Skip to main content
SpringerLink
Log in

Differential Equations Modeling Crowd Interactions

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

Nonlocal conservation laws are used to describe various realistic instances of crowd behaviors. First, a basic analytic framework is established through an ad hoc well-posedness theorem for systems of nonlocal conservation laws in several space dimensions interacting nonlocally with a system of ODEs. Numerical integrations show possible applications to the interaction of different groups of pedestrians and also with other agents.

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

Similar content being viewed by others

References

  • Borsche, R., Colombo, R.M., Garavello, M.: On the coupling of systems of hyperbolic conservation laws with ordinary differential equations. Nonlinearity 23(11), 2749–2770 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • Borsche, R., Colombo, R.M., Garavello, M.: Mixed systems: ODEs—balance laws. J. Differ. Equ. 252(3), 2311–2338 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Borsche, R., Klar, A., Kühn, S., Meurer, A.: Coupling traffic flow networks to pedestrian motion. Math. Models Methods Appl. Sci. 24(2), 359–380 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  • Borsche, R., Meurer, A.: Interaction of road networks and pedestrian motion at crosswalks. Discrete Contin. Dyn. Syst. Ser. S 7(3), 363–377 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  • Bressan, A., Piccoli, B.: Introduction to the Mathematical Theory of Control, Volume 2 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2007)

    Google Scholar 

  • Colombo, R.M., Garavello, M., Lécureux-Mercier, M.: A class of nonlocal models for pedestrian traffic. Math. Models Methods Appl. Sci. 22(4), 1150023–1150034 (2012)

    Article  MathSciNet  Google Scholar 

  • Colombo, R.M., Lécureux-Mercier, M.: An analytical framework to describe the interactions between individuals and a continuum. J. Nonlinear Sci. 22(1), 39–61 (2012a)

    Article  MATH  MathSciNet  Google Scholar 

  • Colombo, R.M., Lécureux-Mercier, M.: Nonlocal crowd dynamics models for several populations. Acta Math. Sci. Ser. B Engl. Ed. 32(1), 177–196 (2012b)

    Article  MATH  MathSciNet  Google Scholar 

  • Colombo, R.M., Pogodaev, N.: Confinement strategies in a model for the interaction between individuals and a continuum. SIAM J. Appl. Dyn. Syst. 11(2), 741–770 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Cristiani, E., Piccoli, B., Tosin, A.: Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model. Simul. 9(1), 155–182 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Etikyala, R., Göttlich, S., Klar, A., Tiwari, S.: Particle methods for pedestrian flow models: from microscopic to nonlocal continuum models. Math. Models Methods Appl. Sci. 24(12), 2503–2523 (2014)

  • Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers Group, Dordrecht (1988). Translated from the Russian

    Book  Google Scholar 

  • Gazis, D.C., Herman, R., Rothery, R.W.: Nonlinear follow-the-leader models of traffic flow. Oper. Res. 9(4), 545–567 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  • Helbing, D., Molnar, P.: Social force model for pedestrian dynamics. Phys. Rev. E 51(5), 4282 (1995)

    Article  Google Scholar 

  • Hughes, R.L.: A continuum theory for the flow of pedestrians. Transp. Res. Part B Methodol. 36(6), 507–535 (2002)

    Article  Google Scholar 

  • Kružkov, S.N.: First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81(123), 228–255 (1970)

    MathSciNet  Google Scholar 

  • Lécureux-Mercier, M.: Improved stability estimates on general scalar balance laws. ArXiv e-prints (2010)

  • LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  • Piccoli, B., Tosin, A.: Pedestrian flows in bounded domains with obstacles. Contin. Mech. Thermodyn. 21, 85–107 (2009). doi:10.1007/s00161-009-0100-x

    Article  MATH  MathSciNet  Google Scholar 

  • Piccoli, B., Tosin, A.: Time-evolving measures and macroscopic modeling of pedestrian flow. Arch. Ration. Mech. Anal. (2010). doi:10.1007/s00205-010-0366-y

  • Sethian, J.A.: Fast marching methods. SIAM Rev. 41(2), 199–235 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edn. Springer, Berlin (2009). A practical introduction

    Book  MATH  Google Scholar 

Download references

Acknowledgments

This work was partially supported by the INDAM–GNAMPA project Conservation Laws: Theory and Applications, by the Graduiertenkolleg 1932 “Stochastic Models for Innovations in the Engineering Sciences” and by the Deutsche Forschungsgemeinschaft (DFG) project “Stochastic Models for Innovations in the Engineering Sciences”.

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Raul Borsche & Anne Meurer

  2. Unità INdAM, c/o DII, Università degli studi di Brescia, Brescia, Italy

    Rinaldo M. Colombo

  3. Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Milan, Italy

    Mauro Garavello

Authors
  1. Raul Borsche
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rinaldo M. Colombo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mauro Garavello
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Anne Meurer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rinaldo M. Colombo.

Additional information

Communicated by Eva Kanso.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Borsche, R., Colombo, R.M., Garavello, M. et al. Differential Equations Modeling Crowd Interactions. J Nonlinear Sci 25, 827–859 (2015). https://doi.org/10.1007/s00332-015-9242-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-015-9242-0

Keywords

Mathematics Subject Classification

Search

Navigation