Cats are part of the problem, including camels, foxes, horses and other invasive species. They went through a lot of trouble to hunt down cats and contain them. Cats wreck native animal populations. They went so far as to fence off areas and train labrador dogs so that they don't hunt native predators such as the (tiny) marsupial "badgers".

But the problem in any of those cases are the rate of population change. As per TFA, mice are fertile at 6 weeks old. What they don't mention is that their gestation period is 3 weeks, females give birth to up to a dozen pups, and can mate immediately after birth.

So in perfectly suited environments like we have here in Australia now, you're looking at a potential order of magnitude growth every 3-4 weeks.

Cats, snakes, raptors, etc have a near-flat population growth, regardless of the amount of additional food around for them.