Nit: If predators periodically make their way to the islands without human assistance, don't the islands have native predators, by virtue of how we've woven ourselves into the definitions?
No because the predators would not survive doing so from their distant original lands. New Zealand is far enough away for this not to be a problem - otherwise it would have happened in the historical past. At least until continental drift brought New Zealand close enough to other land mass. Whether the species then alive would have evolved enough to survive is unknown.
