It’s not quite the Caribbean, but I live in the Florida Keys (Key Largo) and population growth is stagnant here too. However, I think the reason is readily understandable. Environmental regulations, many of them to preserve mangroves, swamps, and coral to mitigate the effects of hurricanes, greatly limit new building and growth.

I don’t know how well that translates to the Caribbean islands but hurricane mitigation is obviously a big concern throughout the region.