Times writer Jonathan Last describes the association between fertility rates and international immigration* by using the flow of people from Puerto Rico to New York City as an example.
This flow, largely taking place between the 1920s and 1940s, does serve as an excellent illustration of the factors that influence population change.
In the 1920s, Puerto Ricans began to trickle into the United States. Their numbers accumulated slowly, and by 1930, there were 50,000 Puerto Ricans living in the U.S. (nearly all in New York City)... But from 1955 to 2010, the number plummeted. Even though the population of Puerto Rico had nearly doubled in that time, the total number of Puerto Ricans moving to the United States in 2010 was only 4,283...What happened is that Puerto Rico's fertility rate imploded.
Puerto Rico has been part of the United States since the end of the Spanish-American war, and Puerto Ricans have been U.S. citizens since 1917. Puerto Ricans couldn't "trickle into the United States" because they were already in the United States.
Despite social commentary to the contrary, Puerto Ricans were migrants, but not immigrants.
*Using the term "international immigration" is redundant, of course, as "immigration" means moving to a foreign country, but I want to be clear in this point.