On population growth with catastrophes

TL;DR

This paper analyzes a class of stochastic population models with catastrophes, combining deterministic growth with random, size-dependent catastrophic events, and investigates their long-term behavior and extinction times.

Contribution

It introduces a detailed analysis of semi-stochastic PDMPs with catastrophes, including recurrence, transience, and extinction time properties, using a novel scale function approach.

Findings

Conditions for recurrence and transience are established.

Finiteness and mean of extinction time are characterized.

Behavior at 0 and infinity states is classified.

Abstract

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.