Xref: utzoo talk.religion.misc:33039 soc.history:2667 sci.bio:4123 Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!turpin From: turpin@cs.utexas.edu (Russell Turpin) Newsgroups: talk.religion.misc,soc.history,sci.bio Subject: Some statistical analysis of "Eve" Summary: Correcting Mr Mayne's intuition Message-ID: <15644@cs.utexas.edu> Date: 10 Dec 90 03:14:53 GMT References: <1990Nov14.204712.15966@wrl.dec.com> <1990Dec3.141344.1486@ujocs.joensuu.fi> <15461@cs.utexas.edu> <0bLsVz600Vpb0ftUp_@andrew.cmu.edu> <15566@cs.utexas.edu> <1613@sun13.scri.fsu.edu> Followup-To: talk.religion.misc,soc.history,sci.bio Organization: U. Texas CS Dept., Austin, Texas Lines: 109 ----- The subject is the theory of an "Eve", one woman who is the matrilineal ancestress of all living humans. Since others may have some interest in this, I have cross-posted it to soc.history and sci.bio. The results of a simple statistical model are included below. ----- 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. In article <1613@sun13.scri.fsu.edu> mayne@vsserv.scri.fsu.edu (William (Bill) Mayne) writes: > Yes, but for populations of significant size N will be very large. Mr Mayne guides his intuitions by one highly improbable case, and later tries incorrectly to extend it to the far more likely case. He writes: > The chance that in a population with n reproducing females n-1 > will have only male offspring is very remote for n of 1000 or > more. The exact probability will depend upon the average number > of offspring and the distribution. ... The probability is only > slightly less that there would arise a generation consisting > of only males, resulting in extinction. ... Actually every Eve > will represent a close brush with extinction, since there will > be only a few females in the generation after Eve. ... All of the above is based on the idea that Eve emerges as a result of all other contemporary women having only sons. This need never happen. Some of Eve's contemporaries might bear only sons, but most of them probably had descendants through matrilineal lines that lasted many generations. As Mr Mayne writes: > Granted, more complex scenarios are possible, such as other > females having female offspring but those female lines becoming > extinct as far as unbroken female lineages go some generations > later. ... This scenario, which is not the one Mr Mayne talks about above, is the *far* more probable one. > ...The calculation of your N is further complicated since human > population is not in a steady state. ... As a first approximation, this a fair assumption for most of human history. For hundreds of thousands of years prior to the neolithic revolution, the human population (like other animal populations) was limited by how much food could be gathered from the natural landscape. > ... 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. 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. The number of generations required to eliminate all but one matrilineal line are shown below. F is the number of women in a steady-state population. (Except for the largest case, I made several runs for each value of F.) #generations (N) #years (15yr/gen) F least most least most ---- ----- ---- ----- ----- 8 5 37 75 1605 128 172 625 2580 9375 1024 2392 4163 35880 62445 Thus, for a population of a few hundred, an Eve appears every few millenia. For a population of two or three thousand, an Eve appears every several tens of thousands of years. One characteristic of matrilineal descent that was made clear from the model is that the presence of living matrilineal descendants from one woman is NOT a stable point, unless she is the sole matrilineal ancestress of the entire population. The greater the portion of the population that is matrilineally descended from her, the greater the likelihood of this portion increasing. But if a lesser portion of the population is matrilineally descended from her, then it tends to decrease. It is far from clear to me that a steady population is the most favorable assumption for the appearance of an Eve. My intuition is that a population that increases and decreases, and where relatives are more likely to survive or die out together, is more conducive to the appearance of an Eve. This is also a more likely description of population change in early human groups. It is not necessary to assume that this relatively small group was ever the only population of humans; only that all current humans are matrilineally descended from them. Other groups might have existed at the same time, but died out (or were killed off) with little matrilineal mixing with this one group. Finally, it should be kept in mind that social structures can impede or promote matrilineal mixing. For example, if women remain with their original group when they mate with men from another group (the man either remaining with his own group or changing to the woman's group), then there is NO matrilineal mixing between groups. This remains so even if people religiously mate outside their own group. This may have been an important factor in determining mitochondrial DNA inheritance. Russell