Wolves have an Alpha male and female with the only mating rights in the pack.
Most primates have a lead male and a lead female as well.
Lions have one male leading the pack, and that one male usually has a favorite female.
Elephants have a matriarchal society
Meerkats have a matriarchal society with one alpha female, and a male counterpart
Lemurs are the same as Meerkats
Dogs like wolves usually have one alpha dog
Seems to be a monarchy regime in most cases.
Though I have to ask you this question: Is this thread a way to post the question of "If animals don't have a set government, then why should people have a set government?"
As shown by examples there are many species of animals who DO have a complex form of government within their particular herd,flock,pack,pride etc.
God set it up that way I think. Within the church there is God (and depending on your denomination) then the pope/archbishop/board of trustees
After those there is the Bishop/Pastor. After those are Priest/Elders and after those the Vestry/Congregation.
Just as within the stereotypical family there is God, Cat, Father, Mother, Children, Plants, and Dog.

Sorry. I couldn't resist (I'm a cat lover)