  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.
