Modelling Interactions Between Active and Passive Agents Moving Through Heterogeneous Environments

  • Matteo Colangeli
  • Adrian Muntean
  • Omar RichardsonEmail author
  • Thi Kim Thoa Thieu
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


We study the dynamics of interacting agents from two distinct intermixed populations: one population includes active agents that follow a predetermined velocity field, while the second population contains exclusively passive agents, i.e., agents that have no preferred direction of motion. The orientation of their local velocity is affected by repulsive interactions with the neighboring agents and environment. We present two models that allow for a qualitative analysis of these mixed systems. We show that the residence times of this type of systems containing mixed populations is strongly affected by the interplay between these two populations. After showing our modelling and simulation results, we conclude with a couple of mathematical aspects concerning the well-posedness of our models.


Crowd dynamics Lattice gas model Fire and smoke dynamics Particle methods Heterogeneous domains 


02.70.Uu 07.05.Tp 05.06.-k. 

MSC 2010:

65Z05 82C80 91E30. 


  1. 1.
    Adams, R.A., Fournier, J.J.: Sobolev Spaces, vol. 140. Academic Press (2003)Google Scholar
  2. 2.
    Anh, N.T.N., Daniel, Z.J., Du, N.H., Drogoul, A., An, V.D.: A hybrid macro-micro pedestrians evacuation model to speed up simulation in road networks. In: International Conference on Autonomous Agents and Multiagent Systems, pp. 371–383. Springer (2011)Google Scholar
  3. 3.
    Bellomo, N., Clarke, D., Gibelli, L., Townsend, P., Vreugdenhil, B.: Human behaviours in evacuation crowd dynamics: from modelling to big data toward crisis management. Physics of Life Reviews 18, 1–21 (2016)CrossRefGoogle Scholar
  4. 4.
    Bellomo, N., Gibelli, L.: Toward a mathematical theory of behavioral-social dynamics for pedestrian crowds. Mathematical Models and Methods in Applied Sciences 25(13), 2417–2437 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bensoussan, A.: Stochastic Navier-Stokes equations. Acta Applicandae Mathematica 38(3), 267–304 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Billingsley, P., Dudley, R., et al.: Convergence of probability measures. Bulletin of the American Mathematical Society 77(1), 25–27 (1971)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bortz, A., Kalos, M., Lebowitz, J.: A new algorithm for Monte Carlo simulation of Ising spin systems. Journal of Computational Physics 17(1), 10–18 (1975)CrossRefGoogle Scholar
  8. 8.
    Boyer, F., Fabrie, P.: Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, vol. 183. Springer Science & Business Media (2012)Google Scholar
  9. 9.
    Cao, S., Song, W., Liu, X., Mu, N.: Simulation of pedestrian evacuation in a room under fire emergency. Proceeding Engineering 71, 403–409 (2014)CrossRefGoogle Scholar
  10. 10.
    Chu, M.L., Parigi, P., Law, K., Latombe, J.C.: Modeling social behaviors in an evacuation simulator. Computer Animation and Virtual Worlds 25(3–4), 373–382 (2014)CrossRefGoogle Scholar
  11. 11.
    Ciallella, A., Cirillo, E.N.M., Curşeu, P.L., Muntean, A.: Free to move or trapped in your group: Mathematical modeling of information overload and coordination in crowded populations. Mathematical Models and Methods in Applied Sciences M3AS (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cirillo, E.N.M., Colangeli, M.: Stationary uphill currents in locally perturbed zero-range processes. Phys. Rev. E 96, 052,137 (2017)CrossRefGoogle Scholar
  13. 13.
    Cirillo, E.N.M., Colangeli, M., Muntean, A.: Blockage-induced condensation controlled by a local reaction. Phys. Rev. E 94, 042,116 (2016)CrossRefGoogle Scholar
  14. 14.
    Colombo, R.M., Lorenz, T., Pogodaev, N.I.: On the modeling of moving populations through set evolution equations. Discrete & Continuous Dynamical Systems - A 35(1), 73–98 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Corbetta, A., Bruno, L., Muntean, A., Toschi, F.: High statistics measurements of pedestrian dynamics. Transportation Research Procedia 2, 96–104 (2014)CrossRefGoogle Scholar
  16. 16.
    Cristiani, E., Piccoli, B., Tosin, A.: Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Modeling & Simulation 9(1), 155–182 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Transactions on automatic control 52(5), 852–862 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press (2014)Google Scholar
  19. 19.
    De Masi, A., Presutti, E.: Mathematical Methods for Hydrodynamic Limits. Lecture notes in mathematics. Springer-Verlag (1991)CrossRefGoogle Scholar
  20. 20.
    Evans, L.: Partial Differential Equations, vol. 19. American Mathematical Society (2010)Google Scholar
  21. 21.
    Evans, L.: An Introduction to Stochastic Differential Equations, vol. 82. American Mathematical Soc. (2012)Google Scholar
  22. 22.
