 05375 Armando G. M. Neves and Carlos H. C. Moreira
 Applications of the GaltonWatson process to human
DNA evolution and demography
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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 GaltonWatson 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 GaltonWatson 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 GaltonWatson 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..
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