At no time do bighorn sheep exhibit any sort of aggressive behavior, such as territoriality, that might result in limiting the population level. The head-butting bouts among rams is not an expression of intolerance to crowding as is the case with aggressive behavior among males in some avian populations. Actually most of the battles may be in the "high school" class of younger rams. It is a less-usual sight to see a big ram asserting his superiority. Fighting does not result in breeding being accomplished by solely by a few of the most virile rams. The aggressive behavior probably probably ensures a flow of good genes throughout the population, but it seems to have no bearing on limiting population density.From The Bighorn Sheep in the United States by Helmut Buechner (1960).
Big Ram Rumble from NatGeo.