The setting of Old Yeller by Fred Gipson is Texas after the Civil War and more specifically a farmstead. When Travis's dad, along with some others, has to leave for months on a cattle drive and leaves his fourteen-year-old son in charge of the farm (as well as Travis's mother and five-year-old brother), there is little doubt that the boy is going to have to do some growing up--and that is exactly what happens in this story.
Obviously Travis is more mature in many ways than today's young teenagers because he grew up on a farm and knows how to shoulder responsibility. In other ways, though, he is still just a young teenager. We watch him develop over the course of the novel, and we see it most through his interactions with Old Yeller.
In the beginning of the novel, Travis sees Old Yeller as nothing more than a nuisance. In all fairness, the dog is rather a nuisance who ruins things and eats things and causes all manner of trouble for Travis. The boy would rather get rid of the dog because the dog is making him work harder than he has to; however, he has to let his little brother keep the stray dog because his mother makes him do so. Time after time, Old Yeller proves to be a nuisance rather than a help for Travis.
One day, however, things start to change. Arliss nearly dies when he is attacked by a bear but is dramatically saved by Old Yeller. Travis demonstrates maturity by recognizing how close he came to losing his brother and how much he loves the boy; he is also forced to admit to himself that something he thought was just an annoyance was actually a valuable source of help.
When they all discover that Old Yeller has been stealing from a neighbor and Travis learns that Searcy is trying to capture the thief, Travis has to become a problem-solver in older to save the dog's life. In another display of maturity, Travis has to sleep outside to protect the crops and now values the companionship of Old Yeller. Strangely, the dog also seems to be maturing, as Old Yeller begins to act as Travis's partner.
When Old Yeller's real owner comes to claim him, Travis and his family are able to convince the man to let Old Yeller stay. Old Yeller's owner takes Travis aside, treating him like the man of the family and warning him about the outbreak of hydrophobia.
In another display of maturity (though this time perhaps he thinks he is more mature than he really is), Travis tries to deal with the vicious wild hogs, something he has done before with his father. When he gets in trouble, Old Yeller again comes to his rescue. To his credit, after his rather foolish act which gets him badly hurt, Travis does all the right things. He tries to help the wounded dog, he goes for help, and he brings help back to the dog that saved his life.
Hydrophobia appears on the farm, and once again Travis demonstrates his maturity. In trying to protect the family, Old Yeller is bitten by a rabid wolf. Travis knows what he has to do and he does it. Travis says:
He made me so mad at first that I wanted to kill him. Then, later, when I had to kill him, it was like having to shoot some of my own folks. That's how much I'd come to think of the big yeller dog.
When Travis's dad comes home, he is proud of Travis for taking such good care of things. This is the final indication that Travis changed from a boy who was annoyed by a stray dog to a young man who was willing to shoot the dog that had become a friend in order to save his family.