On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms

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September 2011, 6(3): 383-399. doi: 10.3934/nhm.2011.6.383

On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms

Nicola Bellomo ^1, and Abdelghani Bellouquid ^2,

1.	Department of Mathematics, Politecnico Torino, Corso Duca degli Abruzzi 24, 10129, Torino
2.	University Cadi Ayyad, Ecole Nationale des Sciences Appliquées, Safi, Morocco

Received December 2010 Revised June 2011 Published August 2011

Abstract
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This paper presents a critical overview on the modeling of crowds and swarms and focuses on a modeling strategy based on the attempt to retain the complexity characteristics of systems under consideration viewed as an assembly of living entities characterized by the ability of expressing heterogeneously distributed strategies.

Keywords: scaling, swarms dynamics, living systems., Crowds, complexity.

Mathematics Subject Classification: Primary: 90B20, 92D50, 92-02, 35L65, 82-0.

Citation: Nicola Bellomo, Abdelghani Bellouquid. On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms. Networks and Heterogeneous Media, 2011, 6 (3) : 383-399. doi: 10.3934/nhm.2011.6.383

References:

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[2]	G. Ajmone Marsan, N. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economical systems by functional subsystems representation, Kinetic Related Models, 1 (2008), 249-278. doi: 10.3934/krm.2008.1.249.
[3]	A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955.
[4]	M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study, Proc. Nat. Acad. Sci., 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.
[5]	R. N. Bearon and K. L. Grünbaum, From individual behavior to population models: A case study using swimming algae, J. Theor. Biol., 251 (2008), 33-42. doi: 10.1016/j.jtbi.2008.01.007.
[6]	N. Bellomo, "Modeling Complex Living Systems. A Kinetic Theory and Stochastic Game Approach," Modeling and Simulation in Science, Engineering, and Technology, Birkhäuser Boston, Inc., Boston, MA, 2008.
[7]	N. Bellomo and A. Bellouquid, On the modelling of vehicular traffic and crowds by the kinetic theory of active particles, in "Mathematical Modelling of Collective Behaviour in Socio-Economics and Life Sciences" (eds. G. Naldi, L. Pareschi and G. Toscani), Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Inc., Boston, MA, (2010), 273-296.
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[9]	N. Bellomo, H. Berestycki, F. Brezzi and J.-P. Nadal, Mathematics and complexity in human and life sciences, Math. Models Methods Appl. Sci., 19 (2009), 1385-1389. doi: 10.1142/S0218202509003826.
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[14]	N. Bellomo and C. Dogbè, On the modelling crowd dynamics from scaling to hyperbolic macroscopic models, Math. Models Methods Appl. Sci., 18 (2008), 1317-1345. doi: 10.1142/S0218202508003054.
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[19]	M. L. Bertotti and M. Delitala, Conservation laws and asymptotic behavior of a model of social dynamics, Nonlinear Anal. RWA, 9 (2008), 183-196. doi: 10.1016/j.nonrwa.2006.09.012.
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[26]	A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini and R. Tavarone, From empirical data to inter-individual interactions: Unveiling the rules of collective animal behavior, Math. Models Methods Appl. Sci., 20 (2010), 1491-1510. doi: 10.1142/S0218202510004660.
[27]	Y. Chjang, M. D'Orsogna, D. Marthaler, A. Bertozzi and L. Chayes, State transition and the continuum limit for 2D interacting, self-propelled particles system, Physica D, 232 (2007), 33-47. doi: 10.1016/j.physd.2007.05.007.
[28]	R. M. Colombo and M. D. Rosini, Existence of nonclassical solutions in a pedestrian flow model, Nonlinear Anal. RWA, 10 (2009), 2716-2728. doi: 10.1016/j.nonrwa.2008.08.002.
[29]	V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics, Math. Models Methods Appl. Sci., 18 (2008), 1217-1247. doi: 10.1142/S0218202508003017.
[30]	E. Cristiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515.
[31]	F. Cucker and Jiu-Gang Dong, On the critical exponent for flocks under hierarchical leadership, Math. Models Methods Appl. Sci., 19 (2009), 1391-1404. doi: 10.1142/S0218202509003851.
[32]	C. F. Daganzo, Requiem for second order fluid approximations of traffic flow, Transp. Research B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z.
[33]	P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.
[34]	S. de Lillo, M. Delitala and C. Salvadori, Modelling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles, Math. Models Methods Appl. Sci., 19 (2009), 1404-1425. doi: 10.1142/S0218202509003838.
[35]	M. Delitala, P. Pucci and C. Salvatori, From methods of the mathematical kinetic theory for active particles to modelling virus mutations, Math. Models Methods Appl. Sci., 21 (2011), 843-870. doi: 10.1142/S0218202511005398.
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[42]	D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys, 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.
[43]	D. Helbing, A. Johansson and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Physical Review E, 75 (2007), 046109. doi: 10.1103/PhysRevE.75.046109.
[44]	D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical feature of escape panic, Nature, 407 (2000), 487-490. doi: 10.1038/35035023.
[45]	D. Helbing, P. Molnár, I. Farkas and K. Bolay, Self-organizing pedestrian movement, Environment and Planning B, 28 (2001), 361-383. doi: 10.1068/b2697.
[46]	D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Phys. Rev. E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.
[47]	D. Helbing and M. Moussaid, Analytical calculation of critical perturbation amplitudes and critical densities by non-linear stability analysis for a simple traffic flow model, Eur. Phys. J. B., 69 (2009), 571-581. doi: 10.1140/epjb/e2009-00042-6.
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show all references

