2011, 4(4): 1025-1047. doi: 10.3934/krm.2011.4.1025

Continuous limit of a crowd motion and herding model: Analysis and numerical simulations

1. 

Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität (WWU) Münster, Einsteinstr. 62, D-48149 Münster

2. 

DAMTP, University of Cambridge, Cambridge CB3 0WA, United Kingdom, United Kingdom

Received  June 2011 Revised  September 2011 Published  November 2011

In this paper we study the continuum limit of a cellular automaton model used for simulating human crowds with herding behaviour. We derive a system of non-linear partial differential equations resembling the Keller-Segel model for chemotaxis, however with a non-monotone interaction. The latter has interesting consequences on the behaviour of the model's solutions, which we highlight in its analysis. In particular we study the possibility of stationary states, the formation of clusters and explore their connection to congestion.
    We also introduce an efficient numerical simulation approach based on an appropriate hybrid discontinuous Galerkin method, which in particular allows flexible treatment of complicated geometries. Extensive numerical studies also provide a better understanding of the strengths and shortcomings of the herding model, in particular we examine trapping effects of crowds behind non-convex obstacles.
Citation: Martin Burger, Peter Alexander Markowich, Jan-Frederik Pietschmann. Continuous limit of a crowd motion and herding model: Analysis and numerical simulations. Kinetic & Related Models, 2011, 4 (4) : 1025-1047. doi: 10.3934/krm.2011.4.1025
References:
[1]

A. Aw and M. Rascle, Resurrection of "second order'' models of traffic flow,, SIAM J. Appl. Math., 60 (2000), 916. doi: 10.1137/S0036139997332099.

[2]

P. Bastian and B. Rivière, Superconvergence and H(div)-projection for discontinuous Galerkin methods,, Int. J. Numer. Meth. Fluids., 42 (2003), 1043. doi: 10.1002/fld.562.

[3]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,, Electron. J. Differential Equations, 2006 ().

[4]

M. Brunnermeier, "Asset Pricing under Asymmetric Information: Bubbles, Crashes, Technical Analysis and Herding,", Oxford University Press, (2001).

[5]

M. Burger, Y. Dolak-Struss and C. Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions,, Commun. Math. Sci., 6 (2008), 1.

[6]

M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst planck equations for ion flux through confined geometries,, preprint, (2010).

[7]

C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton,, Physica A: Statistical Mechanics and its Applications, 295 (2001), 507.

[8]

J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion,, Math. Models Methods Appl. Sci., 21 (2011), 515. doi: 10.1142/S0218202511005131.

[9]

B. Chopard and M. Droz, Cellular automata model for the diffusion equation,, Journal of Statistical Physics, 64 (1991), 859. doi: 10.1007/BF01048321.

[10]

B. Chopard and M. Droz, "Cellular Automata Modeling of Physical Systems,", Collection Aléa-Saclay: Monographs and Texts in Statistical Physics, (1998).

[11]

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.

[12]

A. Devenow and I. Welch, Rational herding in financial economics,, Papers and Proceedings of the Tenth Annual Congress of the European Economic Association, 40 (1996), 603. doi: 10.1016/0014-2921(95)00073-9.

[13]

M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case,, Journal of Differential Equations, 250 (2011), 1334. doi: 10.1016/j.jde.2010.10.015.

[14]

M. Di Francesco and J. Rosado, Fully parabolic Keller-Segel model for chemotaxis with prevention of overcrowding,, Nonlinearity, 21 (2008), 2715. doi: 10.1088/0951-7715/21/11/012.

[15]

E. W. Dijkstra, A note on two problems in connexion with graphs,, Numer. Math., 1 (1959), 269. doi: 10.1007/BF01386390.

[16]

C. Dogbé, Modeling crowd dynamics by the mean-field limit approach,, Mathematical and Computer Modelling, 52 (2010), 1506.

[17]

Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity,, SIAM J. Appl. Math., 66 (2005), 286. doi: 10.1137/040612841.

