There is a common misconception that wild wolves, and by extension dogs, form packs in which there is a dominant "alpha male." This idea was based on incorrect research. The research was carried out on wolves that were kept in captivity in conditions that for humans would be considered those of a prison. Wolves in this unnatural situation do not behave as they would in the wild. In the wild, packs are normally family units, and there is dominance only in the same sense that the parents in a human family tend to dominate their children.

Dogs have some behavioral and biological differences from wolves, but the notion of an alpha male and a strict dominance hierarchy is no more correct for them than for wolves. Most of the world's dogs are not house pets but rather they have a lifestyle in which they form more casual alliances and contacts with humans, who give them food. Examples are village dogs in Africa or dogs that live in garbage dumps in Mexico City. These dogs guard resources from each other to some extent, but they do not form dominance hierarchies.

So if you bring new dogs into your household, there is no particular reason to expect that they will fight for dominance because of some behavioral programming encoded in their DNA. Conflict is possible, but can be dealt with by common-sense measures such as making sure that each dog's food dish is far away from the others'.

References

https://www.washingtonpost.com/opinions/five-myths-about-dogs/2014/02/10/c53c6970-8f46-11e3-b46a-5a3d0d2130da_story.html

Mech, L. David. 1999. Alpha status, dominance, and division of labor in wolf packs. Canadian Journal of Zoology 77:1196-1203. Jamestown, ND: Northern Prairie Wildlife Research Center, https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1343&context=usgsnp

There is a common misconception that wild wolves, and by extension dogs, form packs in which there is a dominant "alpha male." This idea was based on incorrect research. The research was carried out on wolves that were kept in captivity in conditions that for humans would be considered those of a prison. Wolves in this unnatural situation do not behave as they would in the wild. In the wild, packs are normally family units, and there is dominance only in the same sense that the parents in a human family tend to dominate their children.

Dogs have some behavioral and biological differences from wolves, but the notion of an alpha male and a strict dominance hierarchy is no more correct for them than for wolves. Most of the world's dogs are not house pets but rather they have a lifestyle in which they form more casual alliances and contacts with humans, who give them food. Examples are village dogs in Africa or dogs that live in garbage dumps in Mexico City. These dogs guard resources from each other to some extent, but they do not form dominance hierarchies.

So if you bring new dogs into your household, there is no particular reason to expect that they will fight for dominance because of some behavioral programming encoded in their DNA. Conflict is possible, but can be dealt with by common-sense measures such as making sure that each dog's food dish is far away from the others'.

References

https://www.washingtonpost.com/opinions/five-myths-about-dogs/2014/02/10/c53c6970-8f46-11e3-b46a-5a3d0d2130da_story.html

Mech, L. David. 1999. Alpha status, dominance, and division of labor in wolf packs. Canadian Journal of Zoology 77:1196-1203. Jamestown, ND: Northern Prairie Wildlife Research Center, https://digitalcommons.unl.edu/usgsnpwrc/381/