Since there was some resistance to my earlier assertion that demographers pretty much agreed that the world's population would begin to fall in the 21st century, I thought maybe I'd better go take a second look. It has been many years since I've done any fertility and childbearing research; the past ten years I was stuck at the other end of the lifespan, studying old people.
Nothing's changed; if anything, the models predict faster declines and lower levels of eventual fertility than I recall from 20 years ago. UNEP's World Population Prospects interactive website is a data geek's dream. I blame Reform Club commenters for the state of my lawn; I really should have been mowing grass yesterday, not constructing life tables.
Much was made of the fact that the UN projects population out to 2050 and at that point population is still rising. That's simply an artifact of the cut-off point that was chosen for the display. There's no mystery to how the predictions are constructed, and it's easy to extend the curves out as long as you'd like. But you don't even need to do that to conclude that the UN asserts just what I asserted: fertility rates will converge towards a rate that is less than replacement. It's right there on the methods page: based on the best models of fertility UNEP has, world-wide, fertility is converging on 1.85 children per woman, but not all countries will get there before 2050. They predict world-wide TFR below replacement (2.1 children per woman) starting in 2039. Annual growth rate peaked in 1965; excess of births over deaths peaked in 1985. Because of the lag between fertility changes and population changes, you don't see this translate into a decline in absolute population levels until later; from the UN data I calculate the growth rate goes below zero, and hence population peaks and begins to decline, in 2070.
The fertility transition is one of the most widely observed, predictable, well-established phenomena in social science. It is not a fantasy of right-wing nut job cornucopian economists; in fact it is such a bedrock of boring mainstream population science that a contrarian like me should be a little embarrassed spending so much time talking about it.