This is not a present perfect. It is a modal construction; the modal is "may".
In this context, "may"* means "it is possible that ... [present or future]"; "may have" means "it is possible that ... [past]".
Thus, "We believe they may reach the island on a raft" is "We believe that it is possible that they will reach the island on a raft".
And "We believe they may have reached the island on a raft." is "We believe that it is possible that they reached the island on a raft".
With the modals, the only two possibilities for tense are the modal alone (making the proposition which follows non-past) and the modal followed by "have" (making the proposition which follows past).
*without the "have".