We can try to answer this on two levels, though the textual information we have on this issue is scant. On one level, we can say the conflict between Duke Frederick and Duke Senior is Person against Person. This makes it an interpersonal conflict: something has gone wrong between two people (in this case, two brothers) and has resulted in a vicious conflict. This conflict, of course, is paralleled in the same brother against brother interpersonal conflict between Oliver and Orlando, though we know more about their conflict than we know about the Dukes' conflict.
On another level, we have to ask what caused the conflict? To that question, we have little textual guidance. We might speculate that the classic (fairy tale-like) theme holds for the cause of their conflict: older brother inherits throne; younger brother is envious and resentful; older brother is loving and benevolent; younger brother's malice makes him evil and malevolent; younger brother steals throne and exiles older brother. [This is reminiscent in some ways of Lear's banishment in King Lear.] Yet, is there textual evidence to support this speculation?
The textual evidence we do have for the cause of the conflict is scant and comes from Celia, Duke Frederick and Duke Senior.
Celia tells us that Frederick acted by force, probably indicating a battle; the transfer of power was not by agreement. Celia also tells us that Frederick has a rough and envious disposition. This means a disposition given to quarrelsomeness and argumentation resulting in open conflict.
for what he hath taken away from thy
father perforce, (I.ii)
My father's rough and envious disposition
Sticks me at heart. (I.ii)
Duke Frederick tells us a different story of himself, yet unintentionally confirms Celia's observations by his actions. He has forcefully summoned Oliver to court to accuse him of being involved with Celia's and Rosalind's escape, with rumors of rendezvous with Orlando attached to it. He tells Oliver it is good that he is a man of mercy or else he would take Orlando's punishment out from Oliver's life. He confiscates all Oliver's goods and sends him out with one year to find and return Orlando dead or alive. Thus while Frederick claims to be a man of mercy, he shows himself to be a hateful, vengeful, quick to anger, quick to violently act villain.
But were I not the better part made mercy,
I should not seek an absent argument
Of my revenge, thou present. (III.i)
Duke Senior adds another clue by suggesting that at court, his life, and those of his courtiers, was in peril (danger) and that the court was a place made dangerous by envy.
Are not these woods
More free from peril than the envious court? (II.i)
So perhaps we do have enough textual evidence to say the conflict between Duke Frederick (the younger brother who enviously stole the throne) and Duke Senior (the older benevolent brother who is the rightful ruler) was based on Frederick's envy and angry, malicious desire to rule in the place of his older brother.