  Heterogeneities in environmental conditions often induce corresponding
heterogeneities in the distribution of species. In the extreme case of a
localized patch of increased growth rates, reproducing populations can become
strongly concentrated at the patch despite the entropic tendency for population
to distribute evenly. Several deterministic mathematical models have been used
to characterize the conditions under which localized states can form, and how
they break down due to convective driving forces. Here, we study the
delocalization of a finite population in the presence of number fluctuations.
We find that any finite population delocalizes on sufficiently long time
scales. Depending on parameters, however, populations may remain localized for
a very long time. The typical waiting time to delocalization increases
exponentially with both population size and distance to the critical wind speed
of the deterministic approximation. We augment these simulation results by a
mathematical analysis that treats the reproduction and migration of individuals
as branching random walks subject to global constraints. For a particular
constraint, different from a fixed population size constraint, this model
yields a solvable first moment equation. We find that this solvable model
approximates very well the fixed population size model for large populations,
but starts to deviate as population sizes are small. The analytical approach
allows us to map out a phase diagram of the order parameter as a function of
the two driving parameters, inverse population size and wind speed. Our results
may be used to extend the analysis of delocalization transitions to different
settings, such as the viral quasi-species scenario.
