On the path to extinction

摘要

Populations can die out in many ways. We investigate one basic form of extinction, stable or intrinsic extinction, caused by individuals on the average not being able to replace themselves through reproduction. The archetypical such population is a sub-critical branching process, i.e., a population of independent, asex-ually reproducing individuals, for which the expected number of progeny per individual is less than one. The main purpose is to uncover a fundamental pattern of nature. Mathematically, this emerges in large systems, in our case subcritical populations, starting from a large number, x, of individuals. First we describe the behavior of the time to extinction T: as x grows to infinity, it behaves like the logarithm of x, divided by r, where r is the absolute value of the Malthusian parameter. We give a more precise description in terms of extreme value distributions. Then we study population size partway (or u-way) to extinction, i.e., at times uT, for 0 < u < 1, e.g., u = 1/2 gives halfway to extinction. (Note that mathematically this is no stopping time.) If the population starts from x individuals, then for large x, the proper scaling for the population size at time uT is x into the power u - 1. Normed by this factor, the population u-way to extinction approaches a process, which involves constants that are determined by life span and reproduction distributions, and a random variable that follows the classical Gumbel distribution in the continuous time case. In the Markov case, where an explicit representation can be deduced, we also find a description of the behavior immediately before extinction.