: 10.1007/978-3-540-71050-9. |
[29] |
T. A. Witten, L. M. Sander, diffusion-limited aggregation, a kinetic critical phenomenon, Phys. Rev. Lett., 47 (1981), 1400-1403.
doi: 10.1103/PhysRevLett.47.1400. |
show all references
References:
[1] |
L. Ambrosio, N. Gigli and G. Savare, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Lectures in Mathematics, ETH Zürich . Birkhäuser Verlag, Basel, 2005. |
[2] |
L. Ambrosio and G. Savare, Gradient flows of probability measures, Handbook of differential equations, Evolutionary equations, III (2007), 1–136. Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam.
doi: 10.1016/S1874-5717(07)80004-1. |
[3] |
M. Badoual, P. Berbez, M. Aubert and B. Grammaticos, Simulating the migration and growth patterns of Bacillus subtilis, Physica A, 388 (2009), 549-559.
doi: 10.1016/j.physa.2008.10.046. |
[4] |
E. Ben-Jacob, From snowflake formation to growth of bacterial colonies II: Cooperative formation of complex colonial patterns, Contemporary Physics, 38 (1997), 205-241.
doi: 10.1080/001075197182405. |
[5] |
E. Ben-Jacob, O. Shochet, A. Tenenbaum, I. Cohen, A. Czirok and T. Vicsek, Generic modeling of cooperative growth patterns in bacterial colonies, Nature, 368 (1994), 46-49.
doi: 10.1038/368046a0. |
[6] |
G. Buttazzo and F. Santambrogio, A mass transportation model for the optimal planning of an urban region SIAM Review, 51 (2009), 593-610.
doi: 10.1137/090759197. |
[7] |
J. Dambrine, B. Maury, N. Meunier and A. Roudneff-Chupin, A congestion model for cell migration, Comm. Pure Appl. Anal., 11 (2012), 243-260.
doi: 10.3934/cpaa.2012.11.243. |
[8] |
E. Feireisl, D. Hilhorst, M. Mimura and R. Weidenfeld, On a nonlinear diffusion system with resource-consumer interaction, Hiroshima Math. J., 33 (2003), 253-295. |
[9] |
H. Fujikawa and M. Matsushita, Fractal growth of Bacillus subtilis on agar plates, J. Phys. Soc. Japan, 58 (1989), 3875-3878.
doi: 10.1143/JPSJ.58.3875. |
[10] |
H. Fujikawa and M. Matsushita, Bacterial fractal growth in the concentration field of a nutrient, J. Phys. Soc. Japan, 60 (1991), 88-94.
doi: 10.1143/JPSJ.60.88. |
[11] |
I. Golding, Y. Kozlovsky, I. Cohen and E. Ben-Jacob, Studies of bacterial branching growth using reaction-diffusion models for colonial development, Physica A, 260 (1998), 510-554.
doi: 10.1016/S0378-4371(98)00345-8. |
[12] |
K. Kawasaki, A. Mochizuki, M. Matsushita, T. Umeda and N. Shigesada, Modeling spatio-temporal patterns generated by bacillus subtilis, Journal of Theoretical Biology, 188 (1997), 177-185. |
[13] |
S. Kitsunezaki, Interface dynamics for bacterial colony formation, J. Phys. Soc. Japan, 66 (1997), 1544-1550.
doi: 10.1143/JPSJ.66.1544. |
[14] |
A. M. Lacasta, I. R. Cantalapiedra, C. E. Auguet, A. Peñaranda and L. Ramìrez-Piscina, Modelling of spatio-temporal patterns in bacterial colonies, Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics, 59 (1999), 7036. |
[15] |
A. Marrocco, H. Henry, I. B. Holland, M. Plapp, S. J. Séror and B. Perthame, Models of self-organizing bacterial communities and comparisons with experimental observations, Math. Model. Nat. Phenom., 5 (2010), 148-162.
doi: 10.1051/mmnp/20105107. |
[16] |
M. Matsushita and H. Fujikawa, Diffusion-limited growth in bacterial colony formation, Physica A, 168 (1990), 498-506.
doi: 10.1016/0378-4371(90)90402-E. |
[17] |
M. Matsushita, F. Hiramatsu, N. Kobayashi, T. Ozawa, Y. Yamasaki and T. Matsuyama, Colony formation in bacteria, experiments and modeling, Biofilms, 1 (2004), 305-317.
doi: 10.1017/S1479050505001626. |
[18] |
M. Matsushita, J. Wakita, H. Itoh, I. Rafols, T. Matsuyama, H. Sakaguchi and M. Mimura, Interface growth and pattern formation in bacterial colonies, Physica A, 249 (1998), 517-524.
doi: 10.1016/S0378-4371(97)00511-6. |
[19] |
M. Matsushita, J. Wakita, H. Itoh, K. Watanabe, T. Arai, T. Matsuyama, H. Sakaguchi and M. Mimura, Formation of colony patterns by a bacterial cell population, Physica A, 274 (1999), 190-199.
doi: 10.1016/S0378-4371(99)00328-3. |
[20] |
M. Mimura, H. Sakaguchi and M. Matsushita, Reaction-diffusion modelling of bacterial colony patterns, Physica A, 282 (2000), 283-303.
doi: 10.1016/S0378-4371(00)00085-6. |
[21] |
B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Math. Mod. Meth. Appl. Sci., 20 (2010), 1787-1821.
doi: 10.1142/S0218202510004799. |
[22] |
B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling, Netw. Heterog. Media, 6 (2011), 485-519.
doi: 10.3934/nhm.2011.6.485. |
[23] |
M. Mimura, Pattern formation in consumer-finite resource reaction-diffusion systems, Publ. RIMS, Kyoto Univ., 40 (2004), 1413-1431.
doi: 10.2977/prims/1145475451. |
[24] |
A. Roudneff-Chupin, "Modélisation Macroscopique De Mouvements De Foule,'' Ph. D thesis, Université Paris-Sud (2011), available at http://tel.archives-ouvertes.fr/tel-00678596. |
[25] |
F. Santambrogio, Gradient flows in Wasserstein spaces and applications to crowd movement, to appear in the proceedings of the Seminar X-EDP, École Polytechnique, Palaiseau, 2010. |
[26] |
N. Sukumar, Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids, Int. J. Numer. Meth. Engng, 57 (2003), 1-34.
doi: 10.1002/nme.664. |
[27] |
C. Villani, "Topics in Optimal Transportation," Grad. Stud. Math., 58 (AMS, Providence 2003).
doi: 10.1007/b12016. |
[28] |
C. Villani, "Optimal Transport, Old and New," Grundlehren der mathematischen Wissenschaften, 338 (2009).
doi: 10.1007/978-3-540-71050-9. |
[29] |
T. A. Witten, L. M. Sander, diffusion-limited aggregation, a kinetic critical phenomenon, Phys. Rev. Lett., 47 (1981), 1400-1403.
doi: 10.1103/PhysRevLett.47.1400. |
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