I try to keep up with this stuff, and I am aware that rich countries
are currently not maintiaining their populations solely by reproduction.
The increase in rich countries is by illegal immigration. (This is not
a complaint, it's an obeservation.) However, the poorer countries,
especially in Latin America and in Asia, have rapidly increasing
populations. Even in Africa and even with the AIDS epidemic, the
populations are growing. The trend you mention actually depends on
increasing wealth, which in turn depends on increased technology.
If the population peaks in 2037 using existing techology with no
advances, it will do so because the average per-capita income reaches
roughly what rich countries consider the poverty level. At this level of
consumption and with no advances in technology, the population will
almost certainly destroy the Earth's ecosystems within 50 years.
The way out of this mess is to continue to deploy new technology,
which should allow us to reach the per-capita levels of wealth that
correlate with zero population growth a much lower demand on the
ecosystems. Of course, IMO we will actually end up with radical
changes to our civilization which render this entire discussion
irrelevant. For example, If I can use nanotech to establish myself
in space, I will make no further demands on the Earth's ecosystems.
If I can upload copies of myself, or if I joinb with one or more
others to create a composite copy, I have no biological constraint on
reporduction. Much more likely, we will spawn a new civilization based
on superintelligence(s), the results of which are incomprehensible.