As long as creatu are on your profile they constantly need feeding to prevent them starving and dying (you can also stock up your food pen to automatically feed them)
However, to take some of the pressure off, you can put your pets into your showroom (click on the lil dog up top left) - pets in the showroom don't require feeding but also won't age at all, so putting them in there essentially freezes them until you remove them. The same applies with pets in rancher shops.
Hope that helps!