Gene genealogy in a population of variable size

The genealogical properties of a small population with continuous overlapping generations that fluctuates randomly in size are studied using a model based on a stochastic birth-death process. The distribution of the coalescence times is presented, as well as a method for computing the expected overall length of the genealogy as a function of the individual birth rate gamma, the individual death rate mu, and the present population size. The relationship between the birth and death rates and the shape of the resulting genealogy is studied. The total length of the genealogy is shown to be maximized when gamma = mu. The joint distribution of the coalescence times is shown to be invariant in gamma and mu, conditional on the current population size, so that exponential growth of a population cannot be distinguished from exponential decline based on the shape of the resulting genealogy. The model is used to predict the probability that all genetic variation is lost from a recent founder population.