Journal of Statistical Physics

, Volume 152, Issue 6, pp 1033–1068 | Cite as

A Hierarchy of Heuristic-Based Models of Crowd Dynamics

  • P. Degond
  • C. Appert-Rolland
  • M. Moussaïd
  • J. Pettré
  • G. Theraulaz


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.


Pedestrian dynamics Behavioral heuristics Rational agents Individual-based models Kinetic model Fluid model Game theory Closure relation Monokinetic von Mises-Fisher distribution Nash equilibrium 



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).


  1. 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. 2.
    Aoki, I.: A simulation study on the schooling mechanism in fish. Bull. Jpn. Soc. Sci. Fish. 48, 1081–1088 (1982) zbMATHCrossRefGoogle Scholar
  3. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Aumann, R.: Existence of competitive equilibria in markets with a continuum of traders. Econometrica 32, 39–50 (1964) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Aw, A., Rascle, M.: Resurrection of second order models of traffic flow. SIAM J. Appl. Math. 60, 916–938 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 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) ADSCrossRefGoogle Scholar
  7. 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. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 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) CrossRefGoogle Scholar
  10. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Blanchet, A.: Variational methods applied to biology and economics. Dissertation for the Habilitation, University Toulouse 1 Capitole (December 2012) Google Scholar
  13. 13.
    Bolley, F., Cañizo, J.A., Carrillo, J.A.: Mean-field limit for the stochastic Vicsek model. Appl. Math. Lett. 25, 339–343 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 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. 15.
    Bouchut, F., James, F.: Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness. Commun. Partial Differ. Equ. 24, 2173–2189 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Chertock, A., Kurganov, A., Polizzi, A., Timofeyev, I.: Pedestrian flow models with slowdown interactions. Math. Models Methods Appl. Sci. (to appear) Google Scholar
  19. 19.
    Colombo, R.M., Rosini, M.D.: Pedestrian flows and nonclassical shocks. Math. Methods Appl. Sci. 28, 1553–1567 (2005) MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Coscia, V., Canavesio, C.: First-order macroscopic modelling of human crowd dynamics. Math. Models Methods Appl. Sci. 18 Suppl, 1217–1247 (2008) MathSciNetCrossRefGoogle Scholar
  21. 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) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Cristiani, E., Piccoli, B., Tosin, A.: Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model. Simul. 9, 155–182 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 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) CrossRefGoogle Scholar
  24. 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. 25.
    Degond, P., Appert-Rolland, C., Pettré, J., Theraulaz, G.: Macroscopic pedestrian models based on synthetic vision (submitted) Google Scholar
  26. 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) ADSCrossRefGoogle Scholar
  27. 27.
    Degond, P., Hua, J.: Self-organized hydrodynamics with congestion and path formation in crowds. J. Comput. Phys. 237, 299–319 (2013) ADSCrossRefGoogle Scholar
  28. 28.
    Degond, P., Hua, J., Navoret, L.: Numerical simulations of the Euler system with congestion constraint. J. Comput. Phys. 230, 8057–8088 (2011) MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 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. 30.
    Degond, P., Motsch, S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci. 18 Suppl, 1193–1215 (2008) MathSciNetCrossRefGoogle Scholar
  31. 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) zbMATHCrossRefGoogle Scholar
  32. 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) CrossRefGoogle Scholar
  33. 33.
    Green, E.J., Porter, R.H.: Noncooperative collusion under imperfect price information. Econometrica 52, 87–100 (1984) zbMATHCrossRefGoogle Scholar
  34. 34.
    Grégoire, G., Chaté, H.: Onset of collective and cohesive motion. Phys. Rev. Lett. 92, 025702 (2004) ADSCrossRefGoogle Scholar
  35. 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. 36.
    Helbing, D.: A mathematical model for the behavior of pedestrians. Behav. Sci. 36, 298–310 (1991) CrossRefGoogle Scholar
