  We consider the dynamics of a population spatially structured in colonies
that are vulnerable to catastrophic events occurring at random times, which
randomly reduce their population size and compel survivors to disperse to
neighboring areas. The dispersion behavior of survivors is critically
significant for the survival of the entire species. In this paper, we consider
an uniform dispersion scheme, where all possible survivor groupings are equally
probable. The aim of the survivors is to establish new colonies, with
individuals who settle in empty sites potentially initiating a new colony by
themselves. However, all other individuals succumb to the catastrophe. We
consider the number of dispersal options for surviving individuals in the
aftermath of a catastrophe to be a fixed value $d$ within the neighborhood. In
this context, we conceptualize the evolution of population dynamics occurring
over a homogeneous tree. We investigate the conditions necessary for these
populations to survive, presenting pertinent bounds for survival probability,
the number of colonized vertices, the extent of dispersion within the
population, and the mean time to extinction for the entire population.
