I found this on the internet:
When Rats Leave a Sinking Ship
During the first century A.D. Pliny the Elder wrote in his Natural History that "when a building is about to fall down, all the rats desert it." A more modern proverb suggests that rats always leave a sinking ship.
This proverbial phenomenon has been observed for centuries and led to the belief that rats possess some mystical power to anticipate disaster. A dread warning that a ship would be making its final voyage was foretold if rats were seen scurrying off the ship before it sailed. In The Tempest, Shakespeare described the boat on which the duke and his infant daughter had been set adrift as so unseaworthy that "the very rats instinctively have quite it."
It is not unusual for rats to leave a ship in great columns or masses if it is sinking. Therefore the sight of hordes of rats scurrying to upper decks of a ship might suggest to fellow passengers that the ship is on its way down. This has nothing to do with the rat’s extrasensory perception or powers of prognostication but with its awareness of what is happening at that very moment.
Being burrow dwellers by nature, rats live in the deepest recesses of the ship, in the bilge. This area is so low as to be almost inaccessible to the sailors. Thus the rats become aware of water entering the ship some time before the crew is alerted. As their nesting places are flooded, the rodents are impelled to flee the ship. Their continuous shrill cries of alarm quickly summon the rest of the rats from the hold. They build up into a large, frightened mass of rodents making a panicky exodus. This is, of course, a final calamity for the rats, since they will try to swim to eternity and usually do.
It is quite natural that a sight such as this would incite the passengers and crew to an equally hasty, harried departure but one that ends less disastrously.
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