Two population models with constrained migrations

We study two models of population with migration. On an island lives an individual whose genealogy is given by a critical Galton–Watson tree. If its offspring ends up consuming all the resources, 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, and then another one when they do. 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.