Most likely you have a stray cat or an animal that is catching rats in the area and bringing the to your doorstep and leaving them. It is not uncommon for an animal to catch a prey item, in this case a rat and play with it until dead and leave it when it no longer will move.
Since the first find that you made was just the Rats head, whatever animal may have left the part of the rat may also have eaten some of it. I would check around the area for stray animals and especially cats, as they will do this quite often.
If the animal feels it is your door step is a safe area they will leave it there. Some animals, like the cat next store may be bringing you a gift of prey. I have seen this quite often, especially with cats. If it was a fox, they may have felt it was a safe place and gotten scared off, before being able to do away with the whole rat. Joan
sorry one more thing. is there anything else I can do to discourage it?
By washing down the area well with some bleach water may help. You also may want to use an enzmatic cleaner to help get rid of any scent (may not be dectable to you) from the prey or other animals.
A motion dector light that goes on when something is at the doorstep may also help so if there is movement, it scares the animal off. Joan