Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

Modelling the evolution of human trail systems


Many human social phenomena, such as cooperation1,2,3, the growth of settlements4, traffic dynamics5,6,7 and pedestrian movement7,8,9,10, appear to be accessible to mathematical descriptions that invoke self-organization11,12. Here we develop a model of pedestrian motion to explore the evolution of trails in urban green spaces such as parks. Our aim is to address such questions as what the topological structures of these trail systems are13, and whether optimal path systems can be predicted for urban planning. We use an ‘active walker’ model14,15,16,17,18,19 that takes into account pedestrian motion and orientation and the concomitant feedbacks with the surrounding environment. Such models have previously been applied to the study of complex structure formation in physical14,15,16, chemical17 and biological18,19 systems. We find that our model is able to reproduce many of the observed large-scale spatial features of trail systems.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Between the straight, paved ways on the university campus in Stuttgart-Vaihingen a trail system has evolved (centre of the picture).
Figure 2: The structure of the emerging trail system (yellow to blue) depends essentially on the attractiveness parameter κ.
Figure 3: The places and walking directions of pedestrians are represented here by arrows.
Figure 4: When pedestrians leave footprints on the ground, trails will develop, and only parts of the ground are used for walking (in contrast to pa.

Similar content being viewed by others


  1. Axelrod, R. & Dion, D. The further evolution of cooperation. Science 242, 1385–1390 (1988).

    Article  ADS  CAS  Google Scholar 

  2. Clearwater, S. H., Huberman, B. A. & Hogg, T. Cooperative solution of constraint satisfaction problems. Science 254, 1181–1183 (1991).

    Article  ADS  CAS  Google Scholar 

  3. Helbing, D. Quantitative Sociodynamics Stochastic Methods and Models of Social Interaction Processes(Kluwer Academic, Dordrecht, (1995)).

    MATH  Google Scholar 

  4. Makse, H. A., Havlin, S. & Stanley, H. E. Modelling urban growth patterns. Nature 377, 608–612 (1995).

    Article  ADS  CAS  Google Scholar 

  5. Herman, R., Lam, T. & Prigogine, I. Multilane vehicular traffic and adaptive human behavior. Science 179, 918–920 (1973).

    Article  ADS  CAS  Google Scholar 

  6. Herman, R. & Prigogine, I. Atwo-fluid approach to town traffic. Science 204, 148–151 (1979).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  7. Helbing, D. Verkehrsdynamik. Neue physikalische Modellierungskonzepte(Springer, Berlin, (1997)).

    Book  Google Scholar 

  8. Henderson, L. F. The statistics of crowd fluids. Nature 229, 381–383 (1971).

    Article  ADS  CAS  Google Scholar 

  9. Henderson, L. F. Sexual differences in human crowd motion. Nature 240, 353–355 (1972).

    Article  ADS  CAS  Google Scholar 

  10. Helbing, D. & Molnár, P. Social force model for pedestrian dynamics. Phys. Rev. E 51, 4282–4286 (1995).

    Article  ADS  CAS  Google Scholar 

  11. Haken, H. Advanced Synergetics 2nd edn(Springer, Berlin, (1987)).

    Google Scholar 

  12. Nicolis, G. & Prigogine, I. Self-Organization in Nonequilibrium Systems. From Dissipative Structures to Order through Fluctuations(Wiley, New York, (1977)).

    MATH  Google Scholar 

  13. Schenk, M. Untersuchungen zum Fußgängerverhalten Thesis, Univ. Stuttgart((1995)).

    Google Scholar 

  14. Kayser, D. R., Aberle, L. K., Pochy, R. D. & Lam, L. Active walker models: tracks and landscapes. Physica A 191, 17–24 (1992).

    Article  ADS  Google Scholar 

  15. Lam, L. Active walker models for complex systems. Chaos Solitons Fractals 6, 267–285 (1995).

    Article  ADS  Google Scholar 

  16. Schweitzer, F. & Schimansky-GGeier, L. Clustering of “active” walkers in a two-component system. Physica A 206, 359–379 (1994).

    Article  ADS  Google Scholar 

  17. Schimansky-Geier, L., Mieth, M., Rosé, H. & Malchow, H. Structure formation by active Brownian particles. Phys. Lett. A 207, 140–146 (1995).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  18. Ben-Jacob, E. et al. Generic modelling of cooperative growth patterns in bacterial colonies. Nature 368, 46–49 (1994).

    Article  ADS  CAS  Google Scholar 

  19. Stevens, A. & Schweitzer, F. in Mechanisms of Cell and Tissue Motion(eds Alt, W., Deutsch, A. & Dunn, G.) 183–192 (Birkhäuser, Basel, (1997)).

    MATH  Google Scholar 

  20. Eigen, M. & Schuster, P.The Hypercycle(Springer, Berlin, (1979)).

    Book  Google Scholar 

  21. Feistel, R. & Ebeling, W. Evolution of Complex Systems. Self-Organization, Entropy and Development(Kluwer, Dordrecht, (1989)).

    Google Scholar 

  22. Timmermans, H., van der Hagen, X. & Borgers, A. Transportation systems, retail environments and pedestrian trip chaining behaviour: Modelling issues and applications. Transportation Res. B 26, 45–59 (1992).

    Article  Google Scholar 

  23. Hillier, B. Space is a Machine: A Configurational Theory of Architecture(Cambridge Univ. Press, (1996)).

    Google Scholar 

Download references


We thank F. Schweitzer for discussions, and W. Weidlich and M. Treiber for reviews of the manuscript.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Dirk Helbing.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Helbing, D., Keltsch, J. & Molnár, P. Modelling the evolution of human trail systems. Nature 388, 47–50 (1997).

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing