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On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms
1. | Department of Mathematics, Politecnico Torino, Corso Duca degli Abruzzi 24, 10129, Torino |
2. | University Cadi Ayyad, Ecole Nationale des Sciences Appliquées, Safi, Morocco |
References:
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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. |
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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. |
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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. |
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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. |
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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. |
[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. |
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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. |
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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. |
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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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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.
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E. Bonabeau, M. Dorigo and G. Theraulaz, "Swarm Intelligence: From Natural to Artificial Systems," Oxford University Press, Oxford, 1999. |
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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.
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S. Buchmuller and U. Weidman, Parameters of pedestrians, pedestrian traffic and walking facilities, ETH Report Nr. 132, October, 2006. |
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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. |
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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.
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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. |
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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. |
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P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.
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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. |
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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. |
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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.
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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.
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C. Dogbè, On the Cauchy problem for macroscopic model of pedestrian flows, J. Math. Anal. Appl., 372 (2010), 77-85.
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D. Helbing, A. Johansson and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Physical Review E, 75 (2007), 046109.
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D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical feature of escape panic, Nature, 407 (2000), 487-490.
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D. Helbing, P. Molnár, I. Farkas and K. Bolay, Self-organizing pedestrian movement, Environment and Planning B, 28 (2001), 361-383.
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D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Phys. Rev. E, 51 (1995), 4282-4286.
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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.
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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.
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