There are always big predators at top of the biological food-energy chain. They are few in numbers. Anyways, I believe populations of cats and rats will fluctuate. Increase in cats will cause decrease in rats and then subsequent decrease in cats. Like phase difference between sine and cosine wave graphs.
I believe introducing some small wild cat population would be a right thing to do, tho I would like to hear opinions of experts.
Edit: I don't think cats are invasive in the way rats are
Trouble with cats is they often destroy bird populations as well, not sure how much of a problem this is in Australia, but in New Zealand feral cats are up there with pests that need to wiped off any island that is to be rehabilitated.
Feral cats are a massive problem in rural Australia. They decimate small native mammals. The idea of introducing them to control rats and save native animals is insane - the cats would wipe out everything smaller than them.
Introducing a foreign species into an ecosystem has failed spectacularly so many times I cannot believe people still propose it.
Unfortunately cats don't eat only rats, they also like birds for instance. Look at the dodo extension mentioned in another thread, domestic animals played an important role there. Then the birds usually keep (nasty) insects population in control, etc and keeping a balance is tricky.
Snakes or other crawling rat predators would probably be a better idea, but then trying to convince the island people to share their space with snakes and weird looking insects at the same time would probably not go well :)
Snakes have bad impact on birds as well. Maybe not adults, but the eggs. I believe there is always a bigger fish in the ocean, everything is just a matter of equilibrium and its rules of self-control. Humans and rats probably are the only species that ignore those rules.
A reasonable model if you assumed that cats exclusively ate rats - and wouldn't wreak havoc on any other native rodent populations (if they still exist).
Not exactly their niche. They might hunt for insects, but it won't be enough in terms of energy acquired. So probably they will reduce in population until rats' population will restore back.
And its good to keep rats busy. They won't wander as freely as they do now and hence eat less insects.
I believe introducing some small wild cat population would be a right thing to do, tho I would like to hear opinions of experts.
Edit: I don't think cats are invasive in the way rats are