Another thing, about early explorers, they weren't necesarily on a mission to find, log and report back. They would probably be well prepared to move to a new land and stay there, or move even farther away. Even if carried to the caribbean accidentally, they had most of what they needed to survive at hand. What I think would add to this debate is a discussion of their cultural tendencies when hunting and fishing. If they did so as families, they would have even less reason to return to their homelands once they found a nice place to set up.
You mean "rising." England is a perfect example, It's going like a see-saw.