In the alpha pack you have the parents as the Alpha male and female. They are usually the leaders of the pack. Alphas that are unrelated may join the pack but majority of the pack are children of the alpha male and female. They are also usually the only ones who are allowed to breed in the pack. They stand at the top of the hierarchy within the pack.
Alpha packs dont actually show a lot of dominance, and if the alphas do choose to it would be mostly related to food. Alphas would beg others for their food by showing submission. By doing that it improves the relationship between the alphas instead of causing conflction. Beta alphas eat after the aplha's, but that isnt always the case. Depending on who's the hungriest in the pack they would eat first and defend their meal until they are done eating.
Alphas have combative relationships with other animals that are predators. Cougars, bears, mountain lions, and foxes are some of the predator animals that are competitive with alphas, Foxes would even prey on the cubs. There are animals like crows who dont fight the alphas yet try to steal some of their food. The crows would follow the alpha pack duirng a hunt and wait after the kill to try and steal bites of their food.