[18]

H. Egger, A class of hybrid mortar finite element methods for interface problems with non-matching meshes,, Technical Report AICES-2009-2, (): 2009.

[19]

, V. for linear stability,, Available from: \url{http://www.jfpietschmann.eu/crowdmotion}., ().

[20]

M. Fukui and Y. Ishibashi, Self-organized phase transitions in CA-models for pedestrians,, J. Phys. Soc. Japan, 8 (1999), 2861. doi: 10.1143/JPSJ.68.2861.

[21]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits,, J. Statist. Phys., 87 (1997), 37. doi: 10.1007/BF02181479.

[22]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Monographs and Studies in Mathematics, 24,, Pitman (Advanced Publishing Program), (1985).

[23]

D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Mod. Phys., 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067.

[24]

D. Helbing and P. Molnar, Social force model for pedestrian dynamics,, Physical Review E, 51 (1995). doi: 10.1103/PhysRevE.51.4282.

[25]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3.

[26]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103.

[27]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II,, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51.

[28]

R. L. Hughes, A continuum theory for the flow of pedestrians,, Transportation Research Part B: Methodological, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7.

[29]

E. Keller and L. Segel, Initiation of slide mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5.

[30]

A. Kirchner, "Modellierung und Statistische Physik Biologischer und Sozialer Systeme,", Ph.D Thesis, (2002).

[31]

A. Kirchner and A. Schadschneider, Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics,, Physica A: Statistical Mechanics and its Applications, 312 (2002), 260.

[32]

J.-M. Lasry and P.-L. Lions, Mean field games,, Jpn. J. Math., 2 (2007), 229. doi: 10.1007/s11537-007-0657-8.

[33]

P. A. Markowich, "The Stationary Semiconductor Device Equations,", Computational Microelectronics, (1986).

[34]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990). doi: 10.1007/978-3-7091-6961-2.

[35]

M. Matsumoto and T. Nishimura, Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator,, ACM Trans. Model. Comput. Simul., 8 (1998), 3. doi: 10.1145/272991.272995.

[36]

B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of the gradient-flow type,, Math. Models Methods Appl. Sci., 20 (2010), 1787. doi: 10.1142/S0218202510004799.

[37]

M. Muramatsu and T. Nagatani, Jamming transition in two-dimensional pedestrian traffic,, Physica A, 275 (2000), 281. doi: 10.1016/S0378-4371(99)00447-1.

[38]

J. R. Nofsinger and R. W. Sias, Herding and feedback trading by institutional and individual investors,, The Journal of Finance, 54 (1999), 2263. doi: 10.1111/0022-1082.00188.

[39]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501.

[40]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow,, Arch. Ration. Mech. Anal., 199 (2011), 707. doi: 10.1007/s00205-010-0366-y.

[41]

R. M. Raafat, N. Chater and C. Frith, Herding in humans,, Trends in Cognitive Sciences, 13 (2009), 420. doi: 10.1016/j.tics.2009.08.002.

[42]

A. Schadschneider, W. Klingsch, H. Kluepfel, T. Kretz, C. Rogsch and A. Seyfried, Evacuation Dynamics: Empirical Results, Modeling and Applications, in "Encyclopedia of Complexity and System Science" (ed. R. A. Meyers), Vol. 3, pp. 3142,, Springer, (2009).

[43]

O. Schenk and K. Gärtner, Solving unsymmetric sparse systems of linear equations with pardiso,, Journal of Future Generation Computer Systems, 20 (2004), 475. doi: 10.1016/j.future.2003.07.011.

[44]

O. Schenk and K. Gärtner, On fast factorization pivoting methods for sparse symmetric indefinite systems,, Elec. Trans. Numer. Anal, 23 (2006), 158.

[45]

J. Schöberl, Netgen an advancing front 2d/3d-mesh generator based on abstract rules,, Computing and Visualization in Science, 1 (1997), 41. doi: 10.1007/s007910050004.

[46]

M. Simpson, K. Landman and B. Hughes, Diffusing populations: Ghosts or folks?,, Australasian Journal of Engineering Education, 15 (2009), 59.

