  Various types of stabilizing controls lead to a deterministic difference
equation with the following property: once the initial value is positive, the
solution tends to the unique positive equilibrium. Introducing additive
perturbations can change this picture: we give examples of difference equations
experiencing additive perturbations which have solutions staying around zero
rather than tending to the unique positive equilibrium. When perturbations are
stochastic with a bounded support, we give an upper estimate for the
probability that the solution can stay around zero. Applying extra conditions
on the behavior of the map function $f$ at zero or on the amplitudes of
stochastic perturbations, we prove that the solution tends to the unique
positive equilibrium almost surely. In particular, this holds either for all
amplitudes when the right derivative of the map $f$ at zero exceeds one or,
independently of the behavior of $f$ at zero, when the amplitudes are not
square summable.
