Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games


Biodiversity is essential to the viability of ecological systems. Species diversity in ecosystems is promoted by cyclic, non-hierarchical interactions among competing populations. Central features of such non-transitive relations are represented by the ‘rock–paper–scissors’ game, in which rock crushes scissors, scissors cut paper, and paper wraps rock. In combination with spatial dispersal of static populations, this type of competition results in the stable coexistence of all species and the long-term maintenance of biodiversity1,2,3,4,5. However, population mobility is a central feature of real ecosystems: animals migrate, bacteria run and tumble. Here, we observe a critical influence of mobility on species diversity. When mobility exceeds a certain value, biodiversity is jeopardized and lost. In contrast, below this critical threshold all subpopulations coexist and an entanglement of travelling spiral waves forms in the course of time. We establish that this phenomenon is robust; it does not depend on the details of cyclic competition or spatial environment. These findings have important implications for maintenance and temporal development of ecological systems and are relevant for the formation and propagation of patterns in microbial populations or excitable media.

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Figure 1: The rules of the stochastic model.
Figure 2: The critical mobility Mc.
Figure 3: Spiralling patterns.
Figure 4: Phase diagram.


  1. 1

    Durrett, R. & Levin, S. Allelopathy in spatially distributed populations. J. Theor. Biol. 185, 165–171 (1997)

    CAS  Article  Google Scholar 

  2. 2

    Durrett, R. & Levin, S. Spatial aspects of interspecific competition. Theor. Pop. Biol. 53, 30–43 (1998)

    CAS  Article  Google Scholar 

  3. 3

    Kerr, B., Riley, M. A., Feldman, M. W. & Bohannan, B. J. M. Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors. Nature 418, 171–174 (2002)

    ADS  CAS  Article  Google Scholar 

  4. 4

    Czárán, T. L., Hoekstra, R. F. & Pagie, L. Chemical warfare between microbes promotes biodiversity. Proc. Natl Acad. Sci. USA 99, 786–790 (2002)

    ADS  Article  Google Scholar 

  5. 5

    Szabó, G. & Fáth, G. Evolutionary games on graphs. Phys. Rep. 446, 97–216 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  6. 6

    Dykhuizen, D. E. Santa rosalia revisited: Why are there so many species of bacteria? Antonie Van Leeuwenhoek 73, 25–33 (1998)

    CAS  Article  Google Scholar 

  7. 7

    Smith, J. M. Evolution and the Theory of Games (Cambridge Univ. Press, Cambridge, 1982)

    Google Scholar 

  8. 8

    Hofbauer, J. & Sigmund, K. Evolutionary Games and Population Dynamics (Cambridge Univ. Press, Cambdrige, 1998)

    Google Scholar 

  9. 9

    Nowak, M. A. Evolutionary Dynamics (Belknap Press, Cambridge, Massachusetts, 2006)

    Google Scholar 

  10. 10

    May, R. M. & Leonard, W. J. Nonlinear aspects of competition between species. SIAM J. Appl. Math. 29, 243–253 (1975)

    MathSciNet  Article  Google Scholar 

  11. 11

    Johnson, C. R. & Seinen, I. Selection for restraint in competitive ability in spatial competition systems. Proc. R. Soc. Lond. B 269, 655–663 (2002)

    Article  Google Scholar 

  12. 12

    Reichenbach, T., Mobilia, M. & Frey, E. Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model. Phys. Rev. E 74, 051907 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  13. 13

    Jackson, J. B. C. & Buss, L. Allelopathy and spatial competition among coral reef invertebrates. Proc. Natl Acad. Sci. USA 72, 5160–5163 (1975)

    ADS  CAS  Article  Google Scholar 

  14. 14

    Sinervo, B. & Lively, C. M. The rock–scissors–paper game and the evolution of alternative male strategies. Nature 380, 240–243 (1996)

    ADS  CAS  Article  Google Scholar 

  15. 15

    Kirkup, B. C. & Riley, M. A. Antibiotic-mediated antagonism leads to a bacterial game of rock–paper–scissors in vivo. Nature 428, 412–414 (2004)

