The man acts upon his instinctive pride rather than his actual instincts. He would much rather prove something and nearly die attempting to cross the trail than wait a day and make it across the trail with ease. The dog acts on the second of his instincts instead of the first, “It knew that it was no time for travelling. Its instinct told it a truer tale than was told to the man by the man’s judgement.” His first instinct is to find a warm place to hide and survive there, but his second instinct is to follow the creature he knows that can give him food and build a fire.
Generally as the temperature goes down as do the animals in a sense that they hibernate. These animals hibernate in different ways depending on their species. For example take an ordinary bear the bear as a species goes into a full hibernation for most of winter, sleeping in warm caves.
On the other hand we have animals like the dog in the story and he does not hibernate. The dogs instinct is to curl us and stay warm conserving his energy and being much less active. Yet when the man comes one he knows will give him food and warmth leaves the dog follows not wanting to be without food and warmth.
So instead of doing as his first instinct would he follows the second instinct which is to go with the man who will give him food and warmth.
As the man is dying he takes his last breath to acknowledge the old man who’s warning he should have heeded, “You were right, old hoss; you were right,” the man mumbled to the old-timer of Sulphur Creek.” The young man then slipped into his final slumber, the dog waiting patiently nearby for the man to make a fire. “Later the dog whined loudly. And still later it crept close to the man and caught the scent of death. This made the animal bristle and back away. Then it turned and trotted up the trail in the direction of the camp it knew, where were the other food-providers and fire-providers.” Once the dog realizes that the man can no longer give him food or make a fire he leaves in search of others that can do the same so that he may survive.