The Russian federation now faces the same problem. The growth rate of the Russian population contines to be negative (despite calls from Putin to bear more children), while the relatively poor Muslim populations--generally non-Russian--are growing. Strategypage summarizes the problem:
If present trends continue, the population of Russia will decline from 143 million to 100 million by 2050. Not only that, but by 2050, most of the population may be Moslem. Currently, about 15 percent of Russians are Moslem, and the average Moslem family has three or more children, while the average non-Moslem family has one or two. While Christian (largely Slavic) Russians have seen their numbers tumble, the Moslem population of Russia has grown over 40 percent since 1989 (from births, migration and conversions). There has also been a religious revival, with the number of mosques growing from under a thousand when the Soviet Union collapsed, to over 8,000 today. That means Moslem men drink a lot less, and live healthier, and longer, lives.
In addition to a higher birth rate, the number of Moslems will increase because of migration. A falling birth rate among the Slavic population will create an enormous labor shortage, and the closest source of additional labor is poor, over populated, Moslem nations.
Dunnigan then suggests that things won't be quite so bad, because of the demographic transition (he doesn't use that term, but he notes that wealth tends to lead to smaller families). I'm not so sure. First, the greatest source of wealth in the next 50 years is likely to be the oil fields of central Asia, and that wealth is not going to be in of hands of the typical family--oil wealth is notorious for supporting cronyism, not general economic devlopment, and it will be outside of the Russian Federation in any case. Second, even if the wealth does get out to more families, these are not the people who would immigrate for low-pay jobs, and that's likely to be all that will be available for them in Russia. Third, the relationship between wealth and population size has an intervening variable--the status of women. (Dunnigan notes this, by the way.) I suspect there's a long way to go before we'll see a significant improvement in gender equality.
In other words, we may witness a major "clash of civilizations" within the Russian Federation, and soon.