DOI: 10.1140/epje/i2002-10112-3
Survival and extinction in cyclic and neutral three-species systems
M. Ifti and B. BergersenDepartment 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 (
,
,
), and
its counterpart: the three-component neutral drift model (
or
2B,
or
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.
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
