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To actually model the quirks of the history and geography of the world you really need a computer simulation.
[Chang, 1999] builds a 2-parent rather than 1-parent model - in pursuit of the real MRCA, rather than just the female-female or male-male one. In his model, if we assume a constant population size, 2 parents per individual, and random mating, then we expect the MRCA to be (log2 of the population size) generations in the past. This is incredibly recent. e.g. Take population size as (a generous) 500 million to estimate the world population over recent history. Then the MRCA is 29 generations ago - say around 1200 AD!
An extreme example of earth's geography would be total isolation. Many human populations, especially in Australia, the Pacific, the Americas and the Arctic, seem to have been isolated from each other until modern times. If populations were truly isolated, then the probability of 2 individuals mating either side of the barrier may truly have been zero for thousands of years. In which case the MRCA for the world would be pushed back to thousands of years ago. Apparently [Nei and Roychoudhury, 1982] and [Goldstein et al, 1995] use DNA to estimate ages for the MRCA of 116,000 and 156,000 years ago. One wonders if they are aware that DNA cannot be used to estimate the MRCA.
In theory, cases of extreme religious isolation (or ethnic or linguistic or social isolation) could also push back the MRCA. But we know that extreme religious (or other ethnic or cultural) reproductive isolation simply does not last for hundreds of years. If people share the same territory, some of them will interbreed no matter what. A tiny minority perhaps, but that's all we need to rapidly get everyone descended from an MRCA. The only thing that will stop people interbreeding is total geographical isolation.
Whatever about the world as a whole, Chang's model does suggest that the MRCA for Europe, where populations constantly mixed, may be well within historical times. Quite likely (as is suggested by other independent evidence on my Royal Descents page) the entire population of the West descends from Charlemagne.
One wonders what Chang's model would predict for the most recent strict female-female or strict male-male ancestor. Comparing this with the DNA figures might give us a handle on how unrealistic random mating is as a model.
In nature, it is obvious that this state must be reached as you go back. This is explained in [Dawkins, 1992] and [Dawkins, 1995]. Just go far enough back. Consider ancestral fish. If I am descended from a particular one, then so are all humans.
In Chang's mathematical model this state is reached very quickly, within about 1.77 times the number of generations of the MRCA, i.e. using our numbers above, perhaps c.700 AD. So it would look like this:
Accepting that it is wrong to draw the above conclusions with locally-mating humans - despite that, these figures are in fact quite plausible (if restricted to the Western world at least).
He makes a serious attempt to model non-random mating. He sets up an abstract model of "continents", "countries" and "towns", which can be viewed not merely as geographic position but more abstractly as the pool from which one is more or less likely to choose a mate - whether that pool be geographic, religious or whatever.
He even simulates the historical growth of the world population - adjusting the birth and survival rate so that population growth matches the real numbers over the centuries from 1000 BC to 2000 AD. Interestingly, he found this made little difference to the MRCA date.
Given a reasonable choice of parameters, he estimates the MRCA for the world at c. 300 AD, with bounds of c. 150 BC to c. 800 AD.
The lowest rate of migration (and hence lowest rate of cross-breeding) he tried was: probability of leaving the "country" 0.05 percent and probability of leaving the "continent" 0.001 percent. Even with this extreme local-breeding model he still gets an MRCA for the whole world in historical times at c. 150 BC.
In other words, if you go back before the ACA point, which may be as recent as classical times, you are descended from around 60 percent of any ancient population that has children.
They rename the ACA point as the IA point - the "identical ancestors" point: the point before which all the family trees of people today are composed of exactly the same individuals (though our DNA inheritance has been different since then): "Given the remaining uncertainties about migration rates and real-world mating patterns, the date of the MRCA for everyone living today cannot be identified with great precision. Nevertheless, our results suggest that the most recent common ancestor for the world's current population lived in the relatively recent past - perhaps within the last few thousand years. And a few thousand years before that, although we have received genetic material in markedly different proportions from the people alive at the time, the ancestors of everyone on the Earth today were exactly the same." Also summarised here: "At the identical ancestors point, then, our ancestors came from every corner of the globe, although those from far afield are unlikely to have made a significant contribution to our genetic make-up."
Return to Common ancestors of all humans.