There's some interesting classes by Al Bartlett on youtube where he discusses population growth and the exponential function in general, he was a professor at B

There's some interesting classes by Al Bartlett on youtube where he discusses population growth and the exponential function in general, he was a professor at Boulder, CO. One of his salient points IIRC was when you're living through an exponential curve that's approaching some crisis threshold, be it population growth leading to civilization collapse, or some resource consumption exhausting its supply, it happens very abruptly because you jump from just halfway to the end, which through a linear lens appears very distant, to at the critical end, in just a single doubling period.