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Soft Matter and Biological Physics
Eur. Phys. J. E 10, 241-248 (2003)
DOI: 10.1140/epje/i2002-10112-3

Survival and extinction in cyclic and neutral three-species systems

M. Ifti and B. Bergersen

Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC, V6T 1Z1 Canada

ita@physics.ubc.ca

(Received 14 August 2002 and Received in final form 14 February 2003 / Published online: 1 April 2003)

Abstract
We study the ABC model ( $A + B \to 2B$, $B + C \to 2C$, $C + A \to 2A$), and its counterpart: the three-component neutral drift model ( $A + B \to 2A$ or 2B, $B + C \to 2B$ or 2C, $C + A \to 2C$ or 2A.) In the former case, the mean-field approximation exhibits cyclic behaviour with an amplitude determined by the initial condition. When stochastic phenomena are taken into account the amplitude of oscillations will drift and eventually one and then two of the three species will become extinct. The second model remains stationary for all initial conditions in the mean-field approximation, and drifts when stochastic phenomena are considered. We analyzed the distribution of first extinction times of both models by simulations of the master equation, and from the point of view of the Fokker-Planck equation. Survival probability vs. time plots suggest an exponential decay. For the neutral model the extinction rate is inversely proportional to the system size, while the cyclic model exhibits anomalous behaviour for small system sizes. In the large system size limit the extinction times for both models will be the same. This result is compatible with the smallest eigenvalue obtained from the numerical solution of the Fokker-Planck equation. We also studied the behaviour of the probability distribution. The exponential decay is found to be robust against certain changes, such as the three reactions having different rates.

PACS
87.10.+e - General theory and mathematical aspects.
87.23.Cc - Population dynamics and ecological pattern formation.

© EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2003