Population problem

PQ, Morale, and Bonus to Growth.

I think the Pacifist party gives you a bonus to pop growth, and there are some bonuses you can "buy" that also give a bonus. Other than that, the planet's quality and/or random colony events determine its growth. It's very beneficial to build a Soil Enhancement as your first Social Project on every planet, because it increases pop growth relatively quickly.

So does Habitat Improvement. A very nice early tech.

In my experience, pop growth is pretty steady whether morale is as low as 55 or the max 100, so having super morale does not fix the problem. Therefore, the cheapest solution is the two low level medical technologies that give +10 growth each, unless you are going to spend empire points to buy a faster growth rate at the game setup. A more expensive fix is to build or trade for aphrodisiacs for +30 more. A much slower long term solution is to have Earth (or your best PQ colony) build every morale improvement possible to keep morale in the breeding range and have it build transports to ship a billion people at a time to low pop planets that need help. The ships then can be re-used for invasions later if that is desired.

As for the theory of how the aliens do it, I have an observation that I have not tested. I notice that some aliens like to keep high homeworld populations at the beginning of the game by lightly loading their colony ships. Of course, they are aliens and who says that they have the same base pop growth as humans? Maybe some of them breed like rabbits!