Reflections on the extinction-explosion dichotomy

TL;DR

This paper examines the conditions under which populations modeled by stochastic processes either go extinct or grow infinitely, exploring relaxations of classical assumptions to understand the dichotomy better.

Contribution

It investigates how relaxing classical assumptions affects the extinction-explosion dichotomy, providing new insights and results on the conditions for this phenomenon.

Findings

Classical assumptions ensure the dichotomy holds.

Relaxing assumptions can lead to both positive and negative results.

The paper identifies conditions under which the dichotomy persists or fails.

Abstract

A wide range of stochastic processes that model the growth and decline of populations exhibit a curious dichotomy: with certainty either the population goes extinct or its size tends to infinity. There is a elegant and classical theorem that explains why this dichotomy must hold under certain assumptions concerning the process. In this note, I explore how these assumptions might be relaxed further in order to obtain the same, or a similar conclusion, and obtain both positive and negative results.