Your pet may be domesticated but it is an accomplished hunter. Even if there are no mice, a cat will hunt birds, rabbits and other small animals. It does not matter if you feed your cat; its natural instincts require it to hunt. You will never have worry about your cat going hungry. It will take care of itself. If you watch your cat you will see that the hunting instinct is paramount. Even when the cat is sleeping it is ready to pounce and seek prey. You can watch your cat dosing on the deck one minute, and in the next minute it will be returning with the evening meal of fresh rabbit.
So what should you do when a cat presents a gift of fresh kill that was lovingly placed at your feet? Show your appreciation! These gifts show that the cat considers you a member of the family. You should pet your cat to show your appreciation, speak to it lovingly and give it a treat. There is nothing that you can do to make a healthy cat stop this behavior. It is not possible to discourage it.
Don't declaw your cat. Hunting is part of its nature. Declawing will make hunting a bit more difficult for your cat, but it will not affect the thrill or the inclination to hunt. If your cat was lucky, hunting lessons may have been provided by its mother. Mother cats will train their kittens if they can. But your cat does not need these lessons to be an accomplished hunter; the hunting instinct is very powerful. Those who learn from their mothers are masters, but the rest are extremely effective amateurs.
Hunting is so essential to a its nature that it uses the activities of hunting as play. Have you ever watched your cat playing with a toy mouse? It helps if the toy moves on its own or if you are moving it. The toy is prey and a cat can get bored when the prey stops moving or reacting. You can tell that by what is left behind after an outdoor hunting session. Some of these animals are not eaten, after the cat killed them, it was no longer interested in it.
Continue to give your cat healthy food, and enjoy their hunting skills by playing with your cat, and continue appreciate the gifts that it delivers to you. Don't try to keep the cat from hunting unless you like working against the inevitable. Hunting is just part of being a cat and the exercise keeps your cat in the pink. So what can be done about the hunting instinct? Absolutely nothing, just love and appreciate your cat.