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Armando G. M. Neves and Carlos H. C. Moreira
Applications of the Galton-Watson process to human
DNA evolution and demography
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..