  Demographic noise causes unlimited population growth in a broad class of
models which, without noise, would predict a stable finite population. We study
this effect on the example of a stochastic birth-death model which includes
immigration, binary reproduction and death. The unlimited population growth
proceeds as an exponentially slow decay of a metastable probability
distribution (MPD) of the population. We develop a systematic WKB theory,
complemented by the van Kampen system size expansion, for the MPD and for the
decay time. Important signatures of the MPD is a power-law tail (such that all
the distribution moments, except the zeroth one, diverge) and the presence in
the solution of two different WKB modes.
