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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.


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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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