  We consider a class of cubic stochastic operators that are motivated by
models for evolution of frequencies of genetic types in populations. We take
populations with three mutually exclusive genetic types.
  The long term dynamics of single maps, starting with a generic initial
condition where in particular all genetic types occur with positive frequency,
is asymptotic to equilibria where either only one genetic type survives, or
where all three genetic types occur.
  We consider a family of independent and identically distributed maps from
this class and study its long term dynamics, in particular its random point
attractors. The long term dynamics of the random composition of maps is
asymptotic, almost surely, to equilibria. In contrast to the deterministic
system, for generic initial conditions these can be equilibria with one or two
or three types present (depending only on the distribution).

Dynamical Systems