I'm not sure Lance Roberts' answer is correct. For the Mayor, the rules say this: "If there is at least one colonist on a tile, it is considered occupied. Only occupied tiles may use their functions; unoccupied tiles never function!" (emphasis theirs).
Given that emphasis on the word 'occupied' on the residence card, and the word 'never' in the mayor description, it could be interpreted as island spaces without a colonist do not contribute towards the residence card's ability, as they are not occupied and do not function.
This ambiguity could be fixed by changing the word 'occupied' in relation to the island tiles on the residence card.