    Faure, S., Maury, B.: Crowd motion from the granular standpoint. Mathematical Models and Methods in Applied Sciences 25(03), 463–493 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier-Stokes equations. Probability Theory and Related Fields 102(3), 367–391 (1995)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Garcimartín, A., Pastor, J., Ferrer, L., Ramos, J., Martín-Gómez, C., Zuriguel, I.: Flow and clogging of a sheep herd passing through a bottleneck. Physical Review E 91(2), 022,808 (2015)CrossRefGoogle Scholar
  25. 25.
    Helbing, D., Farkas, I., Vicsek, T.: Simulating dynamical features of escape panic. Nature 407(6803), 487 (2000)CrossRefGoogle Scholar
  26. 26.
    Horiuchi, S., Murozaki, Y., Hukugo, A.: A case study of fire and evacuation in a multi-purpose office building, Osaka, Japan. Fire Safety Science 1, 523–532 (1986)CrossRefGoogle Scholar
  27. 27.
    Hughes, R.: A continuum theory for the flow of pedestrians. Transportation Research Part B: Methodological 36(6), 507–535 (2002)CrossRefGoogle Scholar
  28. 28.
    Hung, N.M., Vinh, H.T., Jean-Charles, R.: Modeling and simulation of fire evacuation in public buildings. Advances in Computer Science: an International Journal 4(6), 1–7 (2015)Google Scholar
  29. 29.
    Jacod, J., Protter, P.: Probability Essentials. Springer Science & Business Media (2004)Google Scholar
  30. 30.
    Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg (1998)zbMATHGoogle Scholar
  31. 31.
    Landau, D., Binder, K.: A Guide to Monte Carlo Simulations in Statistical Physics. Cambridge University Press, New York, NY, USA (2005)CrossRefGoogle Scholar
  32. 32.
    Majmudar, T.S., Sperl, M., Luding, S., Behringer, R.P.: Jamming transition in granular systems. Phys. Rev. Lett. 98, 058,001 (2007)CrossRefGoogle Scholar
  33. 33.
    Meunier, H., Leca, J.B., Deneubourg, J.L., Petit, O.: Group movement decisions in capuchin monkeys: the utility of an experimental study and a mathematical model to explore the relationship between individual and collective behaviours. Behaviour 143(12), 1511–1527 (2006)CrossRefGoogle Scholar
  34. 34.
    Nguyen, M.H., Ho, T.V., Zucker, J.D.: Integration of smoke effect and blind evacuation strategy (sebes) within fire evacuation simulation. Simulation Modelling Practice and Theory 36, 44–59 (2013)CrossRefGoogle Scholar
  35. 35.
    Pankavich, S., Michalowski, N.: A short proof of increased parabolic regularity. Electronic Journal of Differential Equations 205 (2005)Google Scholar
  36. 36.
    Pavliotis, G.A.: Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations, vol. 60. Springer (2014)Google Scholar
  37. 37.
    Richardson, O.: Mercurial. (2015). Python framework for building, running and post-processing crowd simulations
  38. 38.
    Richardson, O.: Large-scale multiscale particle models in inhomogeneuous domains: Modelling and implementation. Master’s thesis, Technische Universiteit Eindhoven (2016)Google Scholar
  39. 39.
    Richardson, O., Jalba, A., A, M.: Effects of environment knowledge in evacuation scenarios involving fire and smoke - a multiscale modelling and simulation approach. ArXiv e-prints (2017). URL
  40. 40.
    Ronchi, E., Fridolf, F., Frantzich, H., Nilsson, D., Walter, A.L., Modig, H.: A tunnel evacuation experiment on movement speed and exit choice in smoke. Fire Safety Journal (2017)Google Scholar
  41. 41.
    Schieborn, D.: Viscosity Solutions of Hamilton-Jacobi Equations of Eikonal Type on Ramified Spaces. Ph.D. thesis, University Tübingen (2006)Google Scholar
  42. 42.
    Tan, L., Hu, M., Lin, H.: Agent-based simulation of building evacuation: Combining human behavior with predictable spatial accessibility in a fire emergency. Information Sciences 295, 53–66 (2015)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Treuille, A., Cooper, S., Popovic, Z.: Continuum crowds. ACM Trans. Graph. 25(3), 1160–1168 (2006)CrossRefGoogle Scholar
  44. 44.
    Tsai, J., Fridman, N., Bowring, E., Brown, M., Epstein, S., Kaminka, G., Marsella, S., Ogden, A., Rika, I., Sheel, A., et al.: Escapes: evacuation simulation with children, authorities, parents, emotions, and social comparison. In: The 10th International Conference on Autonomous Agents and Multiagent Systems-Volume 2, pp. 457–464. International Foundation for Autonomous Agents and Multiagent Systems (2011)Google Scholar
  45. 45.
    Voter, A.F.: Introduction to the Kinetic Monte Carlo Method. In: K.E. Sickafus, E.A. Kotomin, B.P. Uberuaga (eds.) Radiation Effects in Solids, pp. 1–23. Springer Netherlands, Dordrecht (2007)Google Scholar
  46. 46.
    Zuriguel, I., Parisi, D.R., Hidalgo, R.C., et al.: Clogging transition of many-particle systems flowing through bottlenecks. Scientific reports 4, 7324 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Matteo Colangeli
    • 1
  • Adrian Muntean
    • 2
  • Omar Richardson
    • 2
    Email author
  • Thi Kim Thoa Thieu
    • 3
  1. 1.Università degli Studi dell’AquilaLAquilaItaly
  2. 2.Karlstad UniversityKarlstadSweden
  3. 3.Gran Sasso Science InstituteLAquilaItaly

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