But there are limits to what our land can support in the way of population increases.

There's plenty more room over there. Some figures (taken from the CIA world fact book):

USA: 280,562,489 people, 9,158,960 sq km = 30 people/sq km
UK: 59,778,002 people, 241,590 sq km = 247 people/sq km
Bangladesh: 133,376,684 people, 133,910 sq km = 996 people/sq km

So, with these figures, you could get another 1,981,700,631 people (that's an 8-fold increase in population) into your country, and still only have the population density of the UK.