Beta Colony, the sort of European welfare state that is already breeding below replacement, has added additional obstacles on top, so that it ought to be breeding far below replacement. (Any families in Europe that have ten or twelve children are doing insufficient to keep up the population, but better than not having them.) Conversely, Pern has a heavily agrarian society where quite small children can be put to work. We have some histories of plague, but not much, and none of starvation. Yet, over the course of centuries, they did not move out and occupy new land to handle their burgeoning population. No one notices any signs of population increase.
But, on top of that, there's the question of reproducing social structures. The Black Death did not just eliminate half of Europe. It produced a powerful amount of clout for the working classes, because it made labor dear. (The upper classes were either amalgamated when one person inherited more, and so needed more state, or fell to a new person -- it didn't just vanish.) The Victorians noticed that ancient noble houses died out. Seems that lords like to marry heiresses. And heiresses tended to have few siblings. And then tended to have as many children as their mother had had. If she were an only child -- which would make her rich, inheriting everything -- and she had a girl rather than a boy. . . . Well, they were always adding new peers, but that in itself produced its own effect.
Totalitarian states have it worse. Hard to cultivate a suitable ruthless ruler when he has to grow up under the watchful eye of the state police. North Korea manages a little better with the dynastic principles, but by the third generation, we saw clear problems where the leader's sons were showing distinct signs of royal brathood -- not desirable in a sovereign. Meanwhile you have strength and prosperity growing in the criminal element.
To be sure, there's the other possibility, of allowing it to change and producing actual -- gasp -- history for the world. The problem with that, above and beyond the way that many writers don't think of it, is that it may not fit the story, which it could easily swamp if the story is about some other drama, not that of social forces producing history.