    ADS  CAS  Article  Google Scholar 

  16. 16

    Levin, S. A. Dispersion and population interactions. Am. Nat. 108, 207–228 (1974)

    Article  Google Scholar 

  17. 17

    Hassell, P. M., Comins, H. N. & May, R. M. Spatial structure and chaos in insect population dynamics. Nature 353, 255–258 (1991)

    ADS  Article  Google Scholar 

  18. 18

    Blasius, B., Huppert, A. & Stone, L. Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399, 354–359 (1999)

    ADS  CAS  Article  Google Scholar 

  19. 19

    King, A. A. & Hastings, A. Spatial mechanism for coexistence of species sharing a common natural enemy. Theor. Pop. Biol. 64, 431–438 (2003)

    Article  Google Scholar 

  20. 20

    Durrett, R. & Levin, S. The importance of being discrete (and spatial). Theor. Pop. Biol. 46, 363–394 (1994)

    Article  Google Scholar 

  21. 21

    Redner, S. A Guide to First-Passage Processes (Cambridge Univ. Press, Cambridge, 2001)

    Google Scholar 

  22. 22

    Hastings, A. Transients: the key to long-term ecological understanding? Trends Ecol. Evol. 19, 39–45 (2004)

    Article  Google Scholar 

  23. 23

    Berg, H. C. E. coli in Motion (Springer, New York, 2003)

    Google Scholar 

  24. 24

    Igoshin, O. A., Welch, R., Kaiser, D. & Oster, G. Waves and aggregation patterns in myxobacteria. Proc. Natl Acad. Sci. USA 101, 4256–4261 (2004)

    ADS  CAS  Article  Google Scholar 

  25. 25

    Siegert, F. & Weijer, C. J. Spiral and concentric waves organize multicellular Dictyostelium mounds. Curr. Biol. 5, 937–943 (1995)

    CAS  Article  Google Scholar 

  26. 26

    Thul, R. & Falcke, M. Stability of membrane bound reactions. Phys. Rev. Lett. 93, 188103 (2004)

    ADS  CAS  Article  Google Scholar 

  27. 27

    Gardiner, C. W. Handbook of Stochastic Methods (Springer, Berlin, 1983)

    Google Scholar 

  28. 28

    Wiggins, S. Introduction to Applied Nonlinear Dynamical Systems and Chaos Ch. 2 and 3 (Springer, New York, 1990)

    Google Scholar 

  29. 29

    Aranson, I. S. & Kramer, L. The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74, 99–143 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  30. 30

    Hauert, C., de Monte, S., Hofbauer, J. & Sigmund, K. Volunteering as red queen mechanism for cooperation in public goods games. Science 296, 1129–1132 (2002)

    ADS  CAS  Article  Google Scholar 

  31. 31

    Gillespie, D. T. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434 (1976)

    ADS  MathSciNet  CAS  Article  Google Scholar 

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We thank M. Bathe and M. Leisner for discussions on the manuscript. Financial support of the German Excellence Initiative via the program “Nanosystems Initiative Munich (NIM)” as well as the SFB “Manipulation of Matter at the Nanometer Length Scale” is gratefully acknowledged. M.M. is grateful to the Alexander von Humboldt Foundation for support through a fellowship.

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Correspondence to Erwin Frey.

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Supplementary information

Supplementary Notes

This file contains Supplementary Notes on the concept of extensivity, on stochastic partial differential equations and the complex Ginzburg-Landau equation as well as on scaling arguments. Details on the Supplementary Videos 1 and 2 are given. This file was corrected on 4 September 2007. (PDF 125 kb)

Supplementary Video 1

This file contains Supplementary Video 1. The video shows the temporal development of the three species when mobility is small and coexistence is stable. (MOV 9805 kb)

Supplementary Video 2

This file contains Supplementary Video 2. The video shows the temporal development of the three species when mobility is near the critical value, coexistence is still stable. (MOV 9794 kb)

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Reichenbach, T., Mobilia, M. & Frey, E. Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games. Nature 448, 1046–1049 (2007).

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