In Australia, where cats are not a native animal, problems arise when domestic moggies escape into the bush and go feral. A breeding population of feral cats will quickly revert to grey-tabby or red-tabby coloration. They tend to be large-sized as the smaller, weaker kittens have a lower survival rate. Predation on native animals can be catastrophic.
Measures such as desexing, fenced cat runs, curfews and triple-belling are part of the response to this problem.
Nevertheless, even a well-fed domestic cat will hunt, because that behaviour is hard-wired into the species. If the cat is permitted outside at night, it will range over a surprisingly large area looking for prey. However, the chase-catch-kill-eat pattern is often truncated, in a pet cat, to chase-catch-bring home for the delectation (eeew!) of my humans.
We used to have one of those "come & go at will" cat flaps for our own pet cat, but it was locked shut permanently after she brought into the house two live venomous snakes, one baby king-brown and one red-bellied black. Now she has to pass a visual inspection at the door before we let her in the house. She is quite capable of miaowing loudly to be let in while clenching a mynah-bird in her jaws!