  In this paper we consider the global qualitative properties of a
stochastically perturbed logistic model of population growth. In this model,
the stochastic perturbations are assumed to be of the white noise type and are
proportional to the current population size. Using the direct Lyapunov method,
we established the global properties of this stochastic differential equation.
In particular, we found that solutions of the equation oscillate around an
interval, and explicitly found the end points of this interval. Moreover, we
found that, if the magnitude of the noise exceeds a certain critical level
(which is also explicitly found), then the stochastic stabilisation
("stabilisation by noise") of the zero solution occurs. In this case, (i) the
origin is the lower boundary of the interval, and (ii) the extinction of the
population due to stochasticity occurs almost sure (a.s.) for a finite time.