  37. 37.
    Helbing, D.: A fluid dynamic model for the movement of pedestrians. Complex Syst. 6, 391–415 (1992) MathSciNetzbMATHGoogle Scholar
  38. 38.
    Helbing, D., Molnàr, P.: Social force model for pedestrian dynamics. Phys. Rev. E 51, 4282–4286 (1995) ADSCrossRefGoogle Scholar
  39. 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. 40.
    Henderson, L.F.: On the fluid mechanics of human crowd motion. Transp. Res. 8, 509–515 (1974) CrossRefGoogle Scholar
  41. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 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) CrossRefGoogle Scholar
  43. 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) CrossRefGoogle Scholar
  44. 44.
    Hughes, R.L.: A continuum theory for the flow of pedestrians. Transp. Res., Part B, Methodol. 36, 507–535 (2002) CrossRefGoogle Scholar
  45. 45.
    Hughes, R.L.: The flow of human crowds. Annu. Rev. Fluid Mech. 35, 169–182 (2003) ADSCrossRefGoogle Scholar
  46. 46.
    Hsu, E.P.: Stochastic Analysis on Manifolds. Graduate Series in Mathematics, vol. 38. American Mathematical Society, Providence (2002) zbMATHGoogle Scholar
  47. 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) ADSCrossRefGoogle Scholar
  48. 48.
    Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2, 229–260 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  49. 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) CrossRefGoogle Scholar
  50. 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. 51.
    Maury, B., Roudneff-Chupin, A., Santambrogio, F., Venel, J.: Handling congestion in crowd motion models. Netw. Heterog. Media 6, 485–519 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  52. 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) Google Scholar
  53. 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) CrossRefGoogle Scholar
  54. 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) ADSCrossRefGoogle Scholar
  55. 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) CrossRefGoogle Scholar
  56. 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. 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. 58.
    Paris, S., Pettré, J., Donikian, S.: Pedestrian reactive navigation for crowd simulation: a predictive approach. Eurographics 26, 665–674 (2007) Google Scholar
  59. 59.
    Payne, H.J.: Models of Freeway Traffic and Control. Simulation Councils Inc., La Jolla (1971) Google Scholar
  60. 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) CrossRefGoogle Scholar
  61. 61.
    Piccoli, B., Tosin, A.: Pedestrian flows in bounded domains with obstacles. Contin. Mech. Thermodyn. 21, 85–117 (2009) MathSciNetADSzbMATHCrossRefGoogle Scholar
  62. 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. 63.
    Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory 2, 65–67 (1973) zbMATHCrossRefGoogle Scholar
  64. 64.
    Schmeidler, D.: Equilibrium points of nonatomic games. J. Stat. Phys. 7, 295–300 (1973) MathSciNetADSzbMATHCrossRefGoogle Scholar
  65. 65.
    van den Berg, J., Overmars, H.: Planning time-minimal safe paths amidst unpredictably moving obstacles. Int. J. Robot. Res. 27, 1274–1294 (2008) CrossRefGoogle Scholar
  66. 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) ADSCrossRefGoogle Scholar
  67. 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) CrossRefGoogle Scholar
  68. 68.
    Watson, G.S.: Distributions on the circle and sphere. J. Appl. Probab. 19, 265–280 (1982) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • P. Degond
    • 1
    • 2
  • C. Appert-Rolland
    • 3
    • 4
  • M. Moussaïd
    • 5
  • J. Pettré
    • 6
  • G. Theraulaz
    • 7
    • 8
  1. 1.UPS, INSA, UT1, UTM, Institut de Mathématiques de ToulouseUniversité de ToulouseToulouseFrance
  2. 2.CNRSInstitut de Mathématiques de Toulouse UMR 5219ToulouseFrance
  3. 3.Laboratoire de Physique ThéoriqueUniversité Paris SudOrsay cedexFrance
  4. 4.CNRS, UMR 8627Laboratoire de Physique ThéoriqueOrsayFrance
  5. 5.Center for Adaptive Behavior and CognitionMax Planck Institute for Human DevelopmentBerlinGermany
  6. 6.INRIA Rennes – Bretagne AtlantiqueRennesFrance
  7. 7.Centre de Recherches sur la Cognition Animale, UMR-CNRS 5169Université Paul SabatierToulouse cedex 9France
  8. 8.CNRSCentre de Recherches sur la Cognition AnimaleToulouseFrance

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