American Institute of Mathematical Sciences

Advanced
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

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 & Heterogeneous Media, 2011, 6 (3) : 383-399. doi: 10.3934/nhm.2011.6.383
References:
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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.  Google Scholar

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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.  Google Scholar

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[9]

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[15]

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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. Google Scholar

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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.  Google Scholar

[20]

A. Bertozzi, D. Grunbaum, P. S. Krishnaprasad, and I. Schwartz, Swarming by nature and by design, 2006., Available from: , ().   Google Scholar

[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.  Google Scholar

[22]

E. Bonabeau, M. Dorigo and G. Theraulaz, "Swarm Intelligence: From Natural to Artificial Systems," Oxford University Press, Oxford, 1999. Google Scholar

[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.  Google Scholar

[24]

S. Buchmuller and U. Weidman, Parameters of pedestrians, pedestrian traffic and walking facilities, ETH Report Nr. 132, October, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

D. Helbing, A mathematical model for the behavior of pedestrians, Behavioral Sciences, 36 (1991), 298-310. doi: 10.1002/bs.3830360405.  Google Scholar

[42]

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