From a rather old book in the series, Morrowind Fauna, we learned that the female kagouti is the "Dominant" of the two sexes in courtship, HOWEVER, ESO has provided interesting snippets into the life of the kagouti to show how exactly this is, but perhaps not in the ways that we expected. What I'm about to share are concrete observations, not speculations, on Kagouti behavior that can be observed in-game.

To summarize in one brief sentence: kagouti reproduce and care for their young in much the same manner as large ground-nesting birds like Ostriches, Emus, and Cassowaries.

The act of courtship for Kagouti begins with adult males congregating in a display ground, such as a hillock or open field, where the bulls and their bright colors can easily be seen by the females. What follows is a violent series of clashes between the males, which can result in serious injury or even death to the participants, in order to establish which bull among them gains control of the display area and thus access to the visiting females. Once a dominant male has established himself king of the display ground, the females begin to visit him.

Although the male is the bigger, stronger, and more aggressive of the two sexes, this is where we learn how the female claims the 'dominant' position in courtship: the female does NOTHING for kagouti reproduction other than mating, and laying the eggs. She does not have to fight like the bulls do, nor does she, even after the eggs have been lain, care for her offspring. Once the females have mated with the dominant bull, they move on, and leave him to care for the nest and the surviving hatchlings entirely by himself. It is the male kagouti who is left with the responsibility of raising the next generation.

And Kagouti bulls are VERY good parents. Although the hatchlings, debatably, with their armored hides and horns, would likely be able to fend for themselves soon after they hatch, better than a guar or alit hatchling, the male kagouti cares for his brood even long after they emerge from their shells; protecting and teaching them how to survive well up into their sub-adult age, until they are nearly as large as he is, a process that could take well over one or two years to see to completion. As a last footnote, it seems that the time when bulls congregate to compete at a display ground is the only time they are truly aggressive with one another or maintain a territory, as when they have hatchlings to care for, the bulls seem much more amicable with one another; feeding near to (but not too closely) to other bulls and their offspring.