I'm generalizing, but they probably could. I know of other instances, though, where a species has become established in a localized area and can exist for quite some time but then go through a period of breakout once the animal gets fully adapted to its new ecosystem.
This is the pattern we saw on green crab. They have been on the east coast of North America for several hundred years but confined to several bays down around Chesapeake and the Potomac area. In the last 50 to 70 years, since the Second World War, they have broken out and are migrating up the coast in a fairly aggressive fashion.