[Qwertious]

There's a really simple solution to this: Have a cultural policy that what's normal/acceptable is to delay having children for quite a while, and then to have only one child per couple.

One child per couple means that the total population caps out at double the current population (a population of 32 would have 16 children, who'd have 8, who'd have 4, who'd 2, then 1 person who's SOL - 32+16+8+4+2+1=63). A single order of magnitude is an acceptable loss, and everyone gets a child of their own.

Note: Not as in "literally force people to have only one child", just make it frowned upon like smoking generally is - most people won't care enough to defy it, and in the long term it should be enough. Although IMO families like the Duggars[1] shouldn't be provided immortality unless they start using birth control.

[1] https://en.wikipedia.org/wiki/19_Kids_and_Counting