The island of Providenciales struck me as pretty sparsely populated for being the most built-up island of the Turks & Caicos Islands so I looked it up and, indeed, TCI has a fairly low population density for a country that doesn’t feature vast yawning wilderness or anything. I wondered if that was typical of the region, and it turns out not to be:
For comparison’s sake, St Martin is like Taiwan or the Palestinian Territories—very dense, denser than any American state. Puerto Rico is about as dense as New Jersey, Lebanon, or South Korea. Jamaica is more like Germany, and Cuba is sparser than Pennsylvania but denser than California. The Bahamas are way down near the bottom, presumably because it’s an archipelago rather than an island, but still denser than 18 states.
My assumption is that for islands there would be major advantages to obtaining very high density. It’s extremely useful to have decent road and rail links to a major container port, but you probably wouldn’t build such a port on an island unless a ton of people lived there.