1. The guy who first came with the 2 Billion figure in the 1920s was the scientist Raymond Pearl, though the copncept of 'carrying capacity' as we understand it today did not exist. Pearl's work was for the most part statistic/economic; It was Eugine Odum who later picked up that earlier work in the 1950s and applied it to the ecology concept of 'carrying capacity' which was independently developed by the observation of animals in natural environments. You can check the standard form in Odum's textbook "Fundamentals of Ecology".

The problem with the original formulation for Carrying Capacity is that it is assumed to be fixed, because animal behavior is governed mostly by insticts. Humans, even if ultimately subject to the Laws of Nature, can show much wider variations in behavior due to culture, availability of technology and many other factors. According to the wikipedia page, UN has several estimations of current carrying capacity, and they vary widely (From 4 to 11B) depending on each researcher biases and methodology.

I personally assume that the results in the higher end of the spectrum come from cornucopians that fail to take into account the economic and political presures that get in the way of implementing the (theoretically) optimal solutions, and therefore assume that actual carrying capacity is closer to the 4-5B range... but then, it's my own bias speaking there.

2. Other concept you can take from Odum is that long term carrying capacity can be eroded by organisims that happen to find a short term way to reproduce beyond the current carrying capacity of the ecosystem they belong to. This is what I was talking about in my previous msg, though I admit it sounds a bit convoluted and ranty in retrospective.

If you have a 7 seat car and you usually drive around with 10 or more people on it, (or with merely 5 fraternity bros that usually behave like baboons on meth) someone is eventually going to break one of the seats - probably the copilot one, which happens to be the least robust one. Then, you end up with a 6 seat car, at least for the lenght of the time that it takes you to fix it. And if you do not fix the seat but keep driving around with the same people on board, you are going to break another seat, and another.