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Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Nov;74(5 Pt 1):051907. Epub 2006 Nov 10.

Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model.

Author information

1
Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 München, Germany.

Abstract

Cyclic dominance of species has been identified as a potential mechanism to maintain biodiversity, see, e.g., B. Kerr, M. A. Riley, M. W. Feldman and B. J. M. Bohannan [Nature 418, 171 (2002)] and B. Kirkup and M. A. Riley [Nature 428, 412 (2004)]. Through analytical methods supported by numerical simulations, we address this issue by studying the properties of a paradigmatic non-spatial three-species stochastic system, namely, the "rock-paper-scissors" or cyclic Lotka-Volterra model. While the deterministic approach (rate equations) predicts the coexistence of the species resulting in regular (yet neutrally stable) oscillations of the population densities, we demonstrate that fluctuations arising in the system with a finite number of agents drastically alter this picture and are responsible for extinction: After long enough time, two of the three species die out. As main findings we provide analytic estimates and numerical computation of the extinction probability at a given time. We also discuss the implications of our results for a broad class of competing population systems.

PMID:
17279939
DOI:
10.1103/PhysRevE.74.051907
[Indexed for MEDLINE]

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