Interesting question. I'll put as many references as I can, but I don't know if there's enough material addressing many of these questions directly. I'm also not a bitcoin pro, but I'll explain as best as I understand. Anyone feel free to correct me if necessary.
Remember that bitcoins are just numbers that are a key to unlocking other numbers stored in the blockchain (
wiki Private_key explains this somewhat.)
If the people take their private keys
and a copy of the blockchain from the time they left, they can transfer coins among each other on the island using their own nodes over their island network, with their new fork of the blockchain. They can also mine on that blockchain. Keep in mind that it is a fork -- they are creating their own transaction history, which sort of means these are no longer Bitcoins, but IslandCoins.
"If multiple different software packages [...] disagree on the protocol and the rules for transactions, this could potentially cause a fork in the block chain, with each faction of users being able to accept only their own version of the history of transactions." In this case it is multiple networks, isolated from each other, but the end result is still two chains with disagreeing history.
I will use the term "mining" for the process of maintaining the blockchain, and "coin generation" for the process of receiving coins in exchange for maintaining the blockchain (as currently happens.)
Mining [...] is the process of adding transaction records to Bitcoin's public ledger of past transactions.The problem would be in trying to take any of the coins back to "civilization." There is no way to transfer their newly-generated coins onto the mainline blockchain -- at least not without re-doing the generation (at the much higher "mainline" difficulty level.) In fact, the generation of IslandCoin will "taint" the island's transactions, so that the only transactions that can be taken back to the internet, are ones that don't touch new coins at all. As time goes on, most of the transactions will probably become tainted by IslandCoins. Because of this, and the fact that the Bitcoin blockchain doesn't have any of the island's history, our carrot-farmer's wallet is useless, at face value. However, if he also takes the island's blockchain back with him, he should be able to replay pure-Bitcoin transactions to his addresses* back to the Bitcoin network, giving him access to any coins that were not tainted by IslandCoins. (*To his addresses, or anyone else's on the island's, for that matter. In fact, if a single transaction sends to both his address and someone else's, he will have to leave it unchanged when importing it to the Bitcoin blockchain. However he still would only be able to spend coins from addresses he owns.)
The wiki on Transaction verification is the best place I can think to back this up, but it's pretty involved.As weex pointed out, mining difficulty would be sky-high to begin with, so they would have to either wait for it to come back down, or maybe modify their nodes to reset the difficulty.
Since coin generation will ruin the value of their pre-owned coins, if there was any chance anyone would want to take their coins back to civilization, it would be best if nobody generated coins on the island. They can subdivide the existing coins as small as they like (well, no smaller than can be taken back to the main blockchain, if that's an issue.)
One last note on mining that just came to mind: I think any mining fees collected on the island would also be unspendable on the mainline chain, since they are assigned through coinbase transactions that can only be created by the block miner, not relayed to anyone on the network.