Extinction in a branching process: Why some of the fittest strategies cannot guarantee survival

Abstract

A standard approach for comparing biological strategies is to examine the mean and variance in reproductive success. These values rely on measures of the first two moments of the offspring distribution. Here we discuss an alternative, comparing strategies by their probability of extinction. We focus on the interplay between extinction and the moments of the offspring distribution. The probability of extinction decreases with increasing odd moments and increases with increasing even moments, a property which is intuitively clear. There is no closed form solution to calculate the probability of extinction in general, and numerical methods are often used to infer its value. Alternatively, one can use analytical approaches to generate bounds on the extinction probability. We discuss these bounds, focusing on the theory of s-convex ordering of random variables, a method primarily used in the field of actuarial sciences. This method can be used to generate "worst case scenario" distributions using the first few moments of the offspring distribution, which then lead to upper bounds on the probability of extinction.