So I wonder, what are the factors that influence how long it takes for the barbarians to take over? A few that come to mind:
(1) Geographical factors. The U.S. and the U.K. are separated by ocean from the Muslim world, so we're in a less exposed position than, say, Syria. The freezing temperatures of Russia could also protect them from invaders who aren't prepared for the cold.
(2) Economic health. According to some accounts, it was bad economic policy that brought down the Roman Empire. A strong economy is a force multiplier, since it can fund production of more and better weapons, along with food and everything else needed to run a military and mobilize vast armies. A country's warmaking potential comes down to not only the number of laborers, but the productivity of that labor, which depends on training and capital. A strong economy, with full employment, also tends to keep the country more politically stable.
(3) Ability to assimilate barbarians (including both immigrants and conquered peoples). This goes hand-in-hand with economic health. If the economy is strong, then the barbarians usually want to join the civilization rather than try to beat it. They usually don't fully culturally assimilate in the first generation, but their kids typically do. When people see success, they tend to want to imitate it if they can.
(4) Technological superiority. Even when the barbarians are numerically superior, guns, germs, and steel can do quite a number on them. The stronger the economy, the more it can afford to spend endless amounts of money on technology to fight the enemy from a distance, while suffering few casualties.
(5) The will to rule. The British Empire probably could have maintained control over India, except that it wasn't willing to resort to ruthless enough tactics. They could have just shot Gandhi and the others who disobeyed them, if they had really wanted to maintain their dominion. But their will to rule had weakened.
Historically, it has been possible for relatively small European civilizations to control vast and populous empires. (In the British Empire, the ratio was
10:1.) So maybe the U.S. can hold on to its territory and sphere of influence for a long time even as its population shrinks.
There are some groups that haven't assimilated well into western culture, but how much of a threat do they really pose, aside from their participation in the political system? Most of the Muslims who try to bomb us fail, and the ones who succeed, do so only because we left gaping holes in our security, e.g. by not securing airline cockpits. That almost falls in the category of lacking the will to rule.
It's almost as though our politicians wanted 9/11 to happen, so that they could get the people to unite behind them and support going to war and taking away a lot of civil liberties. It's almost as though Osama bin Laden was a pawn in a game of 5D chess. Hmm.
Anyway, the U.S., E.U., and Russia could, even now, with their low fertility rates, still team up to totally clobber the whole Arab world, take everything they have, and utterly suppress Islam worldwide, if they wanted to. It would require only the will to rule. But, that sounds like a lot of work, and the west has become decadent and lazy.
Also, the western world has adopted a strong belief that every life and culture is precious and must be nurtured, rather than destroyed, so that it can reach its full potential, even if it doesn't seem all that promising (or maybe ESPECIALLY if it doesn't seem all that promising). This was something Ayn Rand wrote a lot about. The Stoddard Temple has to be converted into the Stoddard Home for Subnormal Children, because their need is greater.
So, there's Muslim terrorism? That just means that they're a problem child that needs extra resources. Therefore, Libya, Egypt, Syria, Palestine, etc. must be taken away from secular leaders and given to the Muslims to rule over. Then we can have peace in our time.
People speak of barbarians coming into the U.S. and corrupting and undermining us, but we could also do the same to them. A lot of non-western countries still have patriarchal cultures that encourage traditional marriage and childbearing, but our culture infiltrates theirs (via Facebook, feminist propaganda, etc.) and they end up becoming as bad as us in those ways.