  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.
