Armando G. M. Neves and Carlos H. C. Moreira
Applications of the Galton-Watson process to human 
DNA evolution and demography
(305K, pdf)

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..