This isn't actually true, the first world for instance have either zero or even declining natural population growth rate when they are far below their carrying capacity, see Japan and Russia's -declining- population.
Their are only two Advanced Economies (which First World has become Synonymous with) that had negative population growth between 2000-2010: Germany and Estonia.
Of the other countries with negative population growth, all but one of them (Greece) is either part of the former Eastern Bloc or a very small island country.
Their are only 2-3 countries that approach population stability, and that's only because they're small and grow slowly.
Japan grew by 1.1 million people over the last decade and the negative population growth only started in 2008.
Russia, while having seen negative growth rates since the 90's (a result of economic collapse, but also immigration to the ex-SSRs), has seen positive growth rate the last few years and has a government that's actively promoting population growth.
And the fact that birth rates rapidly declines towards replacement rate in every developing country (see India or Brazil's birth rate over the last 50 years with no state imposed population control).
That is a myth, while birth rates do decrease as a result of their no longer being a 50% chance of kids dying before they're adults and not needing to birth a village to survive economically, the idea that they all will become low is not correct, afterall Africa itself is'nt following the trend and in the post-industrial countries birth rates ar starting to either stabilize or grow.
IIRC replacement level is 2.1
The replacement level is 2.1 in the West/Developed countries, however globally it's 2.33.