In the original Nintendogs, the puppy will eventually come back with a present once the player has resumed the game. Items obtained from a present at home are usually low in value, such as the Tissue Box.
In Nintendogs + Cats, puppies can no longer run away, but kittens can. If left unattended, a cat will leave and return with a present. Cat presents contain materials and essentials like shampoo most of the time. But very rarely, they may contain items like the Promise Ring or Gold Bar. Furthermore, this is the only other method Chicken Jerky can be obtained other than using the Pedometer.