What happens to the bitcoin network when the miners all stop in the future? What happens to the bitcoin network when the miners all stop, years in the future after all the bitcoins have been mined?  How will the network continue to function?  Won't bitcoins then be useless?  What would be the incentive for an individual to continue using computational power to service all the transactions?  Isn't this like a ticking time bomb or is there something I'm not getting?
Bitcoin becomes very insecure if miners stop mining. Think of how easy a 51% attack would be to pull off.

However, I disagree with your assumption that miners will stop. And certainly that if Bitcoin dies it would be because miners stop. I would instead think that miners would only stop if something else already killed Bitcoin. Bitcoin is designed to always give miners an incentive to keep mining and secure the network. If miners don't have an incentive to mine, then Bitcoin has already failed.

Many mining pools don't pay out income from transaction fees and the whole thing is often glossed over. But mining income is newly minted coins PLUS fees from the transactions you include in the blocks you generate.

Have a look at the tx fee statistics at http://blockchain.info/no/charts/transaction-fees

As I write this at the end of march 2013, it looks like tx fees were about 4 BTC per day a year ago and are now about 50 BTC. At the same time the newly minted coins in each block has dropped from 50 BTC to 25 BTC.

With 6 blocks per hour this means one year ago we mined 7200 BTC in new coins and 4 BTC in fees per day. Now we mine daily 3600 BTC in new coins and 50 BTC in tx fees. So tx fees went from being 0.00056% of the income a year back to the 1.39% of mining income they are today.

I'm sure you can see where this is going. No, mining does not stop when the minting of new coins stops.