Rats are terrible all right, though stoats are worse. There has been some success with new locally developed trap technologies. These ones use a CO2 cylinder to strike the rats or stoats, then reset themselves automatically. http://www.scoop.co.nz/stories/BU1402/S00802/unprecedented-0.... Much better if you're a city boy who doesn't really like dealing with traps and dead animals too much.
We use a bigger version to catch possums. Its hard to tell how many we catch though, as the feral pigs like to carry off and eat the dead ones. Its really a fascinating logistical problem trying to eradicate a broad spectrum of pests.
That's basically what rats do. I've got pet rats and they're cute and intelligent, but they're rodents optimized for breeding. That can breed so fast it isn't even funny and they destroy everything with their teeth to either eat it or build nests from the bits and pieces or just to test it out with their teeth.
Rats are crazy awesome. They're not mice or hamsters or gerbils. They are so much more intelligent and can have a personality. And I'm convinced that hamster personality is a scale of how likely it is to bite you
That being said, wild rats are very destructive because of their hardiness and intelligence. Raccoons may be the top animal for intelligence in urban environments, but they can't match rats for their destructive potential.
Many predators live in balance with the population of their preys. When the predators are too successful, the prey population will diminish, which causes the predator population to shrink, which is good for the preys, etc.
But omnivores like rats can wipe out an entire prey population and then just switch to some other source of food.
Rats devastate island populations in particular because those populations aren't evolved to deal with them. Relatively remote islands create a sort of pocket ecosystem where new organisms can't easily arrive from outside, so all the resident species evolve to deal with only the local threats and to fit snugly into their niche. But rats are the end product of a ongoing continent-wide evolutionary battle; they're flexible, they're great at dealing with new situations, and the locals aren't.