Until pretty recently, mankind never really wasted a thought about the overpopulation topic, not without a connection to war at the same time - humans always conquered territory of other tribes and national states if they didn't have enough of their own -, also the powers controlling society and state lived according to what religions said and they always spoke like "man shall subdue earth", not he shall take care of it 'cause he's only been given one.
Humans... lived with an image of themselves in mind - the powerful ones! - like it's their holy self-evident right to spread, to breed and to make the earth suit their needs. Still there are enough heads in high positions out there who live by those principles and only act on the outside like they don't.
So... it's hard to tell what could happen if mankind really suddenly put efforts into this, controlling how their own population develops and all other things that are linked to it.
Until now mankind has only seen laissez faire, or even "always spread, always invest in more", some even abusing the toxic circumstances that result from that way of keeping the development of things.

What I wanna say is... Have they really tried before of shaping the fate of their species on their own?
Before they only lived like the big egoists, feeding themselves on sumptuous circumstances, taking for granted that they never cease to exist. The principle also worked out for a pretty long time, only it's starting slowly to get short in it.
So... I don't know what it might look like if they really understood the seriousness of their situation and effectively did something about it than fooling themselves all the time, pretending it isn't so or burying the head in the sand.