Does your cat hunt - and how do you deal with it?

Our indoor/outdoor-fenced-yard cat hunts indoors. I have to admit she has caught a few mice over the past few years. She is normally an older, somewhat languid kitty but if she detects a little mouse-rustling, it is amazing to see her snap into a tense, on-point hunter mode and scout around for the source of the noise. We don't mind if she does this indoors. Mice can carry hantavirus which is no joke for humans to contact. When she kills a mouse, she does bring it to us and I dispose of it in the backyard.

Cats have a deleterious effect on songbird populations. However, I've never either seen songbirds on the ground (we do have elevated feeders) or have seen her catch a bird of any kind. That's her (Carolina) in my avatar, incidentally.