Unfortunately, I think the more realistic outcome would be that the rest of the world would be turned into farmland or pastureland.
The population trajectory appears to be ballistic, with us hitting the apex in the next 5-15 years (educated women have less children and prolong having children until later in life; almost all first world countries are already below replacement rate, India and China won't be far behind).
The problem is not population (or even food production) but energy consumption. But that effects our first world lifestyle so no one in the first world wants to take it seriously.
The first world is rapidly getting more efficient with regards to energy usage, and while I can't speak to the rest of the world, the US is quickly deploying renewables. With wind generation alone, we can supply 10x our annual energy consumption.
Enough sunlight hits the earth ever 5 minutes to power the entire world for a year. Its not an energy problem, its a capture and distribution problem.