title: Reflections on the extinction-explosion dichotomy.
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 an 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.
authors.affiliation: Biomathematics Research Centre, University of Canterbury, Private Bag 4800, Christchurch, New Zealand. Electronic address: mathmomike@gmail.com.
authors.forename: Mike
authors.initials: M
authors.lastname: Steel
mesh_terms: D053476:Extinction, Biological; D006801:Humans; D008390:Markov Chains; D008954:Models, Biological; D011157:Population Dynamics; D013269:Stochastic Processes
keywords: Borel–Cantelli lemma; Coupling; Extinction; Markov chain; Population size