Well, there's nothing that says the Amish are homogeneous or must stay that way.

Even among Amish, there will be some sub-groups where fertility falls, and others where it stays the same or rises.

So, the Amish who reduce fertility (like those you quoted) will eventually be outnumbered by those other groups of Amish who continued to reproduce at higher rates.

Even if the differential is tiny, and the high-population group is initially a single individual, it will eventually dominate. Heck, even with zero high-fertility sub-population, one will eventually mutate into existence and then dominate. This is evolution.

Maybe it'll be 50 years, or 150, or 1,000 years. What you're saying could affect the timing of the wheat-and-chessboard outcome, but it doesn't change that outcome.

>Demographics are just not a one-way street. Certainly there are strong incentives and traditions to pop out lots of children in some contexts. But those incentives change, and do people's behaviors.

You're still imagining that the base "rules" like human nature are invariant. For example, you're talking about "incentives", as though people respond to incentives in fixed ways.

But they don't. Even if the incentives change to encourage low populations, some subgroup will quasi-irrationally ignore those incentives any over-reproduce anyway. Eventually, that subgroup and their quasi-irrational characteristics will dominate.

>We're going to be overrun by _________!' is a common anxiety that has been refuted over and over again by anthropological studies. It only makes sense in models where you assume everyone acts independently and populations and individuals don't really interact with each other.

Well first, you seem to be making some implied xenophobia accusation. Nothing in my comment identified any "we" nor any "_blank_" that "we" would be overrun by. This has nothing to do with fear. I simply said something about population dynamics that applied to all organisms of all species everywhere.

In addition, usually the identity lines aren't that clear and my point doesn't require them to be. I clearly talked about sub-populations who come to dominate their identity group, which is different from one identity group dominating another. Both can occur.

I'd ask that you please don't imply in some accusation of irrational immoral xenophobia on my part, and don't attribute emotions like xenophobic "anxiety" to me without me stating them. It's just not productive.

Regarding your narrow point, 'We're going to be overrun by _________!' has not been refuted. Quite the opposite - historically, it's come true many, many times: Native Americans got overrun by fast-breeding Europeans. Ainu were overrun by fast-breeding Japanese. Berbers were overrun by Arabs. Many ancient tribes overran other tribes (e.g. Zulu overran many other tribes). And so on.

>When someone shows me a picture of exponential growth I always ask whether they're sure it isn't a sigmoid curve.

I agree in general. But here's why I'm sure this isn't a sigmoid curve: Life began on Earth 3.8 billion years ago and population has always been exponential for every species, anywhere, ever.

The only major exception I can think of is humans since the industrial revolution, and evidence shows that this is temporary. I'd be happy to hear about other examples (though I suspect they'd also be temporary or very strange exceptions that prove the rule).

If we want it to be a sigmoid curve, we must choose to make it a sigmoid curve using policy.