This is certainly not true on a regional basis, and I can hardly say with confidence that stagnating population levels in Europe really warrant African families having upwards of 10 children on average when the land can hardly feed the current population as is.
We are talking about huge numbers here if African population really does increase by 3 billion during this century. FWIW, I'm not going to stop driving and start walking so someone else's children can do the pollution instead of me and it levels out.
Your numbers are out of date. There's not a single country where the TFR >= 7. Most of Africa is under 5. That's still too high, but the numbers are dropping, and they are especially dropping in countries with increased development and urbanization.
As to the rest of your comment, I understand that people have fewer children when they trust that their children aren't going to die young, and that's the direction we're headed in.
I don't think there's a rule for that. Is it in the Scripture (because I know that less well)?