[47]

A. Sopasakis and M. A. Katsoulakis, Stochastic modeling and simulation of traffic flow: Asymmetric single exclusion process with Arrhenius look-ahead dynamics,, SIAM Journal on Applied Mathematics, 66 (2006), 921. doi: 10.1137/040617790.

[48]

, V. visualization environment,, Available from: \url{http://www.llnl.gov/visit/home.html}., ().

[49]

U. Weidmann, "Transporttechnik der Fussgänger-Transporttechnische Eigenschaften des Fussgängerverkehrs (Literaturstudie),", in German, 90 (1993).

[50]

D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding,, Nonlinear Analysis, 59 (2004), 1293.

show all references

References:
[1]

A. Aw and M. Rascle, Resurrection of "second order'' models of traffic flow,, SIAM J. Appl. Math., 60 (2000), 916. doi: 10.1137/S0036139997332099.

[2]

P. Bastian and B. Rivière, Superconvergence and H(div)-projection for discontinuous Galerkin methods,, Int. J. Numer. Meth. Fluids., 42 (2003), 1043. doi: 10.1002/fld.562.

[3]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,, Electron. J. Differential Equations, 2006 ().

[4]

M. Brunnermeier, "Asset Pricing under Asymmetric Information: Bubbles, Crashes, Technical Analysis and Herding,", Oxford University Press, (2001).

[5]

M. Burger, Y. Dolak-Struss and C. Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions,, Commun. Math. Sci., 6 (2008), 1.

[6]

M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst planck equations for ion flux through confined geometries,, preprint, (2010).

[7]

C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton,, Physica A: Statistical Mechanics and its Applications, 295 (2001), 507.

[8]

J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion,, Math. Models Methods Appl. Sci., 21 (2011), 515. doi: 10.1142/S0218202511005131.

[9]

B. Chopard and M. Droz, Cellular automata model for the diffusion equation,, Journal of Statistical Physics, 64 (1991), 859. doi: 10.1007/BF01048321.

[10]

B. Chopard and M. Droz, "Cellular Automata Modeling of Physical Systems,", Collection Aléa-Saclay: Monographs and Texts in Statistical Physics, (1998).

[11]

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.

[12]

A. Devenow and I. Welch, Rational herding in financial economics,, Papers and Proceedings of the Tenth Annual Congress of the European Economic Association, 40 (1996), 603. doi: 10.1016/0014-2921(95)00073-9.

[13]

M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case,, Journal of Differential Equations, 250 (2011), 1334. doi: 10.1016/j.jde.2010.10.015.

[14]

M. Di Francesco and J. Rosado, Fully parabolic Keller-Segel model for chemotaxis with prevention of overcrowding,, Nonlinearity, 21 (2008), 2715. doi: 10.1088/0951-7715/21/11/012.

[15]

E. W. Dijkstra, A note on two problems in connexion with graphs,, Numer. Math., 1 (1959), 269. doi: 10.1007/BF01386390.

[16]

C. Dogbé, Modeling crowd dynamics by the mean-field limit approach,, Mathematical and Computer Modelling, 52 (2010), 1506.

[17]

Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity,, SIAM J. Appl. Math., 66 (2005), 286. doi: 10.1137/040612841.

[18]

H. Egger, A class of hybrid mortar finite element methods for interface problems with non-matching meshes,, Technical Report AICES-2009-2, (): 2009.

[19]

, V. for linear stability,, Available from: \url{http://www.jfpietschmann.eu/crowdmotion}., ().

[20]

M. Fukui and Y. Ishibashi, Self-organized phase transitions in CA-models for pedestrians,, J. Phys. Soc. Japan, 8 (1999), 2861. doi: 10.1143/JPSJ.68.2861.

[21]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits,, J. Statist. Phys., 87 (1997), 37. doi: 10.1007/BF02181479.

[22]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Monographs and Studies in Mathematics, 24,, Pitman (Advanced Publishing Program), (1985).

[23]

D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Mod. Phys., 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067.

