I have cats, Gene, and they only go part way in removing rodents --
though they drag a fine, fat rat up to the porch from time to time, I
still have rats tunneling under the barn.
Depends on the cats. Some would let a mouse build a nest on their
head. Others stalk anything that moves. When I raise a kitten I tease
with freshly trapped mice, then give them to them. Some become great mousers.