References:

[1]	K. Anguige and C. Schmeiser, A one-dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion, J. Math. Biol., 58 (2009), 395-427. doi: 10.1007/s00285-008-0197-8.
[2]	G. Ajmone Marsan, N. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economical systems by functional subsystems representation, Kinetic Related Models, 1 (2008), 249-278. doi: 10.3934/krm.2008.1.249.
[3]	A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955.
[4]	M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study, Proc. Nat. Acad. Sci., 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.
[5]	R. N. Bearon and K. L. Grünbaum, From individual behavior to population models: A case study using swimming algae, J. Theor. Biol., 251 (2008), 33-42. doi: 10.1016/j.jtbi.2008.01.007.
[6]	N. Bellomo, "Modeling Complex Living Systems. A Kinetic Theory and Stochastic Game Approach," Modeling and Simulation in Science, Engineering, and Technology, Birkhäuser Boston, Inc., Boston, MA, 2008.
[7]	N. Bellomo and A. Bellouquid, On the modelling of vehicular traffic and crowds by the kinetic theory of active particles, in "Mathematical Modelling of Collective Behaviour in Socio-Economics and Life Sciences" (eds. G. Naldi, L. Pareschi and G. Toscani), Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Inc., Boston, MA, (2010), 273-296.
[8]	N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis from binary mixtures of multicellular growing systems, Math. Models Methods Appl. Sci., 20 (2010), 1179-1207. doi: 10.1142/S0218202510004568.
[9]	N. Bellomo, H. Berestycki, F. Brezzi and J.-P. Nadal, Mathematics and complexity in human and life sciences, Math. Models Methods Appl. Sci., 19 (2009), 1385-1389. doi: 10.1142/S0218202509003826.
[10]	N. Bellomo, H. Berestycki, F. Brezzi and J.-P. Nadal, Mathematics and complexity in human and life sciences, Math. Models Methods Appl. Sci., 20 (2010), 1391-1395. doi: 10.1142/S0218202510004702.
[11]	N. Bellomo, C. Bianca and M. S. Mongiovi, On the modeling of nonlinear interactions in large complex systems, Applied Mathematical Letters, 23 (2010), 1372-1377. doi: 10.1016/j.aml.2010.07.001.
[12]	N. Bellomo, C. Bianca and M. Delitala, Complexity analysis and mathematical tools towards the modelling of living systems, Phys. Life Rev., 6 (2009), 144-175. doi: 10.1016/j.plrev.2009.06.002.
[13]	N. Bellomo and B. Carbonaro, Towards a mathematical theory of living systems focusing on developmental biology and evolution: a review and perpectives, Phys. Life Reviews, 8 (2011), 1-18. doi: 10.1016/j.plrev.2010.12.001.
[14]	N. Bellomo and C. Dogbè, On the modelling crowd dynamics from scaling to hyperbolic macroscopic models, Math. Models Methods Appl. Sci., 18 (2008), 1317-1345. doi: 10.1142/S0218202508003054.
[15]	N. Bellomo and C. Dogbè, On the modelling of traffic and crowds - a survey of models, speculations, and perspectives, SIAM Review, 53 (2011), 409-463. doi: 10.1142/S0218202508003054.
[16]	A. Bellouquid, E. De Angelis and L. Fermo, Towards the modeling of Vehicular traffic as a complex system: A kinetic theory approach, Math. Models Methods Appl. Sci., 22 (2012), to appear.
[17]	A. Bellouquid and M. Delitala, "Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach," Modeling and Simulation Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2006.
[18]	A. Bellouquid and M. Delitala, Asympotic limits of a discrete kinetic theory model of vehicular traffic, Appl. Math. Letters, 24 (2011), 672-678. doi: 10.1016/j.aml.2010.12.004.
[19]	M. L. Bertotti and M. Delitala, Conservation laws and asymptotic behavior of a model of social dynamics, Nonlinear Anal. RWA, 9 (2008), 183-196. doi: 10.1016/j.nonrwa.2006.09.012.
[20]	A. Bertozzi, D. Grunbaum, P. S. Krishnaprasad, and I. Schwartz, Swarming by nature and by design, 2006. Available from: http://www.ipam.ucla.edu/programs/swa2006.
[21]	V. J. Blue and J. L. Adler, Cellular automata microsimulation of bidirectional pedestrian flows, Transp. Research Board, 1678 (2000), 135-141. doi: 10.3141/1678-17.
[22]	E. Bonabeau, M. Dorigo and G. Theraulaz, "Swarm Intelligence: From Natural to Artificial Systems," Oxford University Press, Oxford, 1999.
[23]	L. Bruno, A. Tosin, P. Tricerri and F. Venuti, Non-local first-order modelling of crowd dynamics: A multidimensional framework with applications, Appl. Math. Model., 35 (2011), 426-445. doi: 10.1016/j.apm.2010.07.007.
