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it's easy to believe that from say, 1500BC to 500BC that at least one guy visited the Americas from Asia, and shared genes. We don't have photos but it's simply not reasonable to say it didn't happen.
Since then, 2,500 / 20 = 125 generations have passed. Remember that this is 125 doublings -- a factor of some 40 digit number. This would spread the guy's genes to every single human in the Americas.

Not at all necesarily true. Among my family, in my generation, two out of three siblings have no children. In my parent's generation, (father's side) one out of two have no children; mother's side, also one out of two have no children. So right now, my four grandparents have a combined total of ONE great-grandchild between them (and probably won't have any more, either.) That's three generations removed. If that kid becomes a priest, or dies young, all four grandparents will be out of the descendents game entirely.

This kind of thing happens a lot, especially in societies with high infant mortality. Which means most of the human race throughout most of its history.

Another way to think about this is that for all species -- including humans -- their ecological niche has what economists call a "carrying capacity", i.e., the highest population that the food supply will support. When a population reaches its carrying capacity, it will stabilize at or just below it, with only minor fluctuations.

Living in an age of population explosion, it's hard to recall that for most of human history, the population was fairly stable. This means that populations were not doubling every generation, they were about the same every generation. In other words the average couple produced and average of two descendants. For every couple that produced four (and this was a sizable fraction of the population) there was another couple that produced none (also a sizable fraction of the population).

Unless a newcoming interloper's genes are better fit (in Darwinian terms) than those of the native population, there is simply no way to be sure that his descendents (if any) will not die out in a few generations or less.

Another way to think about this is that for all species -- including humans -- their ecological niche has what economists call a "carrying capacity", i.e., the highest population that the food supply will support. When a population reaches its carrying capacity, it will stabilize at or just below it, with only minor fluctuations.

Living in an age of population explosion, it's hard to recall that for most of human history, the population was fairly stable. This means that populations were not doubling every generation, they were about the same every generation. In other words the average couple produced and average of two descendants. For every couple that produced four (and this was a sizable fraction of the population) there was another couple that produced none (also a sizable fraction of the population).

Unless a newcoming interloper's genes are better fit (in Darwinian terms) than those of the native population, there is simply no way to be sure that his descendents (if any) will not die out in a few generations or less.

#21, #22, well put (again).