By definition dominant (dominance) and submissive (submission) refers to a role in a relationship. So technically it is possible for a dog to be dominant in one relationship and submissive in a different relationship.
Dominance and Submission mostly determine who has first right to a limited resource. With canines there is also a strong correlation between dominance and leadership. Leadership refers to who makes the decisions for the group as a whole.
Although it is not a definition found in most text books, dogs are commonly referred to as having "dominant" or "submissive" personalities as a trait. Think of this as short for "dominant secure" and "submissive secure". What this means is that certain dogs have a tendency to take the dominant role and others the submissive role more predictably in certain situations or when it is not clear what role they have. Dogs seek to know their role as quickly as possible because it helps them feel secure. After all, understanding who has what role is what reduces aggression and conflict. These terms, like most traits, are relative with grey areas.
Just like with people, there is more likely to be conflict between two dogs of "dominant" personality types.
This video gives more insight to the rules of dominance and submission as it pertains to canines:
Both of my dogs may fight over a toy for about 5sec. But the male dog just takes the toy away from the 1yr old female. The 1.5 male appears to be the dominate one. Even when it comes to drinking water. He wants to go first. The female will wait until he finishes before she attempts to drink water.
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