# American Institute of Mathematical Sciences

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

## 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

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.
Citation: Bertrand Maury, Aude Roudneff-Chupin, Filippo Santambrogio. Congestion-driven dendritic growth. Discrete and 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. [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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##### 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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