  We study the dynamics of colonization of a territory by a stochastic
population at low immigration pressure. We assume a sufficiently strong Allee
effect that introduces, in deterministic theory, a large critical population
size for colonization. At low immigration rates, the average pre-colonization
population size is small thus invalidating the WKB approximation to the master
equation. We circumvent this difficulty by deriving an exact zero-flux solution
of the master equation and matching it with an approximate non-zero-flux
solution of the pertinent Fokker-Planck equation in a small region around the
critical population size. This procedure provides an accurate evaluation of the
quasi-stationary probability distribution of population sizes in the
pre-colonization state, and of the mean time to colonization, for a wide range
of immigration rates. At sufficiently high immigration rates our results agree
with WKB results obtained previously. At low immigration rates the results can
be very different.
