Fact #1: Countries with advanced economies ALL have low population growth.
Fact #2: Countries with underdeveloped economies pretty much ALL have high population growth.
It's economics. As a group, people will have lots of children when it pays -- and vice versa.
In poor countries -- an additional child is basically just an extra mouth to feed (cheap as heck) -- and they can start helping out in just a few years.
In rich countries -- people have higher expectations (colleges, etc.) and children are expensive as heck to raise.
People who fret that population growth will just continue until humanity consumes the whole Earth have a fundamental misunderstanding of population growth. Poor folks consume very little. But as their economy industrializes and they grow wealthier -- they reduce the number of children dramatically. Pretty much always. Economics.
Remember when we used to fret about folks in Spain, Portugal, Ireland and Italy multiplying like rabbits? Well, now that they are relatively rich and developed, they have some of the LOWEST birth rates in all Europe!! And in Asia, newly rich countries like Taiwan, Singapore, S. Korea now have extremely low birth rates as well. China and India's newly emerging upper-middle class also have very low birth rates. Economics.
The BEST way to solve high population growth in any country is to develop (industrialize) the economy -- at least to middle-income status!! That's it!
By the time a country has developed/industrialized its economy, high population growth would have solved itself. The problem facing some rich countries is often a different one: curbing high per-capita consumption (a big problem in the US).
Back to Steve O's 2 scenarios -- taking the US as a whole, scenario 1 is quite common whereas scenario 2 is actually quite uncommon -- for the reasons stated.