Meaning: The fox stopped trying to catch the rabbit and went away from the forest.
Choice of Words: Gave up means stopped trying or pursuing something. Left means departed from a place.
Meaning: This means the fox stopped trying to stay in the forest and left.
Choice of Words: The fox stopped fighting and left the forest.
Meaning: This means the fox decided it didn't want to be in the forest anymore and left.
Choice of Words: The fox gave up its right to be in the forest and left.