  Random population dynamics with catastrophes (events pertaining to possible
elimination of a large portion of the population) has a long history in the
mathematical literature. In this paper we study an ergodic model for random
population dynamics with linear growth and binomial catastrophes: in a
catastrophe, each individual survives with some fixed probability,
independently of the rest. Through a coupling construction, we obtain sharp
two-sided bounds for the rate of convergence to stationarity which are applied
to show that the model exhibits a cutoff phenomenon.
