All my cats are farm cats, and hunting is in their nature. They live to catch things, and enjoy it. Weasels, stoats, squirrels, all types of birds, rats, mice, its a constant line of cats in the field carrying things back to the caravan. One caught a bat one day - managed to rescue that and it was ok!!! Another one, climbed a big conifer tree. I saw her coming back down, reversing, and carrying a pigeon egg in her mouth. Don't ask how she knew there was a pigeon up there anyway, and knew what was in the egg. She dropped it on the wall, and broke it, and devoured the baby pigeon,.
I do not mind the rats etc. but hate the birds they catch. Not a lot you can do, unless keep the cat in, which is impossible with mine.