05-375 Armando G. M. Neves and Carlos H. C. Moreira

Applications of the Galton-Watson process to human DNA evolution and demography (305K, pdf) Nov 1, 05

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Abstract. 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 $\overline N$ is constrained to a small range depending on the probability $p$ that a child is a female. Assuming that the value $p \approx 0.488$ valid nowadays has remained fixed for thousands of generations, the range where a mitochondrial Eve occurs with sizeable probability is $2.0492< \overline N < 2.0510$. 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 $\overline N$ 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..

Files: 05-375.src( 05-375.keywords , eve_adam.pdf.mm )