yes they can survive without food for a week, but here is a suggestion you can purchase automaic fish feeder for a week. The good one can be a little pricey but you wouldn't have to worry for a week without feeding just a suggestion.
I have left mine for over a week without any problems. I just feed them like I would normally feed them right before I leave. Also, if you decide to buy an automatic feeder be sure you test it out for a couple days before you leave so that way you know it is going to work correctly and it's not going to dump all the food in the tank too quickly or not dump any food in at all! :thud:
A human can be more than fine for a week, but would you want to go a week without food? i don't think so. I know fish are different than us, but there's no one that could come over and feed them? what about those fish pyramids? just because something CAN last without food for a week, doesn't mean it's good for it!