Common Ancestors in a Generalized Moran model

The Moran model is used in population genetics as a simple way to describe the stochastic evolution of a biological population. The model assumes that the population has a fixed size N and that it evolves through random reproduction events, in which one individual dies and is replaced by a newborn. Parents of the newborn individual are chosen at random from the population. In the classical Moran model, only one parent is chosen. In this paper we introduce a generalized version of the model such that, with a probability 1− p, the new individual will have two parents. The model is inspired by a two-parent version of the Wright–Fisher model. We study the case where p is strictly less than 1. If we trace the ancestry of a present-day population backwards in time, we will find ancestors who are common to the entire population. We show that the time to the most recent common ancestor is, for large N , closely distributed around 2 1−p ln N generations. Further back in history, there will be several common ancestors alive at the same time. We do not need to go further back than about 3 1−p ln N generations to reach the state where everyone is either a common ancestor of all or not an ancestor of anyone. At this time, the number of common ancestors has a probability distribution which can be expressed in explicit form.