  Catastrophe Markov chain population models have received a lot of attention
in the recent past. We herewith consider two special cases of such models
involving total disasters, both in discrete and in continuous-time. Depending
on the parameters range, the two models can show up a recurrence/transience
transition and, in the critical case, a positive/null recurrence transition.
The collapse transition probabilities are chosen in such a way that the models
are exactly solvable and, in case of positive recurrence, intimately related to
the extended Sibuya and Pareto-Zipf distributions whose divisibility and
self-decomposability properties are shown relevant. The study includes:
existence and shape of the invariant measure, time-reversal, return time to the
origin, contact probability at the origin, extinction probability, height and
length of the excursions, a renewal approach to the fraction of time spent in
the catastrophic state, scale function, first time to collapse and
first-passage times, divisibility properties.
