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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:
[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.
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.
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.
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.
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.
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, (2008).
|
[7] |
N. Bellomo and A. Bellouquid, On the modelling of vehicular traffic and crowds by the kinetic theory of active particles,, in, (2010), 273.
|
[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.
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.
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.
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.
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.
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.
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.
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.
doi: 10.1142/S0218202508003054. |
[16] |
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A. Bellouquid and M. Delitala, "Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach,", Modeling and Simulation Science, (2006).
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A. Bellouquid and M. Delitala, Asympotic limits of a discrete kinetic theory model of vehicular traffic,, Appl. Math. Letters, 24 (2011), 672.
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.
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: , (). Google Scholar |
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V. J. Blue and J. L. Adler, Cellular automata microsimulation of bidirectional pedestrian flows,, Transp. Research Board, 1678 (2000), 135.
doi: 10.3141/1678-17. |
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E. Bonabeau, M. Dorigo and G. Theraulaz, "Swarm Intelligence: From Natural to Artificial Systems,", Oxford University Press, (1999). Google Scholar |
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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.
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, (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.
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.
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.
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.
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.
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.
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.
doi: 10.1142/S0218202509003851. |
[32] |
C. F. Daganzo, Requiem for second order fluid approximations of traffic flow,, Transp. Research B, 29 (1995), 277.
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.
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.
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), 21 (2011), 843.
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.
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.
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.
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.
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.
doi: 10.1016/j.mbs.2008.06.002. |
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D. Helbing, A mathematical model for the behavior of pedestrians,, Behavioral Sciences, 36 (1991), 298.
doi: 10.1002/bs.3830360405. |
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D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phys, 73 (2001), 1067.
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).
doi: 10.1103/PhysRevE.75.046109. |
[44] |
D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical feature of escape panic,, Nature, 407 (2000), 487.
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.
doi: 10.1068/b2697. |
[46] |
D. Helbing and P. Molnár, Social force model for pedestrian dynamics,, Phys. Rev. E, 51 (1995), 4282.
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.
doi: 10.1140/epjb/e2009-00042-6. |
[48] |
L. F. Henderson, On the fluid mechanic of human crowd motion,, Transp. Research, 8 (1975), 509.
doi: 10.1016/0041-1647(74)90027-6. |
[49] |
R. L. Hughes, The flow of human crowds,, Annual Rev. Fluid Mech., 35 (2003), 169.
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.
doi: 10.1016/0022-5193(71)90050-6. |
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A. Kirman and J. Zimmermann, eds., "Economics with Heterogeneous Interacting Agents,", Lecture Notes in Economics and Mathematical Systems, 503 (2001).
|
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K. Lerman, A. Martinoli and A. Galstyan, A review of probabilistic macroscopic models for swarm robotic systems,, in, (2005), 143. Google Scholar |
[53] |
B. Maury, A. Roudneff-Chupin and F. Stantambrogio, A macroscopic crowd motion modelof gradient flow type,, Math. Models Methods Appl. Sci., 20 (2010), 1899.
doi: 10.1142/S0218202510004799. |
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A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, J. Math. Biol., 38 (1999), 534.
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M. Moussaid, D. Helbing, S. Garnier, A. Johanson, M. Combe and G. Theraulaz, Experimental study of the behavioral underlying mechanism underlying self-organization in human crowd,, Proc. Royal Society B: Biological Sciences, 276 (2009), 2755. Google Scholar |
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B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow,, Arch. Rat. Mech. Anal., 199 (2011), 707.
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J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses,, PLoS Computational Biology, 6 (2010).
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F. Venuti, L. Bruno and N. Bellomo, Crowd dynamics on a moving platform: Mathematical modelling and application to lively footbridges,, Mathl. Comp. Modelling, 45 (2007), 252.
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F. Venuti and L. Bruno, Crowd structure interaction in lively footbridges under synchronous lateral excitation: A literature review,, Phys. Life Rev., 6 (2009), 176.
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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.
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.
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.
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.
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.
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, (2008).
|
[7] |
N. Bellomo and A. Bellouquid, On the modelling of vehicular traffic and crowds by the kinetic theory of active particles,, in, (2010), 273.
|
[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.
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.
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.
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.
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.
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.
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.
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.
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). Google Scholar |
[17] |
A. Bellouquid and M. Delitala, "Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach,", Modeling and Simulation Science, (2006).
|
[18] |
A. Bellouquid and M. Delitala, Asympotic limits of a discrete kinetic theory model of vehicular traffic,, Appl. Math. Letters, 24 (2011), 672.
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.
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: , (). Google Scholar |
[21] |
V. J. Blue and J. L. Adler, Cellular automata microsimulation of bidirectional pedestrian flows,, Transp. Research Board, 1678 (2000), 135.
doi: 10.3141/1678-17. |
[22] |
E. Bonabeau, M. Dorigo and G. Theraulaz, "Swarm Intelligence: From Natural to Artificial Systems,", Oxford University Press, (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.
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, (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.
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.
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.
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.
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.
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.
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.
doi: 10.1142/S0218202509003851. |
[32] |
C. F. Daganzo, Requiem for second order fluid approximations of traffic flow,, Transp. Research B, 29 (1995), 277.
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.
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.
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), 21 (2011), 843.
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.
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.
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.
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.
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.
doi: 10.1016/j.mbs.2008.06.002. |
[41] |
D. Helbing, A mathematical model for the behavior of pedestrians,, Behavioral Sciences, 36 (1991), 298.
doi: 10.1002/bs.3830360405. |
[42] |
D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phys, 73 (2001), 1067.
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).
doi: 10.1103/PhysRevE.75.046109. |
[44] |
D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical feature of escape panic,, Nature, 407 (2000), 487.
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.
doi: 10.1068/b2697. |
[46] |
D. Helbing and P. Molnár, Social force model for pedestrian dynamics,, Phys. Rev. E, 51 (1995), 4282.
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.
doi: 10.1140/epjb/e2009-00042-6. |
[48] |
L. F. Henderson, On the fluid mechanic of human crowd motion,, Transp. Research, 8 (1975), 509.
doi: 10.1016/0041-1647(74)90027-6. |
[49] |
R. L. Hughes, The flow of human crowds,, Annual Rev. Fluid Mech., 35 (2003), 169.
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.
doi: 10.1016/0022-5193(71)90050-6. |
[51] |
A. Kirman and J. Zimmermann, eds., "Economics with Heterogeneous Interacting Agents,", Lecture Notes in Economics and Mathematical Systems, 503 (2001).
|
[52] |
K. Lerman, A. Martinoli and A. Galstyan, A review of probabilistic macroscopic models for swarm robotic systems,, in, (2005), 143. Google Scholar |
[53] |
B. Maury, A. Roudneff-Chupin and F. Stantambrogio, A macroscopic crowd motion modelof gradient flow type,, Math. Models Methods Appl. Sci., 20 (2010), 1899.
doi: 10.1142/S0218202510004799. |
[54] |
A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, J. Math. Biol., 38 (1999), 534.
doi: 10.1007/s002850050158. |
[55] |
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