The alpha/beta competitive dynamic isn't a gay/straight thing, it's not even a male/female thing. It's a human thing.
I spent 25+ years working in a variety of furniture retailing environments, from RTA to custom, and most of my co-workers for most of my career were women. Believe me when I say that they can be just as assertive and competitive as men.
If the straight guy you work with is aware of your sexuality and isn't comfortable with it, he can try and pull all manner of alpha-male tricks to attempt to put you "in your place". As the supervisor/manager, it's your responsibility to make sure he knows who's boss (and it ain't him :wink
; some guys get along just fine being part of a cooperative team, others don't. If his conduct is undermining the effectiveness of your team, you need to alert your superiors, as it's clearly unprofessional.
If there's room to grow within the larger organization, you might take a positive approach and encourage him to apply either for the in-house training required for moves upward (and away from you) or to just suggest he apply for lateral/upward moves as is. That way, he's more likely to see you as a mentor and less as an obstruction.
If there's no room to grow, then he needs reminding as to who's boss and who frankly isn't. When I managed sales teams, I led by example much more than by decree, and tried to make the individuals in my team feel as empowered and independent as they could
while still obeying the dynamics of teamwork. Regardless of the circumstances, I made certain that I was always the most productive member of any team I led, not by capitalizing on my position but through the dint of sheer hard work. I rarely had trouble keeping folks in line.
Those who'd chafed under the yoke of micro-managers and by-the-bookers in the past always appreciated the difference of my approach. Those who didn't almost always found ways of getting away from me, whether through transfer or by quitting. But everyone understood who was boss, and those who either disrespected me or tried to undermine my authority were eventually gone one way of the other.