Skip to main content
SpringerLink
Log in

A Hierarchy of Heuristic-Based Models of Crowd Dynamics

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We derive a hierarchy of kinetic and macroscopic models from a noisy variant of the heuristic behavioral Individual-Based Model of Ngai et al. (Disaster Med. Public Health Prep. 3:191–195, 2009) where pedestrians are supposed to have constant speeds. This IBM supposes that pedestrians seek the best compromise between navigation towards their target and collisions avoidance. We first propose a kinetic model for the probability distribution function of pedestrians. Then, we derive fluid models and propose three different closure relations. The first two closures assume that the velocity distribution function is either a Dirac delta or a von Mises-Fisher distribution respectively. The third closure results from a hydrodynamic limit associated to a Local Thermodynamical Equilibrium. We develop an analogy between this equilibrium and Nash equilibria in a game theoretic framework. In each case, we discuss the features of the models and their suitability for practical use.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Al-nasur, S., Kashroo, P.: A microscopic-to-macroscopic crowd dynamic model. In: Proceedings of the IEEE Intelligent Transportation Systems Conference, pp. 606–611 (2006)

    Google Scholar 

  2. Aoki, I.: A simulation study on the schooling mechanism in fish. Bull. Jpn. Soc. Sci. Fish. 48, 1081–1088 (1982)

    Article  MATH  Google Scholar 

  3. Appert-Rolland, C., Degond, P., Motsch, S.: Two-way multi-lane traffic model for pedestrians in corridors. Netw. Heterog. Media 6, 351–381 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aumann, R.: Existence of competitive equilibria in markets with a continuum of traders. Econometrica 32, 39–50 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aw, A., Rascle, M.: Resurrection of second order models of traffic flow. SIAM J. Appl. Math. 60, 916–938 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ballerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani, E., Giardina, I., Lecomte, V., Orlandi, A., Parisi, G., Procaccini, A., Viale, M., Zdravkovic, V.: Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proc. Natl. Acad. Sci. USA 105, 1232–1237 (2008)

    Article  ADS  Google Scholar 

  7. Barbaro, A., Degond, P.: Phase transition and diffusion among socially interacting self-propelled agent. Discrete Contin. Dyn. Syst., Ser. B (to appear). arXiv:1207.1926

  8. Bellomo, N., Bellouquid, A.: On the modeling of crowd dynamics: looking at the beautiful shapes of swarms. Netw. Heterog. Media 6, 383–399 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bellomo, N., Dogbé, C.: On the modelling crowd dynamics from scaling to hyperbolic macroscopic models. Math. Models Methods Appl. Sci. 18 Suppl, 1317–1345 (2008)

    Article  Google Scholar 

  10. Bellomo, N., Dogbé, C.: On the modeling of traffic and crowds: a survey of models, speculations and perspectives. SIAM Rev. 53, 409–463 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Berres, S., Ruiz-Baier, R., Schwandt, H., Tory, E.M.: An adaptive finite-volume method for a model of two-phase pedestrian flow. Netw. Heterog. Media 6, 401–423 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Blanchet, A.: Variational methods applied to biology and economics. Dissertation for the Habilitation, University Toulouse 1 Capitole (December 2012)

  13. Bolley, F., Cañizo, J.A., Carrillo, J.A.: Mean-field limit for the stochastic Vicsek model. Appl. Math. Lett. 25, 339–343 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bouchut, F.: On zero pressure gas dynamics. In: Perthame, B. (ed.) Advances in Kinetic Theory and Computing, pp. 171–190. World Scientific, Singapore (1994)

    Google Scholar 

  15. Bouchut, F., James, F.: Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness. Commun. Partial Differ. Equ. 24, 2173–2189 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Burger, M., Markowich, P., Pietschmann, J.-F.: Continuous limit of a crowd motion and herding model: analysis and numerical simulations. Kinet. Relat. Models 4, 1025–1047 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Carlen, E., Degond, P., Wennberg, B.: Kinetic limits for pair-interaction driven master equations and biological swarm models. Math. Models Methods Appl. Sci. 23, 1339–1376 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chertock, A., Kurganov, A., Polizzi, A., Timofeyev, I.: Pedestrian flow models with slowdown interactions. Math. Models Methods Appl. Sci. (to appear)

