Spoiler (click to reveal)
The enmity on Patmos (while it remains unconquered) acts as a minimum enmity for all islands, and that enmity behaves the way enmity of those islands normally behaves -- it makes raids more difficult, prices more expensive, etc.
One thing to add to Gary's example is to be careful to include both stickers and tokens when comparing which enmity to use -- compare the total enmity on Patmos (both your stickers and any tokens you've placed) to total enmity on any given island (both your stickers and your tokens).
One thing this might mean is that when you raid an island with less "real" enmity than Patmos, the token you place won't make your next raid any more difficult (until that island's enmity catches up to Patmos). It also means that placing a token on Patmos counts against you everywhere.