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Does this mean that at one point, there were 1,300 humans that were the only population around that could have given rise eventually to today's population?
Or, that there were 1,300 individuals that happened to give rise to today's population, and a whole bunch more that didn't, but that would have if those specific 1,300 weren't around?
I wasn't able to tell from this article, and maybe the original study is a little more clever in its analysis, but there's no clear indication of which it is in this article -- which makes me think that maybe it's #2 and the person that wrote it just hasn't realized that #1 and #2 are very different statements.
(Or, maybe #1 gets more clicks, and they're well aware of that.)
It's #1, but there's plenty of room for error, and the exact number the researchers came up with was 1280. That's just what the genetic sampling says. They did the analysis on the samples that they have. It's not like they could travel back in time and take a genetic sample of every populated area in the world, so it isn't a precise number. However, they would expect more variation if there were more of the species actually living at that time. The number they came up with is useful, though. It tells you there weren't a lot of people alive to reproduce at that time and they know there was a higher population of the species before that time. Also, outside of this article, the same evidence is cited along with how it is known that there was probably a huge population collapse thanks to other pieces of evidence. Anyway, plenty of specialists in multiple fields can confirm that environmental conditions were very harsh, making it very difficult for most forms of life to survive. So, the number, as outrageously small as it might seem, is pretty close to reality.
What's their evidence that it's #1, and not #2? I read the links but I still don't see a reason why. I'm not saying it is #2, just that as you said:
Yes, which means that the only populations that are "visible" to this technique are the humans that left descendants to the modern day. Maybe there were other populations that didn't leave descendants to the modern day, specifically because of competition with other humans (but which would have survived, if not for the competition for their same niche).
Again, I'm not saying that that did happen. Just that it instantly jumps out based on a cursory reading of this that it sounds pretty possible, and I'm curious to see if there's an explanation for why it's not possible.
I'm sure there was a huge population collapse; as I understand it, that's an established fact based on a sudden absence of humans in the fossil record. But, it's a very different statement to say "there was a population collapse" versus the statement "there was a population collapse down to 1,280 individuals." One is not proof of the other.