Back in the day, AA used to be by far the most dominant carrier in the Caribbean with their 35ish Airbus A300s. At 267 seats and a huge cargo capacity, they were just about everywhere (including a particular JFK-SDQ flight in November 2001 ). The type was retired in 2009.

As diburning noted, JetBlue entered the market with lower fares than AA and others, and arguably better service. Many Dominican travelers prefer to purchase tickets close-in to departure and in person, which is why JetBlue opened ticket offices in downtown locations. However, over the years (as discussed on other threads) that service level has degraded and fares have gone up. But it is at the point where many people don't even think about other airlines -- if they are flying, it is on JetBlue.

-J.