[24]

D. Helbing and P. Molnar, Social force model for pedestrian dynamics,, Physical Review E, 51 (1995). doi: 10.1103/PhysRevE.51.4282.

[25]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3.

[26]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103.

[27]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II,, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51.

[28]

R. L. Hughes, A continuum theory for the flow of pedestrians,, Transportation Research Part B: Methodological, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7.

[29]

E. Keller and L. Segel, Initiation of slide mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5.

[30]

A. Kirchner, "Modellierung und Statistische Physik Biologischer und Sozialer Systeme,", Ph.D Thesis, (2002).

[31]

A. Kirchner and A. Schadschneider, Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics,, Physica A: Statistical Mechanics and its Applications, 312 (2002), 260.

[32]

J.-M. Lasry and P.-L. Lions, Mean field games,, Jpn. J. Math., 2 (2007), 229. doi: 10.1007/s11537-007-0657-8.

[33]

P. A. Markowich, "The Stationary Semiconductor Device Equations,", Computational Microelectronics, (1986).

[34]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990). doi: 10.1007/978-3-7091-6961-2.

[35]

M. Matsumoto and T. Nishimura, Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator,, ACM Trans. Model. Comput. Simul., 8 (1998), 3. doi: 10.1145/272991.272995.

[36]

B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of the gradient-flow type,, Math. Models Methods Appl. Sci., 20 (2010), 1787. doi: 10.1142/S0218202510004799.

[37]

M. Muramatsu and T. Nagatani, Jamming transition in two-dimensional pedestrian traffic,, Physica A, 275 (2000), 281. doi: 10.1016/S0378-4371(99)00447-1.

[38]

J. R. Nofsinger and R. W. Sias, Herding and feedback trading by institutional and individual investors,, The Journal of Finance, 54 (1999), 2263. doi: 10.1111/0022-1082.00188.

[39]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501.

[40]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow,, Arch. Ration. Mech. Anal., 199 (2011), 707. doi: 10.1007/s00205-010-0366-y.

[41]

R. M. Raafat, N. Chater and C. Frith, Herding in humans,, Trends in Cognitive Sciences, 13 (2009), 420. doi: 10.1016/j.tics.2009.08.002.

[42]

A. Schadschneider, W. Klingsch, H. Kluepfel, T. Kretz, C. Rogsch and A. Seyfried, Evacuation Dynamics: Empirical Results, Modeling and Applications, in "Encyclopedia of Complexity and System Science" (ed. R. A. Meyers), Vol. 3, pp. 3142,, Springer, (2009).

[43]

O. Schenk and K. Gärtner, Solving unsymmetric sparse systems of linear equations with pardiso,, Journal of Future Generation Computer Systems, 20 (2004), 475. doi: 10.1016/j.future.2003.07.011.

[44]

O. Schenk and K. Gärtner, On fast factorization pivoting methods for sparse symmetric indefinite systems,, Elec. Trans. Numer. Anal, 23 (2006), 158.

[45]

J. Schöberl, Netgen an advancing front 2d/3d-mesh generator based on abstract rules,, Computing and Visualization in Science, 1 (1997), 41. doi: 10.1007/s007910050004.

[46]

M. Simpson, K. Landman and B. Hughes, Diffusing populations: Ghosts or folks?,, Australasian Journal of Engineering Education, 15 (2009), 59.

[47]

A. Sopasakis and M. A. Katsoulakis, Stochastic modeling and simulation of traffic flow: Asymmetric single exclusion process with Arrhenius look-ahead dynamics,, SIAM Journal on Applied Mathematics, 66 (2006), 921. doi: 10.1137/040617790.

[48]

, V. visualization environment,, Available from: \url{http://www.llnl.gov/visit/home.html}., ().

[49]

U. Weidmann, "Transporttechnik der Fussgänger-Transporttechnische Eigenschaften des Fussgängerverkehrs (Literaturstudie),", in German, 90 (1993).

[50]

D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding,, Nonlinear Analysis, 59 (2004), 1293.

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