Two population models with constrained migrations

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.