Who are the sovereigns of the Falklands?

Between Argentina and England there is this dispute over the Island Falklands (Malvinas). The islands are a self-governing British Overseas Territory, with the United Kingdom responsible for their defense and foreign affairs. Since 1833 generations of British have gone to live there and made their lives out of fishing and the other resources they can get from the islands. 61% of the population is Falkland Islander, 29% British, 6.5% Chilean & Other

The island is a paradise because everyone has a job, get paid really good amount of money, children go to a British school for free and they are offered scholarships to study in England for their college and university degrees. People speak English, they drive on the left side of the road, have post office boxes and telephone booths just like in England.

            So, who are the sovereigns of the Falklands? The people that have been living there for ages, long generations or the ones that claim territory because they are near and discovered they had the potential to start exploiting the land and sea for oil?