So in my theoretical example, a group of 50 of so people takes 100kBTC and goes to live on a desert island.  They have a local network, but no connection to the internet as a whole.  They attempt to run an economy.

What problems might arise?  Would mining still work?  Could transactions be verified?

Now consider that someone takes a copy of their local wallet with 500BTC that they earned growing carrots back into the world and tries to spend it.  Does it work? 

Is there any way to adapt Bitcoin to solve any problems with this scenario?

Can someone point me to a reference that would answer this type of question?