  19. Colombo, R.M., Rosini, M.D.: Pedestrian flows and nonclassical shocks. Math. Methods Appl. Sci. 28, 1553–1567 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Coscia, V., Canavesio, C.: First-order macroscopic modelling of human crowd dynamics. Math. Models Methods Appl. Sci. 18 Suppl, 1217–1247 (2008)

    Article  MathSciNet  Google Scholar 

  21. Couzin, I.D., Krause, J., James, R., Ruxton, G.D., Franks, N.R.: Collective memory and spatial sorting in animal groups. J. Theor. Biol. 218, 1–11 (2002)

    Article  MathSciNet  Google Scholar 

  22. Cristiani, E., Piccoli, B., Tosin, A.: Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model. Simul. 9, 155–182 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cutting, J.E., Vishton, P.M., Braren, P.A.: How we avoid collisions with stationary and moving objects. Psychol. Rev. 102, 627–651 (1995)

    Article  Google Scholar 

  24. Degond, P.: Macroscopic limits of the Boltzmann equation: a review. In: Degond, P., et al. (eds.) Modeling and Computational Methods for Kinetic Equations, pp. 3–57. Birkhaüser, Basel (2003)

    Google Scholar 

  25. Degond, P., Appert-Rolland, C., Pettré, J., Theraulaz, G.: Macroscopic pedestrian models based on synthetic vision (submitted)

  26. Degond, P., Frouvelle, A., Liu, J.-G.: Macroscopic limits and phase transition in a system of self-propelled particles. J. Nonlinear Sci. 23, 427–456 (2013)

    Article  ADS  Google Scholar 

  27. Degond, P., Hua, J.: Self-organized hydrodynamics with congestion and path formation in crowds. J. Comput. Phys. 237, 299–319 (2013)

    Article  ADS  Google Scholar 

  28. Degond, P., Hua, J., Navoret, L.: Numerical simulations of the Euler system with congestion constraint. J. Comput. Phys. 230, 8057–8088 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Degond, P., Liu, J.-G., Ringhofer, C.: A Nash equilibrium macroscopic closure for kinetic models coupled with mean-field games. arXiv:1212.6130

  30. Degond, P., Motsch, S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci. 18 Suppl, 1193–1215 (2008)

    Article  MathSciNet  Google Scholar 

  31. Di Francesco, M., Markowich, P.A., Pietschmann, J.-F., Wolfram, M.-T.: On the Hughes’ model for pedestrian flow: the one-dimensional case. J. Differ. Equ. 250, 1334–1362 (2011)

    Article  MATH  Google Scholar 

  32. Gautrais, J., Ginelli, F., Fournier, R., Blanco, S., Soria, M., Chaté, H., Theraulaz, G.: Deciphering interactions in moving animal groups. PLoS Comput. Biol. 8, e1002678 (2012)

    Article  Google Scholar 

  33. Green, E.J., Porter, R.H.: Noncooperative collusion under imperfect price information. Econometrica 52, 87–100 (1984)

    Article  MATH  Google Scholar 

  34. Grégoire, G., Chaté, H.: Onset of collective and cohesive motion. Phys. Rev. Lett. 92, 025702 (2004)

    Article  ADS  Google Scholar 

  35. Guy, S.J., Chhugani, J., Kim, C., Satish, N., Lin, M.C., Manocha, D., Dubey, P.: Clearpath: highly parallel collision avoidance for multi-agent simulation. In: ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 177–187 (2009)

    Google Scholar 

  36. Helbing, D.: A mathematical model for the behavior of pedestrians. Behav. Sci. 36, 298–310 (1991)

    Article  Google Scholar 

  37. Helbing, D.: A fluid dynamic model for the movement of pedestrians. Complex Syst. 6, 391–415 (1992)

    MathSciNet  MATH  Google Scholar 

  38. Helbing, D., Molnàr, P.: Social force model for pedestrian dynamics. Phys. Rev. E 51, 4282–4286 (1995)

    Article  ADS  Google Scholar 

  39. Helbing, D., Molnàr, P.: Self-organization phenomena in pedestrian crowds. In: Schweitzer, F. (ed.) Self-Organization of Complex Structures: From Individual to Collective Dynamics, pp. 569–577. Gordon and Breach, London (1997)

