Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2015, Article ID 872807, 7 pages
http://dx.doi.org/10.1155/2015/872807
Research Article

A Mathematical Model of the Formation of Lanes in Crowds of Pedestrians Moving in Opposite Directions

School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA

Received 5 May 2015; Accepted 9 June 2015

Academic Editor: Tetsuji Tokihiro

Copyright © 2015 Guillermo H. Goldsztein. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. Camazine, Self-Organization in Biological Systems, Princeton University Press, Princeton, NJ, USA, 2003.
  2. D. J. T. Sumpter, Collective Animal Behavior, Princeton University Press, 2010.
  3. T. Vicsek and A. Zafeiris, “Collective motion,” Physics Reports, vol. 517, no. 3-4, pp. 71–140, 2012. View at Publisher · View at Google Scholar · View at Scopus
  4. T. Kretz, M. Wölki, and M. Schreckenberg, “Characterizing correlations of flow oscillations at bottlenecks,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2006, no. 2, article P02005, 2006. View at Google Scholar
  5. D. Helbing, J. Keltsch, and P. Molnár, “Modelling the evolution of human trail systems,” Nature, vol. 388, no. 6637, pp. 47–50, 1997. View at Publisher · View at Google Scholar · View at Scopus
  6. M. Moussaïd, N. Perozo, S. Garnier, D. Helbing, and G. Theraulaz, “The walking behaviour of pedestrian social groups and its impact on crowd dynamics,” PLoS ONE, vol. 5, no. 4, Article ID e10047, 2010. View at Publisher · View at Google Scholar · View at Scopus
  7. L. Bruno and F. Venuti, “Crowd-structure interaction in footbridges: modelling, application to a real case-study and sensitivity analyses,” Journal of Sound and Vibration, vol. 323, no. 1-2, pp. 475–493, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. L. F. Henderson, “The statistics of crowd fluids,” Nature, vol. 229, no. 5284, pp. 381–383, 1971. View at Publisher · View at Google Scholar · View at Scopus
  9. L. F. Henderson, “On the fluid mechanics of human crowd motion,” Transportation Research, vol. 8, no. 6, pp. 509–515, 1974. View at Publisher · View at Google Scholar · View at Scopus
  10. D. Helbing, “Traffic and related self-driven many-particle systems,” Reviews of Modern Physics, vol. 73, no. 4, pp. 1067–1141, 2001. View at Publisher · View at Google Scholar · View at Scopus
  11. M. Moussaïd, E. G. Guillot, M. Moreau et al., “Traffic instabilities in self-organized pedestrian crowds,” PLoS Computational Biology, vol. 8, no. 3, Article ID e1002442, 2012. View at Publisher · View at Google Scholar · View at Scopus
  12. S. Hoogendoorn and W. Daamen, “Self-organization in pedestrian flow,” in Traffic and Granular Flow '03, pp. 373–382, Springer, Berlin, Germany, 2005. View at Publisher · View at Google Scholar
  13. M. Moussaïd, D. Helbing, S. Garnier, A. Johansson, M. Combe, and G. Theraulaz, “Experimental study of the behavioural mechanisms underlying self-organization in human crowds,” Proceedings of the Royal Society B: Biological Sciences, vol. 276, no. 1668, pp. 2755–2762, 2009. View at Publisher · View at Google Scholar · View at Scopus
  14. J. Ma, W.-G. Song, J. Zhang, S.-M. Lo, and G.-X. Liao, “k-Nearest-Neighbor interaction induced self-organized pedestrian counter flow,” Physica A: Statistical Mechanics and its Applications, vol. 389, no. 10, pp. 2101–2117, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. F. Weifeng, Y. Lizhong, and F. Weicheng, “Simulation of bi-direction pedestrian movement using a cellular automata model,” Physica A: Statistical Mechanics and its Applications, vol. 321, no. 3-4, pp. 633–640, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. V. J. Blue and J. L. Adler, “Cellular automata microsimulation for modeling bi-directional pedestrian walkways,” Transportation Research Part B: Methodological, vol. 35, no. 3, pp. 293–312, 2001. View at Publisher · View at Google Scholar · View at Scopus
  17. M. Muramatsu, T. Irie, and T. Nagatani, “Jamming transition in pedestrian counter flow,” Physica A: Statistical Mechanics and its Applications, vol. 267, no. 3, pp. 487–498, 1999. View at Publisher · View at Google Scholar · View at Scopus
  18. H. Kuang, X.-L. Li, Y.-F. Wei, T. Song, and S.-Q. Dai, “Effect of following strength on pedestrian counter flow,” Chinese Physics B, vol. 19, no. 7, Article ID 070517, 2010. View at Publisher · View at Google Scholar · View at Scopus
  19. P. G. Gipps and B. Marksjö, “A micro-simulation model for pedestrian flows,” Mathematics and Computers in Simulation, vol. 27, no. 2-3, pp. 95–105, 1985. View at Publisher · View at Google Scholar · View at Scopus
  20. D. Helbing and P. Molnár, “Social force model for pedestrian dynamics,” Physical Review E, vol. 51, no. 5, pp. 4282–4286, 1995. View at Publisher · View at Google Scholar · View at Scopus
  21. D. Helbing, “A mathematical model for the behavior of pedestrians,” Behavioral Science, vol. 36, no. 4, pp. 298–310, 1991. View at Publisher · View at Google Scholar
