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

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	N. Bellomo, "Modeling Complex Living Systems. A Kinetic Theory and Stochastic Game Approach,", Modeling and Simulation in Science, (2008). Google Scholar
[7]	N. Bellomo and A. Bellouquid, On the modelling of vehicular traffic and crowds by the kinetic theory of active particles,, in, (2010), 273. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. 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. 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, (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. Google Scholar
[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. 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. 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. doi: 10.1016/j.physd.2007.05.007. Google Scholar
[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. Google Scholar
[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. 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. 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. 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. 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. doi: 10.1142/S0218202508003005. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
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[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. 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. 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. doi: 10.1002/bs.3830360405. Google Scholar
[42]	D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phys, 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067. Google Scholar
[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. Google Scholar
[44]	D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical feature of escape panic,, Nature, 407 (2000), 487. doi: 10.1038/35035023. Google Scholar
[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. Google Scholar
[46]	D. Helbing and P. Molnár, Social force model for pedestrian dynamics,, Phys. Rev. E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282. Google Scholar
[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. Google Scholar
[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. Google Scholar
[49]	R. L. Hughes, The flow of human crowds,, Annual Rev. Fluid Mech., 35 (2003), 169. doi: 10.1146/annurev.fluid.35.101101.161136. Google Scholar
[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. Google Scholar
[51]	A. Kirman and J. Zimmermann, eds., "Economics with Heterogeneous Interacting Agents,", Lecture Notes in Economics and Mathematical Systems, 503 (2001). Google Scholar
[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. Google Scholar
[54]	A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, J. Math. Biol., 38 (1999), 534. doi: 10.1007/s002850050158. Google Scholar
[55]	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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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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	N. Bellomo, "Modeling Complex Living Systems. A Kinetic Theory and Stochastic Game Approach,", Modeling and Simulation in Science, (2008). Google Scholar
[7]	N. Bellomo and A. Bellouquid, On the modelling of vehicular traffic and crowds by the kinetic theory of active particles,, in, (2010), 273. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. 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. 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, (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. Google Scholar
[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. 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. 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. doi: 10.1016/j.physd.2007.05.007. Google Scholar
[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. Google Scholar
[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. 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. 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. 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. 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. doi: 10.1142/S0218202508003005. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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. 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. doi: 10.1002/bs.3830360405. Google Scholar
[42]	D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phys, 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067. Google Scholar
[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. Google Scholar
[44]	D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical feature of escape panic,, Nature, 407 (2000), 487. doi: 10.1038/35035023. Google Scholar
[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. Google Scholar
[46]	D. Helbing and P. Molnár, Social force model for pedestrian dynamics,, Phys. Rev. E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282. Google Scholar
[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. Google Scholar
[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. Google Scholar
[49]	R. L. Hughes, The flow of human crowds,, Annual Rev. Fluid Mech., 35 (2003), 169. doi: 10.1146/annurev.fluid.35.101101.161136. Google Scholar
[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. Google Scholar
[51]	A. Kirman and J. Zimmermann, eds., "Economics with Heterogeneous Interacting Agents,", Lecture Notes in Economics and Mathematical Systems, 503 (2001). Google Scholar
[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. Google Scholar
[54]	A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, J. Math. Biol., 38 (1999), 534. doi: 10.1007/s002850050158. Google Scholar
[55]	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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