Populations in environments with a soft carrying capacity are eventually extinct

Peter Jagers,Sergei Zuyev

doi:10.1007/s00285-020-01527-5

Peter Jagers, Sergei Zuyev

Open Access

https://doi.org/10.1007/s00285-020-01527-5

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Abstract

Consider a population whose size changes stepwise by its members reproducing or dying (disappearing), but is otherwise quite general. Denote the initial (non-random) size by Z_0 and the size of the nth change by C_n, n= 1, 2, ldots . Population sizes hence develop successively as Z_1=Z_0+C_1, Z_2=Z_1+C_2 and so on, indefinitely or until there are no further size changes, due to extinction. Extinction is thus assumed final, so that Z_n=0 implies that Z_{n+1}=0, without there being any other finite absorbing class of population sizes. We make no assumptions about the time durations between the successive changes. In the real world, or more specific models, those may be of varying length, depending upon individual life span distributions and their interdependencies, the age-distribution at hand and intervening circumstances. We could consider toy models of Galton–Watson type generation counting or of the birth-and-death type, with one individual acting per change, until extinction, or the most general multitype CMJ branching processes with, say, population size dependence of reproduction. Changes may have quite varying distributions. The basic assumption is that there is a carrying capacity, i.e. a non-negative number K such that the conditional expectation of the change, given the complete past history, is non-positive whenever the population exceeds the carrying capacity. Further, to avoid unnecessary technicalities, we assume that the change C_n equals -1 (one individual dying) with a conditional (given the past) probability uniformly bounded away from 0. It is a simple and not very restrictive way to avoid parity phenomena, it is related to irreducibility in Markov settings. The straightforward, but in contents and implications far-reaching, consequence is that all such populations must die out. Mathematically, it follows by a supermartingale convergence property and positive probability of reaching the absorbing extinction state.

Highlights

Almost a century and a half have passed since Galton (1873) and Galton and Watson (1875) introduced their famous simple branching process followed by the infamous conclusion that all families (“surnames”) must die out: “All surnames tend to extinction [...] and this result might have been anticipated, for a surname lost can never be recovered.”
We prove almost sure extinction of quite general, stepwise changing populations, which can reach any size but live in a habitat with a carrying capacity, interpreted as a border line where reproduction becomes sub-critical, but which may be crossed by population size, i.e. a soft, carrying capacity
What happens is that the population size process becomes a super-martingale, when crossing the carrying capacity, and extinction follows from a combination of martingale properties

Summary

Introduction

Almost a century and a half have passed since Galton (1873) and Galton and Watson (1875) introduced their famous simple branching process followed by the infamous conclusion that all families (“surnames”) must die out: “All surnames tend to extinction [...] and this result might have been anticipated, for a surname lost can never be recovered.” Since long it is textbook knowledge, that the extinction probability of supercritical Galton–Watson (and more general) branching processes is less than one, the alternative to extinction being unbounded exponential growth. What happens is that the population size process becomes a super-martingale, when crossing the carrying capacity, and extinction follows from a combination of martingale properties.

Results

Conclusion