After I met my girlfriends boyfriend the first time (we had encountered each other a couple of times but never actually had a conversation) I was thrilled. He was cool, I was cool, everything was obviously a'ok. After the meeting I told her that I liked him and how glad I was that we had gotten along so well. She relayed to me later that he agreed and said "Yah, I don't think Alpha Males really work in polyamory"
That still rings very true to me. The type of competition in an Alpha Male type personality is not going to work out very well in any relationship I'm involved in. I'm not in this to prove myself, to out flank an opponent, or to strut my tail feathers and try to impress my mate.
Hopefully the competitive streak you picked up on is just in your head. Or possibly just his social awkwardness from being nervous himself. If either of those are the case then those are issues which can be worked through. Either way, your emotional approach and your boyfriends logical approach will likely need to sit down and talk about your feelings. It will remain very important (imo) that the two of you are able to express your thoughts to each other safely.