    Google Scholar 

  40. Henderson, L.F.: On the fluid mechanics of human crowd motion. Transp. Res. 8, 509–515 (1974)

    Article  Google Scholar 

  41. Hoogendoorn, S., Bovy, P.H.L.: Simulation of pedestrian flows by optimal control and differential games. Optim. Control Appl. Methods 24, 153–172 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. Huang, W.H., Fajen, B.R., Fink, J.R., Warren, W.H.: Visual navigation and obstacle avoidance using a steering potential function. Robot. Auton. Syst. 54, 288–299 (2006)

    Article  Google Scholar 

  43. Huang, L., Wong, S.C., Zhang, M., Shu, C.-W., Lam, W.H.K.: Revisiting Hughes’ dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm. Transp. Res., Part B, Methodol. 43, 127–141 (2009)

    Article  Google Scholar 

  44. Hughes, R.L.: A continuum theory for the flow of pedestrians. Transp. Res., Part B, Methodol. 36, 507–535 (2002)

    Article  Google Scholar 

  45. Hughes, R.L.: The flow of human crowds. Annu. Rev. Fluid Mech. 35, 169–182 (2003)

    Article  ADS  Google Scholar 

  46. Hsu, E.P.: Stochastic Analysis on Manifolds. Graduate Series in Mathematics, vol. 38. American Mathematical Society, Providence (2002)

    MATH  Google Scholar 

  47. Jiang, Y.-q., Zhang, P., Wong, S.C., Liu, R.-x.: A higher-order macroscopic model for pedestrian flows. Physica A 389, 4623–4635 (2010)

    Article  ADS  Google Scholar 

  48. Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2, 229–260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  49. Lemercier, S., Jelic, A., Kulpa, R., Hua, J., Fehrenbach, J., Degond, P., Appert-Rolland, C., Donikian, S., Pettré, J.: Realistic following behaviors for crowd simulation. Comput. Graph. Forum 31, 489–498 (2012)

    Article  Google Scholar 

  50. Lighthill, M.J., Whitham, J.B.: On kinematic waves, I: flow movement in long rivers; II: a theory of traffic flow on long crowded roads. Proc. R. Soc. A 229, 1749–1766 (1955)

    Google Scholar 

  51. Maury, B., Roudneff-Chupin, A., Santambrogio, F., Venel, J.: Handling congestion in crowd motion models. Netw. Heterog. Media 6, 485–519 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  52. Motsch, S., Moussaïd, M., Guillot, E.G., Lemercier, S., Pettré, J., Theraulaz, G., Appert-Rolland, C., Degond, P.: Dynamics of cluster formation and traffic efficiency in pedestrian crowds (submitted)

  53. Moussaïd, M., Guillot, E.G., Moreau, M., Fehrenbach, J., Chabiron, O., Lemercier, S., Pettré, J., Appert-Rolland, C., Degond, P., Theraulaz, G.: Traffic instabilities in self-organized pedestrian crowds. PLoS Comput. Biol. 8, e1002442 (2012)

    Article  Google Scholar 

  54. Moussaïd, M., Helbing, D., Theraulaz, G.: How simple rules determine pedestrian behavior and crowd disasters. Proc. Natl. Acad. Sci. USA 108, 6884–6888 (2011)

    Article  ADS  Google Scholar 

  55. Ngai, K.M., Burkle, F.M. Jr., Hsu, A., Hsu, E.B.: Human stampedes: a systematic review of historical and peer-reviewed sources. Disaster Med. Public Health Prep. 3, 191–195 (2009)

    Article  Google Scholar 

  56. Nishinari, K., Kirchner, A., Namazi, A., Schadschneider, A.: Extended floor field CA model for evacuation dynamics. IEICE Transp. Inf. Syst. E 87-D, 726–732 (2004)

