Nearly all first world nations have a birth rate insufficient to sustain their populations. It seems that a few generations after people are wealthy enough to ensure their children's survival, they stop having more than 1 or 2 kids.
Unfortunately that's not exactly true. Even at a measly 1% (about the U.S. rate). It only takes 72 years to double a population. But the bigger problem is 2nd and 3rd world nations. The planet cannot support their reaching the economic levels of 1st world nations in order to achieve this natural population leveling.
You are correct to point out that the U.S. population is still growing, but you are incorrect to assume that the birth rate is responsible for the increase. The current U.S. birth rate is 1.88 children per woman, bellow replacement levels. Europe is the same, birth rate insufficient to sustain their population.
Many first world nation still have growing populations which is due to 2 causes:
1) Immigration which, because we are talking about global population levels, is another way of saying that developing and third world countries have high birth rates.
2) People are living longer, which unless we solve mortality, is only a temporary factor.
So yes, poor areas have high birth rates. Perhaps those areas will stay poor or keep their high birth rates after becoming wealthy; those are valid points to make, but I think that as those area become less poor their birthrates will drop and thus global population will stabilize.
Nearly all first world nations have a birth rate insufficient to sustain their populations. It seems that a few generations after people are wealthy enough to ensure their children's survival, they stop having more than 1 or 2 kids.