Red River - DP

Although it seems at first like there are two alpha male cowboys in Red River, I cannot help but conclude that there is only one alpha male.  Tom Dunson is the true alpha male of the story, because he represents all of the typical characteristics of an alpha male cowboy.  Matt Garth at first seems to display the same personality, but with close attention, one can see what separates the two from one another.
Tom exemplifies the alpha male because of his life style.  Like Matheson states, Tom is an absolute fanatic about his work.  He tirelessly pushes on the cattle drive refusing to even aknowledge the difficulties along the way.  He also refuses all temptation, especially that of women.  He leaves a women who loves him dearly early on in the film because, "The west is no place for a woman."  He also adds to it when he refuses Tess' offering of her body in return for not harming Matt.  An Alpha male cowboy shys away from women, and certainly does not participate in sexual encounters.  He is also incredible fearless and persistant.  He is injured several times during the film and yet he presses on.  He lives by his own moral standards as well, he wishes to seek out Matt after he takes over the cattle drive and claim his dominance. 
Matt Garth does sport some qualities of an alpha male.  He stands up for himself, he is a fighter, and a tireless worker.  He, however, posseses the compassion that Tom lacks, he does not wish to hang the men that Tom wanted to, and tends resort to less drastic and rash measures.  He refuses to draw upon Tom showing he is more realistic and does not have the inhierent need to be dominant.  The biggest reason he cannot be the alpha male is the fear he expresses.  He shows his fear when Tom vows to come after him, and according to the text and Matheson, the alpha male cowboy never shows emotion.  Emotion is something the alpha male keeps inside, and he certaintly would not show fear, a sign of inferiority.  Also, later in the film Matt ends up with a woman, which is something very unlike the typical alpha male cowboy.