Islands that are occupied by other players or computer players can be taken over by other players in two different ways.
Takeover by war
A player has to first declare war on the owner of the island. Then the player needs to use military ships to attack the main harbor of the island. After the morale of the island reaches a critical point the player is prompted with 2 options. Either gain all the shares for that island or conquer that island. If the player chooses to conquer the island the player will take over the island and claim it.
If a player buys all the shares of an island the player will have the possibility to conquer the island by buying it. After claiming all 5 shares the player can select the middle of the share diagram. This is very costly.
Small trading outposts that are by default can't be taken over in any way.
If 2 or more players are attacking the same island together there will be no option to conquer the island. The only option is to accept shares for the island distributed based on how much each player contributed to the attack.