In Hatchet, much of Brian's personality development can be explained by the S.T.E.A.L. model of characterization: Speech, Thoughts, Effects on other characters, Actions, Looks.
In terms of speech, Brian's speech patterns when the plane first goes down in the wilderness communicates uncertainty and fear. He talks about what happened and what he is going to do in a hurried, staccato way. This can be seen in the opening of chapter 3 when Brian repeats that he is "gonna die." However, as the narrative progresses, Brian's speech patterns become more focused. He reminds himself of Perpich's words about taking "one thing at a time." This creates more deliberate speech patterns. This is one way in which his speech reflects the development within Brian's character as he undergoes his ordeal.
In terms of thought, Brian's thoughts become more driven towards survival. Brian moves from thoughts that compel him to cry "in frustration and agony" towards thoughts of a "tough hope" that he is not going to let the conditions of the forest beat him. Brian's thoughts become trained towards survival. As time passes, his thoughts move towards a stronger mental attitude that drives him towards living. The novel's epilogue communicates this since Brian "had become more thoughtful ...."
In terms of emotions, at the start of the narrative, Brian is overcome, burdened by "the secret." He dreams about it and the impact it has on him. Brian is angry over the divorce and about the man in the station wagon. Brian's emotions develop with his character as he determines not to dwell on these "secrets." Even at the end of the novel, Brian never tells his father about the secret. Brian has developed as a character along with his emotional condition; he has learned to place his emotions in a more complete context because of what he has experienced.
In terms of actions, when Brian first lands the plane in the wild, his actions are based in uncertainty because ignorance of his true situation. For example, he swats at the mosquitoes, thinking that this will help him. He learns though that such action is wasteful. Over time, his actions become more resourceful and efficient. Building a food shelf, creating fire, developing more intricate weapons, and learning to use the elements around him are examples of actions that Brian takes as he develops and realizes the enormity of his true situation. Even when Brian experiences set backs, such as with the moose attack and the shelter being destroyed, his actions are geared towards rebuilding; his actions show he is developing resilience.
In terms of looks, Brian's looks change. He becomes more gaunt; his face more taut, reflecting exposure to the natural elements. Brian "lost more than 17 percent of his body weight" as a result of being in the wild. The change in Brian's fundamental appearance mirrors his development as a character and his acquisition of a grittier approach to the realities of the world.