Colonization and Collapse on Homogeneous Trees

摘要

We investigate a metapopulation model referring to populations that are spatially structured in colonies. Each colony thrives during a random time until a catastrophe when only a random amount of individuals of that colony survives. These survivors try independently establishing new colonies at neighbour sites, randomly. If the chosen site is occupied, that individual dies, otherwise the individual founds there a new colony. Here we consider this metapopulation model subject to two schemes: (i) Poisson growth, during an exponential time, for each colony and geometric catastrophe, and (ii) Yule growth, during an exponential time, for each colony and binomial catastrophe. We study conditions on the set of parameters for these processes to survive, present relevant bounds for the probability of survival, for the number of vertices that were colonized and for the reach of the colonies compared to the starting point. As a byproduct we study convergence of sequence of branching processes.