There is not just one conflict in the novel. Several events, or conflicts, occur to help carry the story along. If you want to put it in a nutshell, you might say the conflict is the hardship of life in 19th century Texas.
The overriding conflict is that the father is away on a cattle drive, leaving his wife and young sons to fend for themselves. Having to be the "man of the house" puts Travis into conflict with his little brother, Arliss, who wants to play and doesn't recognize Travis as an authority figure. Arliss creates another conflict when he finds the "old yeller dog" and begs to keep him. Travis is set against keeping the dog, but eventually he begins to grow attached to the dog as well. And that attachment creates another conflict when Yeller is infected with rabies.
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The main problem is that Travis must protect Old Yeller from hydrophobia because how much he supported the family.