  22. L. Jian, Y. Lizhong, and Z. Daoliang, “Simulation of bi-direction pedestrian movement in corridor,” Physica A: Statistical Mechanics and its Applications, vol. 354, no. 1–4, pp. 619–628, 2005. View at Publisher · View at Google Scholar · View at Scopus
  23. A. Treuille, S. Cooper, and Z. Popović, “Continuum crowds,” ACM Transactions on Graphics, vol. 25, no. 3, pp. 1160–1168, 2006. View at Publisher · View at Google Scholar
  24. D. Helbing, L. Buzna, A. Johansson, and T. Werner, “Self-organized pedestrian crowd dynamics: experiments, simulations, and design solutions,” Transportation Science, vol. 39, no. 1, pp. 1–24, 2005. View at Publisher · View at Google Scholar · View at Scopus
  25. P. Degond and J. Hua, “Self-organized hydrodynamics with congestion and path formation in crowds,” Journal of Computational Physics, vol. 237, pp. 299–319, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. T. Vicsek, A. Czirók, I. J. Farkas, and D. Helbing, “Application of statistical mechanics to collective motion in biology,” Physica A: Statistical Mechanics and its Applications, vol. 274, no. 1, pp. 182–189, 1999. View at Publisher · View at Google Scholar · View at Scopus
  27. P. Degond, C. Appert-Rolland, M. Moussaïd, J. Pettré, and G. Theraulaz, “A hierarchy of heuristic-based models of crowd dynamics,” Journal of Statistical Physics, vol. 152, no. 6, pp. 1033–1068, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. D. Helbing, “A fluid dynamic model for the movement of pedestrians,” http://arxiv.org/abs/cond-mat/9805213.
  29. N. Bellomo, C. Bianca, and M. Delitala, “Complexity analysis and mathematical tools towards the modelling of living systems,” Physics of Life Reviews, vol. 6, no. 3, pp. 144–175, 2009. View at Publisher · View at Google Scholar · View at Scopus
  30. C. Appert-Rolland, P. Degond, and S. Motsch, “Two-way multi-lane traffic model for pedestrians in corridors,” Networks and Heterogeneous Media, vol. 6, no. 3, pp. 351–381, 2011. View at Publisher · View at Google Scholar
  31. R. L. Hughes, “The flow of human crowds,” Annual Review of Fluid Mechanics, vol. 35, no. 1, pp. 169–182, 2003. View at Google Scholar
  32. Y.-Q. Jiang, P. Zhang, S. C. Wong, and R.-X. Liu, “A higher-order macroscopic model for pedestrian flows,” Physica A: Statistical Mechanics and Its Applications, vol. 389, no. 21, pp. 4623–4635, 2010. View at Publisher · View at Google Scholar · View at Scopus
  33. B. Piccoli and A. Tosin, “Time-evolving measures and macroscopic modeling of pedestrian flow,” Archive for Rational Mechanics and Analysis, vol. 199, no. 3, pp. 707–738, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  34. N. Bellomo and C. Dogbe, “On the modelling crowd dynamics from scaling to hyperbolic macroscopic models,” Mathematical Models and Methods in Applied Sciences, vol. 18, supplement 1, pp. 1317–1345, 2008. View at Google Scholar
  35. E. Cristiani, B. Piccoli, and A. Tosin, “Multiscale modeling of granular flows with application to crowd dynamics,” Multiscale Modeling & Simulation, vol. 9, no. 1, pp. 155–182, 2011. View at Publisher · View at Google Scholar · View at Scopus
  36. A. Chertock, A. Kurganov, A. Polizzi, and I. Timofeyev, “Pedestrian flow models with slowdown interactions,” Mathematical Models and Methods in Applied Sciences, vol. 24, no. 2, pp. 249–275, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. M. Di Francesco and M. Rosini, “Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit,” Archive for Rational Mechanics and Analysis, 2015. View at Publisher · View at Google Scholar
  38. G. G. Løvås, “Modeling and simulation of pedestrian traffic flow,” Transportation Research Part B: Methodological, vol. 28, no. 6, pp. 429–443, 1994. View at Publisher · View at Google Scholar · View at Scopus
  39. S. Hoogendoorn and P. H. Bovy, “Simulation of pedestrian flows by optimal control and differential games,” Optimal Control Applications & Methods, vol. 24, no. 3, pp. 153–172, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  40. S. J. Older, “Movement of pedestrians on footways in shopping streets,” Traffic Engineering & Control, vol. 10, pp. 160–163, 1968. View at Google Scholar
  41. S. Milgram and H. Toch, “Collective behavior: crowds and social movements,” in The Handbook of Social Psychology, vol. 4, 1969. View at Google Scholar
  42. T. Kretz, A. Grünebohm, M. Kaufman, F. Mazur, and M. Schreckenberg, “Experimental study of pedestrian counterflow in a corridor,” Journal of Statistical Mechanics: Theory and Experiment, no. 10, Article ID P10001, 2006. View at Publisher · View at Google Scholar · View at Scopus
  43. D. Helbing and T. Vicsek, “Optimal self-organization,” New Journal of Physics, vol. 1, no. 1, article 13, 1999. View at Google Scholar
  44. G. F. Lawler, Random Walk and the Heat Equation, vol. 55, American Mathematical Society, 2010. View at Publisher · View at Google Scholar · View at MathSciNet