  This paper deals with extinction of an isolated population caused by
intrinsic noise. We model the population dynamics in a "refuge" as a Markov
process which involves births and deaths on discrete lattice sites and random
migrations between neighboring sites. In extinction scenario I the zero
population size is a repelling fixed point of the on-site deterministic
dynamics. In extinction scenario II the zero population size is an attracting
fixed point, corresponding to what is known in ecology as Allee effect.
Assuming a large population size, we develop WKB (Wentzel-Kramers-Brillouin)
approximation to the master equation. The resulting Hamilton's equations encode
the most probable path of the population toward extinction and the mean time to
extinction. In the fast-migration limit these equations coincide, up to a
canonical transformation, with those obtained, in a different way, by Elgart
and Kamenev (2004). We classify possible regimes of population extinction with
and without an Allee effect and for different types of refuge and solve several
examples analytically and numerically. For a very strong Allee effect the
extinction problem can be mapped into the over-damped limit of theory of
homogeneous nucleation due to Langer (1969). In this regime, and for very long
systems, we predict an optimal refuge size that maximizes the mean time to
extinction.
