18 research outputs found
Crossover Behaviour of 3-Species Systems with Mutations or Migrations
We study the ABC model in the cyclic competition and neutral drift versions,
with mutations and migrations introduced into the model. When stochastic
phenomena are taken into account, there are three distinct regimes in the
model. (i) In the "fixation" regime, the first extinction time scales with the
system size N and has an exponential distribution, with an exponent that
depends on the mutation/migration probability per particle. (ii) In the
"diversity" regime, the order parameter remains nonzero for very long times,
and becomes zero only rarely, almost never for large system sizes. (iii) In the
critical regime, the first extinction time has a power-law distribution with
exponent -1. The transition corresponds to a crossover from diffusive behaviour
to Gaussian fluctuations about a stable solution. The analytical results are
checked against computer simulations of the model.Comment: 2nd version revised and refereed 14 pages, 5 figure
Survival and Extinction in Cyclic and Neutral Three--Species Systems
We study the ABC model (A + B --> 2B, B + C --> 2C, C + A --> 2A), and its
counterpart: the three--component neutral drift model (A + B --> 2A or 2B, B +
C --> 2B or 2C, C + A --> 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 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 long--time behaviour of the probability
distribution. The exponential decay is found to be robust against certain
changes, such as the three reactions having different rates.Comment: 19 pages, 11 figures Final versio
Correlations in Ising chains with non-integrable interactions
Two-spin correlations generated by interactions which decay with distance r
as r^{-1-sigma} with -1 <sigma <0 are calculated for periodic Ising chains of
length L. Mean-field theory indicates that the correlations, C(r,L), diminish
in the thermodynamic limit L -> \infty, but they contain a singular structure
for r/L -> 0 which can be observed by introducing magnified correlations,
LC(r,L)-\sum_r C(r,L). The magnified correlations are shown to have a scaling
form F(r/L) and the singular structure of F(x) for x->0 is found to be the same
at all temperatures including the critical point. These conclusions are
supported by the results of Monte Carlo simulations for systems with sigma
=-0.50 and -0.25 both at the critical temperature T=Tc and at T=2Tc.Comment: 13 pages, latex, 5 eps figures in a separate uuencoded file, to
appear in Phys.Rev.