Consider a density-dependent birth-death process

*X*_{N} on a finite state
space of size

*N*. Let

*P*_{N} be the law (on

*D*([0,

*T*]) where

*T*>0 is
arbitrary) of the density process

*X*_{N}/

*N* and let

be the unique
stationary distribution (on [0,1]) of

*X*_{N}/

*N*, if it exists. Typically,
these distributions converge weakly to a degenerate distribution as

,
so the probability of sets not containing the
degenerate point will tend to 0; large deviations is concerned with
obtaining the exponential decay rate of these probabilities. Friedlin-Wentzel
theory is used to establish the large deviations behaviour (as

)
of

*P*_{N}. In the one-dimensional case, a large deviations principle
for the stationary distribution

is obtained by elementary explicit
computations. However, when the birth-death process has an absorbing
state at 0 (so

no longer exists), the same elementary computations
are still applicable to the quasi-stationary distribution, and we show
that the quasi-stationary distributions obey the same large deviations
principle as in the recurrent case. In addition, we address some questions
related to the estimated time to absorption and obtain a large deviations
principle for the invariant distribution in higher dimensions by studying
a quasi-potential.