On population growth with catastrophes

On decay–surge population models

“…Arguing as in Sections 2.4 and 2.5 of [16], a direct consequence of the above duality is the fact that for all t < t 0 (x),…”

“…In this subsection, we exhibit a natural relationship between decay-surge population models, as studied here, and growth-collapse models as developed in Boxma et al [4], Goncalves et al [16], Gripenberg [17], and Hanson and Tuckwell [18]. Growth-collapse models describe deterministic population growth where at random jump times the population undergoes a catastrophe and falls to a random fraction of its previous size.…”

“…In Section 2, we introduce our model and state some first results. Most importantly, we establish a simple relationship between decay-surge models and growth-collapse models as studied in Goncalves, Huillet and Löcherbach [16], that allows us to obtain explicit representations of the law of the first jump time and of the associated speed measure without any further study. Section 3 is devoted to the proof of the existence of the scale function (Proposition 3) together with the study of first moments of hitting times, which are shown to be finite if the speed density is integrable at +∞ (Proposition 6).…”

“…Proposition 1. The process X t is a growth-collapse process as studied in [16] with triple α, β, h given by…”

“…To ensure that P(T x < ∞) = 1, in accordance with Assumption 1 of [16] we will impose the following condition.…”