This is not a planet-scale phenomenon, though. Any student of geography would know that as nations become more mature their birth rates stabilize to more or les

This is not a planet-scale phenomenon, though. Any student of geography would know that as nations become more mature their birth rates stabilize to more or less match the death rate. It is the less-developed countries that would suffer first in a world where our planet can no longer support our population, in which case I would expect that either they would go to war with each other over whatever resources can be taken from each other (and also reducing their populations in the process) or else face starvation (again, reducing population). Either way there's a large difference between starvation and saturation. The human race is not endangered by the fact that we'll soon saturate our planet, and those that do starve as a result would simply join a long line of people who have already met the same fate.