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April  2014, 34(4): 1575-1604. doi: 10.3934/dcds.2014.34.1575

Congestion-driven dendritic growth

Bertrand Maury 1, , Aude Roudneff-Chupin 1, and  Filippo Santambrogio 2,

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

Received  October 2012 Revised  March 2013 Published  October 2013

In order to observe growth phenomena in biology where dendritic shapes appear, we propose a simple model where a given population evolves feeded by a diffusing nutriment, but is subject to a density constraint. The particles (e.g., cells) of the population spontaneously stay passive at rest, and only move in order to satisfy the constraint $\rho\leq 1$, by choosing the minimal correction velocity so as to prevent overcongestion. We treat this constraint by means of projections in the space of densities endowed with the Wasserstein distance $W_2$, defined through optimal transport. This allows to provide an existence result and suggests some numerical computations, in the same spirit of what the authors did for crowd motion (but with extra difficulties, essentially due to the fact that the total mass may increase). The numerical simulations show, according to the values of the parameter and in particular of the diffusion coefficient of the nutriment, the formation of dendritic patterns in the space occupied by cells.
Keywords: Wasserstein distance, congestion, continuity equation., Dendritic growth.
Mathematics Subject Classification: 35K57, 49J45, 49M25, 92B0.
Citation: Bertrand Maury, Aude Roudneff-Chupin, Filippo Santambrogio. Congestion-driven dendritic growth. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1575-1604. doi: 10.3934/dcds.2014.34.1575
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[13]

S. Kitsunezaki, Interface dynamics for bacterial colony formation, J. Phys. Soc. Japan, 66 (1997), 1544-1550. doi: 10.1143/JPSJ.66.1544.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[27]

C. Villani, "Topics in Optimal Transportation," Grad. Stud. Math., 58 (AMS, Providence 2003). doi: 10.1007/b12016.  Google Scholar

[28]

C. Villani, "Optimal Transport, Old and New," Grundlehren der mathematischen Wissenschaften, 338 (2009). doi: 10.1007/978-3-540-71050-9.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[13]

S. Kitsunezaki, Interface dynamics for bacterial colony formation, J. Phys. Soc. Japan, 66 (1997), 1544-1550. doi: 10.1143/JPSJ.66.1544.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[27]

C. Villani, "Topics in Optimal Transportation," Grad. Stud. Math., 58 (AMS, Providence 2003). doi: 10.1007/b12016.  Google Scholar

[28]

C. Villani, "Optimal Transport, Old and New," Grundlehren der mathematischen Wissenschaften, 338 (2009). doi: 10.1007/978-3-540-71050-9.  Google Scholar

[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.  Google Scholar

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