Wait, wait. Do you mean that the rate of increase of global population is declining? This is the other place you lose me. How do we correlate longer life with fewer children, automatically? Governments worldwide step in? Or the third world catches up to the first world in terms of birth rate? I wouldn't see this happening naturally for hundreds of years.The result of these transitions will be a human species that have very few children (if any) and live much longer (150-200) and those are conservative estimates.
The rate of increase in human population growth peaked in 1962 at 2.2%. It has been declining ever since (currently at 1.1%). All statistical models posit that 2050 will be when we plateau and the whole world will be at replacement level fertility. (Almost done the post detailing the reasons why this is happening). Also, longer life does not directly mean fewer children. However, what is leading to decreasing fertility is not going to go away (again post coming soon). Many people in our generation in developed countries will not have children. This will be the first time this has happened in human history. The more affluent, more urbanized, more educated people become... the fewer children they have. Government do not need to step in for this to happen. This is just how humans do. Of course, this is all contingent on the developing world continuing to develop at their current rates (which would see most people living at developed world standard of living by 2050). I suggest watching some of Hans Rosling's TED talks on future demographics. So I conclude personally that if we continue to decide to procreate way less - and we have technologies that will allow us to live much longer - then we will have a population with longer life and fewer children. Which will allow us to keep a stable population.