I agree that it is a matter of time before humans must universally accept some system of "family planning" that discourages families from having more than two biological children. ... But presently I can't imagine what disastrous events would have to occur, to make people consider an over-breeding problem with humans in the same terms they would consider an over-breeding problem with cats: Where the only solution is to physically restrain them, against their will, from bearing additional children.
And say one nation adopts it, while a neighbor refuses -- and then an illegal immigration problem develops, when their standards of living diverge ... This is an argument strong enough to trigger a world war. When millions die in that conflict, what will the victors do? The population has, after all, declined by the killing. Why uphold your restricted breeding program, and deny yourself children, when you can instead kill a foreign devil, and make more room?
I honestly don't know what anyone can do to alter this situation. We'll be fighting elbow-to-elbow, shedding blood for arable land, until the very last day when the Earth bursts into flame and goes tumbling into the sun.
It's disappointing to see human nature, to look at my own two hands and know that I personally am part of our collective undoing. But if a single lifetime must end -- must stop well within the bounds of infinite time just like it began -- why not the same for the whole damned human race? Yes, the idea is fatalistic -- but so what? We don't need to be eternal as gods, we just need to do our best as humans.
... And failure at that, we have no excuse for.