Our local shelter has a category for "Barn Cats." These are cats that don't want to be indoor cats but who need a safe stable living situation. You provide food, shelter and health care as needed. There are semi-feral cats in my neighborhood, and usually one "adopts" me. They don't bother my full grown chickens, but I'm pretty sure that other feral cats were responsible for the demise of many of the unexpected chicks that one hen hatched under a tree. We don't have rats or mice, or, for that matter, any lizards anymore. Make sure that people in your neighborhood are not using rodent poisons - rodenticides - for pest control, because there is the danger that your cat will eat poisoned rodents and be poisoned herself. I learned yesterday that the reason I don't have squirrels damaging my vegetable garden is that the chickens chase them -I have never seen a squirrel move so fast, and there was one of my hens in hot pursuit, she treed the squirrel and the only way it could get away was by climbing the tree and leaping to a nearby wall. I guess everything has a nemesis!
