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 ≈ 0.488 valid nowadays has remained fixed for thousands of generations, the range where a mitochondrial Eve occurs with sizeable probability is 2.0492 < N < 2.0510. We also consider the problem of joint existence of a mitochondrial Eve and a Y chromosome Adam. We remark why this problem may not be treated by two independent Galton–Watson processes and present some simulation results suggesting that joint existence of Eve and Adam occurs with sizeable probability in the same N range. Finally, we show that the Galton–Watson process may be a useful approximation in treating biparental population models, allowing us to reproduce some results previously obtained by Chang and Derrida et al..