  In this paper we study a particular class of Piecewise deterministic Markov
processes (PDMP's) which are semi-stochastic catastrophe versions of
deterministic population growth models. In between successive jumps the process
follows a flow describing deterministic population growth. Moreover, at random
jump times, governed by state-dependent rates, the size of the population
shrinks by a random amount of its current size, an event possibly leading to
instantaneous local (or total) extinction. A special separable shrinkage
transition kernel is investigated in more detail, including the case of total
disasters. We discuss conditions under which such processes are recurrent
(positive or null) or transient. To do so, we introduce a modified scale
function which is used to compute, when relevant, the law of the height of
excursions and to decide if the process is recurrent or not. The question of
the finiteness of the time to extinction is investigated together with the
evaluation of the mean time to extinction when the last one is finite. Some
information on the embedded jump chain of the PDMP is also required when
dealing with the classification of states 0 and infinity that we exhibit.
