  We study two models of population with migration. We assume that we are given
infinitely many islands with the same number r of resources, each individual
consuming one unit of resources. On an island lives an individual whose
genealogy is given by a critical Galton-Watson tree. If all the resources are
consumed, any newborn child has to migrate to find new resources. In this
sense, the migrations are constrained, not random. We will consider first a
model where resources do not regrow, so the r first born individuals remain on
their home island, whereas their children migrate. In the second model, we
assume that resources regrow, so only r people can live on an island at the
same time, the supernumerary ones being forced to migrate. In both cases, we
are interested in how the population spreads on the islands, when the number of
initial individuals and available resources tend to infinity. This mainly
relies on computing asymptotics for critical random walks and functionals of
the Brownian motion.
