The short answer, according to speculation by veterinary medicine professor Alan Beck, is that it would be a disaster. Cats play a critical role in keeping the rodent population down. If there were no cats, we'd be overrun with rats:
Link -via @AlexisMadrigal | Photo: Flickr user wapico
By killing mice and rats in barns and grain storage areas, cats are vital for keeping those pests in check. In India, Beck said, cats are believed to play a significant role in lessening the amount of grain loss caused by consumption or contamination by rodents. In other words, it may be true that humans feed cats, but without cats, humans would have less food in the first place. [...]
And if the rodent population shot up, this would of course trigger a cascade of other ecological effects. On that same island in New Zealand, for instance, ecologists observed that, as rat numbers increased in the absence of cats, the population of seabirds whose eggs rats preyed upon declined. If the approximately 220 million domestic cats in the world all bit the dust, seabird populations would likely fall worldwide, while the populations of non-cat predators that prey on rats would be expected to increase.
Link -via @AlexisMadrigal | Photo: Flickr user wapico
WIN WIN