Applications of the Galton–Watson process to human DNA evolution and demography

We show that the problem of existence of a mitochondrial Eve can be understood as an application of the Galton–Watson process and presents interesting analogies with critical phenomena in Statistical Mechanics. In the approximation of small survival probability, and assuming limited progeny, we are able to find for a genealogic tree the maximum and minimum survival probabilities over all probability distributions for the number of children per woman constrained to a given mean. As a consequence, we can relate existence of a mitochondrial Eve to quantitative demographic data of early mankind. In particular, we show that a mitochondrial Eve may exist even in an exponentially growing population, provided that the mean number of children per woman N¯ is constrained to a small range depending on the probability p   that a child is a female. Assuming that the value p≈0.488p≈0.488 valid nowadays has remained fixed for thousands of generations, the range where a mitochondrial Eve occurs with sizeable probability is 2.0492