December  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. Google Scholar

[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. Google Scholar

[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 (). Google Scholar

[4]

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

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[9]

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

[10]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

[18]

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

[19]

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

[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. Google Scholar

[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. Google Scholar

[22]

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

[23]

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

[24]

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

[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. Google Scholar

[26]

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

[27]

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

[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. Google Scholar

[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. Google Scholar

[30]

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

[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. Google Scholar

[32]

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

[33]

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

[34]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39]

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

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[44]

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

[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. Google Scholar

[46]

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

[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. Google Scholar

[48]

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

[49]

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

[50]

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

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. Google Scholar

[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. Google Scholar

[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 (). Google Scholar

[4]

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

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[9]

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

[10]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

[18]

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

[19]

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

[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. Google Scholar

[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. Google Scholar

[22]

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

[23]

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

[24]

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

[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. Google Scholar

[26]

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

[27]

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

[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. Google Scholar

[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. Google Scholar

[30]

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

[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. Google Scholar

[32]

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

[33]

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

[34]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39]

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

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[44]

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

[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. Google Scholar

[46]

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

[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. Google Scholar

[48]

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

[49]

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

[50]

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

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