  We study large deviations of the time-averaged size of stochastic populations
described by a continuous-time Markov jump process. When the expected
population size $N$ in the steady state is large, the large deviation function
(LDF) of the time-averaged population size can be evaluated by using a WKB
(after Wentzel, Kramers and Brillouin) method, applied directly to the master
equation for the Markov process. For a class of models that we identify, the
direct WKB method predicts a giant disparity between the probabilities of
observing an unusually small and an unusually large values of the time-averaged
population size. The disparity results from a qualitative change in the
"optimal" trajectory of the underlying classical mechanics problem. The direct
WKB method also predicts, in the limit of $N\to \infty$, a singularity of the
LDF, which can be interpreted as a second-order dynamical phase transition. The
transition is smoothed at finite $N$, but the giant disparity remains. The
smoothing effect is captured by the van-Kampen system size expansion of the
exact master equation near the attracting fixed point of the underlying
deterministic model. We describe the giant disparity at finite $N$ by
developing a different variant of WKB method, which is applied in conjunction
with the Donsker-Varadhan large-deviation formalism and involves
subleading-order calculations in $1/N$.
