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From: elturner@phoenix.Princeton.EDU (Edwin L Turner)
Newsgroups: talk.religion.misc,soc.history,sci.bio
Subject: Re: Some statistical analysis of "Eve"
Message-ID: <4616@idunno.Princeton.EDU>
Date: 10 Dec 90 17:01:18 GMT
References: <15566@cs.utexas.edu> <1613@sun13. <15644@cs.utexas.edu>
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In article <15644@cs.utexas.edu> turpin@cs.utexas.edu (Russell Turpin) writes:
>The subject is the theory of an "Eve", one woman who is the
>matrilineal ancestress of all living humans.  
>In a previous posting, I described how for any fixed population,
>an Eve is expected to emerge every N generations, where N is a
>number that is a statistical function of population and mating
>patterns.  

This problem is best thought of as one of asexual reproduction; basicly we
are tracing the evolutionary tree of mitochondria, which could be thought of
as asexual symbiotes living in human cells.  It has a variety of analogies
in other problems (e.g., the dwindling of the pool of family names in a
culture which only passes on one gender's family name to offspring).

>In article <1613@sun13.scri.fsu.edu> mayne@vsserv.scri.fsu.edu (William (Bill) Mayne) writes:
>> ...The calculation of your N is further complicated since human
>> population is not in a steady state. ...

In a sufficiently rapidly growing population (see below), N goes to infinity;
in other words, mulitple pure maternal lines survive indefinitely.

>> ... Still, I would venture the conjecture that for a population
>> with 1000 females N would exceed the number of generations
>> since humans first appeared, probably by a huge factor. ...
>
>Mr Mayne guesses wrong.  

It sounds as though Mr. Mayne may be thinking vaguely about some sort of
combinatorics problem, but this is actually a sort of random walk in log
space problem, so N is not so sensitive to the population size.

>An analytic solution is beyond me, but writing a probabilistic
>model for this problem was a simple exercise, and provided me a
>good excuse to try out a new C compiler on my Macintosh.

I often wonder if computers will put an end to mathematics (as in that old
Asimov story, title?).  Anyway, a friend of mine and I worked out the
problem when we first heard about the Eve stuff about 10 years ago.
Unfortunately, I haven't easily been able to locate the calculation in my
notes, but I remember the answer:  A population with F females (or more
accurately, F independent mitochondrial lines) will carry only a single
maternal line (only a single mitochondrial) lines after just N=xF
generations where x is a small numerical factor (2, 3/2, pi, ...) whose exact
value I forget.  Thus, a thousand females will produce an Eve in well less
than 10^5 years.

We did not solve the general problem in a growing population but did show
that no Eve was ever likely to emerge if the population were growing 
geometricly with a doubling time short compared xF generations (which still
allows very slow growth).

I am sure all of this must be well known to biologists studying population
dynamics.

One interesting thing we do know about Eve is that she had two or more
daughters both of whom had one or more daughters ...

Ed Turner			"Do you want to know a secret,
phoenix!elturner		 Just between you and me?
				 I don't know where I'm going;
				 Yes, I don't know who I'm going to be."
or
elturner@phoenix.Princeton.EDU                 - The Other Side of This Life