Any research done on relationship between fertility rate and wealth? I am discussing about evolution theory with a philosophy student and she observes that wealthier families tend to have fewer children, which kind of goes against natural selection, i.e. the genes of smarter/healthier people do not necessarily pass along the generations. 

Now I am thinking:


Is there really such a pattern? If yes, 
Does it vary across different environments (e.g. developed vs developing countries)?
What is the economics explanation?


References to papers are very welcome. 

I am not sure how to tag this question, feel free to edit.
Population changes slowly over time but has large effects in the long-run.
Here are some well-studied facts about fertility: 


Before the industrial revolution, the rich had more surviving children than the poor.
The transition from stagnation to economic growth is accompanied by a demographic transition from high to low fertility.
Today, both within and across countries, rich and educated mothers have less children than poor and unskilled ones.
Most of the literature finds that the income of the father positively affects fertility, while the income of the mother negatively affects fertility.


Economists would stress that one explanation among others is Parents' income: Opportunity cost of child-rearing time is high for high income/education mothers. They are other reasons.

David de la Croix works on this topic and offers interesting lectures on Fertility, Education, Growth and Sustainability here. You will find references for the above facts and other reasons for the decline in fertility.

An academic reference is: De la Croix D. and M. Doepke, Inequality and growth: why differential fertility matters, American Economic Review, 93, 1091-1113, 2003.