[24]	S. Buchmuller and U. Weidman, Parameters of pedestrians, pedestrian traffic and walking facilities, ETH Report Nr. 132, October, 2006.
[25]	J. A. Carrillo, A. Klar, S. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force, Math. Models Methods Appl. Sci., 20 (2010), 1533-1552. doi: 10.1142/S0218202510004684.
[26]	A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini and R. Tavarone, From empirical data to inter-individual interactions: Unveiling the rules of collective animal behavior, Math. Models Methods Appl. Sci., 20 (2010), 1491-1510. doi: 10.1142/S0218202510004660.
[27]	Y. Chjang, M. D'Orsogna, D. Marthaler, A. Bertozzi and L. Chayes, State transition and the continuum limit for 2D interacting, self-propelled particles system, Physica D, 232 (2007), 33-47. doi: 10.1016/j.physd.2007.05.007.
[28]	R. M. Colombo and M. D. Rosini, Existence of nonclassical solutions in a pedestrian flow model, Nonlinear Anal. RWA, 10 (2009), 2716-2728. doi: 10.1016/j.nonrwa.2008.08.002.
[29]	V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics, Math. Models Methods Appl. Sci., 18 (2008), 1217-1247. doi: 10.1142/S0218202508003017.
[30]	E. Cristiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515.
[31]	F. Cucker and Jiu-Gang Dong, On the critical exponent for flocks under hierarchical leadership, Math. Models Methods Appl. Sci., 19 (2009), 1391-1404. doi: 10.1142/S0218202509003851.
[32]	C. F. Daganzo, Requiem for second order fluid approximations of traffic flow, Transp. Research B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z.
[33]	P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.
[34]	S. de Lillo, M. Delitala and C. Salvadori, Modelling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles, Math. Models Methods Appl. Sci., 19 (2009), 1404-1425. doi: 10.1142/S0218202509003838.
[35]	M. Delitala, P. Pucci and C. Salvatori, From methods of the mathematical kinetic theory for active particles to modelling virus mutations, Math. Models Methods Appl. Sci., 21 (2011), 843-870. doi: 10.1142/S0218202511005398.
[36]	M. Delitala and A. Tosin, Mathematical modelling of vehicular traffic: A discrete kinetic theory approach, Math. Models Methods Appl. Sci., 17 (2007), 901-932. doi: 10.1142/S0218202507002157.
[37]	C. Detrain and J,-L. Doneubourg, Self-organized structures in a superorganism: Do ants "behave" like molecules?, Physics of Life, 3 (2006), 162-187. doi: 10.1016/j.plrev.2006.07.001.
[38]	M. Di Francesco, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case, J. Diff. Equations, 250 (2011), 1334-1362. doi: 10.1016/j.jde.2010.10.015.
[39]	C. Dogbè, On the Cauchy problem for macroscopic model of pedestrian flows, J. Math. Anal. Appl., 372 (2010), 77-85. doi: 10.1016/j.jmaa.2010.06.044.
[40]	D. Grünbaum, K. Chan, E. Tobin and M. T. Nishizaki, Non-linear advection-diffusion equations approximate swarming but not schooling population, Math. Biosci., 214 (2008), 38-48. doi: 10.1016/j.mbs.2008.06.002.
[41]	D. Helbing, A mathematical model for the behavior of pedestrians, Behavioral Sciences, 36 (1991), 298-310. doi: 10.1002/bs.3830360405.
[42]	D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys, 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.
[43]	D. Helbing, A. Johansson and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Physical Review E, 75 (2007), 046109. doi: 10.1103/PhysRevE.75.046109.
[44]	D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical feature of escape panic, Nature, 407 (2000), 487-490. doi: 10.1038/35035023.
[45]	D. Helbing, P. Molnár, I. Farkas and K. Bolay, Self-organizing pedestrian movement, Environment and Planning B, 28 (2001), 361-383. doi: 10.1068/b2697.
[46]	D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Phys. Rev. E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.
[47]	D. Helbing and M. Moussaid, Analytical calculation of critical perturbation amplitudes and critical densities by non-linear stability analysis for a simple traffic flow model, Eur. Phys. J. B., 69 (2009), 571-581. doi: 10.1140/epjb/e2009-00042-6.
[48]	L. F. Henderson, On the fluid mechanic of human crowd motion, Transp. Research, 8 (1975), 509-515. doi: 10.1016/0041-1647(74)90027-6.
[49]	R. L. Hughes, The flow of human crowds, Annual Rev. Fluid Mech., 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136.
[50]	E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoretical Biology, 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.
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