  The effect of small noise in a smooth dynamical system is negligible on any
finite time interval. Here we study situations when it persists on intervals
increasing to infinity. Such asymptotic regime occurs when the system starts
from initial condition, sufficiently close to an unstable fixed point. In this
case, under appropriate scaling, the trajectory converges to solution of the
unperturbed system, started from a certain {\em random} initial condition. In
this paper we consider the case of one dimensional diffusions on the positive
half line, which often arise as scaling limits in population dynamics.