    Google Scholar 

  57. Ondrej, J., Pettré, J., Olivier, A.H., Donikian, S.: A synthetic-vision based steering approach for crowd simulation. In: SIGGRAPH’10 (2010)

    Google Scholar 

  58. Paris, S., Pettré, J., Donikian, S.: Pedestrian reactive navigation for crowd simulation: a predictive approach. Eurographics 26, 665–674 (2007)

    Google Scholar 

  59. Payne, H.J.: Models of Freeway Traffic and Control. Simulation Councils Inc., La Jolla (1971)

    Google Scholar 

  60. Pettré, J., Ondřej, J., Olivier, A.-H., Cretual, A., Donikian, S.: Experiment-based modeling, simulation and validation of interactions between virtual walkers. In: SCA ’09: Proceedings of the 2009 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 189–198 (2009)

    Chapter  Google Scholar 

  61. Piccoli, B., Tosin, A.: Pedestrian flows in bounded domains with obstacles. Contin. Mech. Thermodyn. 21, 85–117 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  62. Reynolds, C.W.: Steering behaviors for autonomous characters. In: Proceedings of Game Developers Conference, San Jose, California, pp. 763–782 (1999)

    Google Scholar 

  63. Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory 2, 65–67 (1973)

    Article  MATH  Google Scholar 

  64. Schmeidler, D.: Equilibrium points of nonatomic games. J. Stat. Phys. 7, 295–300 (1973)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  65. van den Berg, J., Overmars, H.: Planning time-minimal safe paths amidst unpredictably moving obstacles. Int. J. Robot. Res. 27, 1274–1294 (2008)

    Article  Google Scholar 

  66. Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229 (1995)

    Article  ADS  Google Scholar 

  67. Warren, W.H., Fajen, B.R.: From optic flow to laws of control. In: Vaina, L.M., Beardsley, S.A., Rushton, S. (eds.) Optic Flow and Beyond, pp. 307–337. Kluwer, Dordrecht (2004)

    Chapter  Google Scholar 

  68. Watson, G.S.: Distributions on the circle and sphere. J. Appl. Probab. 19, 265–280 (1982)

    Article  Google Scholar 

Download references

Acknowledgements

This work has been supported by the French ‘Agence Nationale pour la Recherche (ANR)’ in the frame of the contracts ‘Pedigree’ (ANR-08-SYSC-015-01) and ‘CBDif-Fr’ (ANR-08-BLAN-0333-01).

Author information

Authors and Affiliations

  1. UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, Université de Toulouse, 31062, Toulouse, France

    P. Degond

  2. CNRS, Institut de Mathématiques de Toulouse UMR 5219, 31062, Toulouse, France

    P. Degond

  3. Laboratoire de Physique Théorique, Université Paris Sud, bâtiment 210, 91405, Orsay cedex, France

    C. Appert-Rolland

  4. CNRS, UMR 8627, Laboratoire de Physique Théorique, 91405, Orsay, France

    C. Appert-Rolland

  5. Center for Adaptive Behavior and Cognition, Max Planck Institute for Human Development, Lentzeallee 94, 14195, Berlin, Germany

    M. Moussaïd

  6. INRIA Rennes – Bretagne Atlantique, Campus de Beaulieu, 35042, Rennes, France

    J. Pettré

  7. Centre de Recherches sur la Cognition Animale, UMR-CNRS 5169, Université Paul Sabatier, Bât 4R3, 118 Route de Narbonne, 31062, Toulouse cedex 9, France

    G. Theraulaz

  8. CNRS, Centre de Recherches sur la Cognition Animale, 31062, Toulouse, France

    G. Theraulaz

Authors
  1. P. Degond
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Appert-Rolland
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Moussaïd
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. Pettré
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. G. Theraulaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Degond.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Degond, P., Appert-Rolland, C., Moussaïd, M. et al. A Hierarchy of Heuristic-Based Models of Crowd Dynamics. J Stat Phys 152, 1033–1068 (2013). https://doi.org/10.1007/s10955-013-0805-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-013-0805-x

Keywords

Search

Navigation