The central conflict of Old Yeller is largely internal. The protagonist, Travis, has been made the man of the house when his father departs for a cattle drive. Fourteen-year-old Travis is still a child, both physically and emotionally, so this deep sense of responsibility is new for him. He has issues getting his five-year-old brother, Arliss, to listen to him, for instance.
The appearance of Old Yeller increases Travis's inner conflict. At first, he doesn't know what to do with the dog and finds the animal more trouble than he's worth as he eats the family's meat. However, Old Yeller becomes a protective figure, saving Arliss from a bear and staving off a rabid wolf. He is almost a fatherly surrogate figure, providing safety and companionship to Travis.
Travis's inner conflict comes to a head when Old Yeller contracts rabies after fighting the wolf. The dog is now a danger to Travis and his family. Travis's duty is to protect his family, but the child part of him loves Old Yeller and does not want to kill him. However, Travis chooses to fully step into adulthood and so shoots Old Yeller, his last link to childhood.