Dying would mean that your pet is taken out of the population entirely (which means your account profile could have 0 pets).
A more realistic approach to this is for the admin to just keep the pet "dead" until it can be revived.
While it's unrealistic, especially for realistic pets, to starve to death and then make it where other pets couldn't interact with it because it's "dead", deleting it would not be the right path to go or it could anger a lot of players who take time away from the game, that all the sudden their level 20 pet has been deleted because it starved to death.
I had a "food stock" system where a player could hold on to so much food, and when they ran out, it would "die" (or be unable to interact with other players). The stock was easy to keep and maintain for as long as you just logged in, and prolonged game play to around 30 days of inactivity.