What is the internal conflict that the father faces in The Road and how is this conflict resolved?

I would argue that the main internal conflict faced by the
father is actually the struggle to cling on to his humanity in a world where the few
remaining humans seem to have abandoned any semblance of the humanity that we know and
appreciate. The father, through trying to ensure the safety and protection of his son,
is forced to kill and to be ruthless to guarantee survival. We see this when the father
shoots the man that discovers them when they are hiding from the convoy of humans. We
can likewise see it when they come across the old man who is travelling by himself. The
father, out of grim experience, initially suspects the old man of being a decoy to
entrap humans such as themselves, and then when they talk to him, the father threatens
him, saying that if it is an ambush, the old man will be killed
first.

It is the son that helps the father cling on to his
humanity, both here and elsewhere in the book. It is the son that asks if they can spend
the night with the old man, and give him some food:

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Maybe we could give him something to
eat.

He stood looking off down the road. Damn, he
whispered. He looked down at the old man. Perhaps he'd turn into a god and they to
trees. All right, he
said.

Thus, arguably the
central conflict that the father faces throughout the novel, both here and also in other
locations, is to stay human and not become a brute, savage beast, killing and destroying
others like so many other humans are depicted as doing in this novel. It is his son that
acts as a check on his moral conscience, reminding him of his
humanity.