Colonization and collapse on Homogeneous Trees

Abstract

We investigate a basic immigration process where colonies grow, during a random time, according to a general counting process until collapse. Upon collapse a random amount of individuals survive. These survivors try independently establishing new colonies at neighbour sites. Here we consider this general process subject to two schemes, Poisson growth with geometric catastrophe and Yule growth with binomial catastrophe. Independent of everything else colonies growth, during an exponential time, as a Poisson (or Yule) process and right after that exponential time their size is reduced according to geometric (or binomial) law. Each survivor tries independently, to start a new colony at a neighbour site of a homogeneous tree. That colony will thrive until its collapse, and so on. 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.