There are many stochastic systems which eventually `die out',
yet appear to be stationary over any reasonable time scale. I will
explore two approaches to modelling this behaviour. The first uses the
idea of a limiting conditional (or quasi-stationary) distribution. I
will explain how this distribution can be used model the long-term
behaviour of these processes. In the second approach, I will describe a
class of stochastic models (density-dependent Markov chains) for which
there are identifiable deterministic analogues. I will delimit
conditions under which a deterministic approximation is justified and
then identify an approximating diffusion process that can be used to
model fluctuations about the deterministic mean path.