Joust

What worked best for me, and has consistently gotten me further ahead, is using the middle floating platform as a ceiling so riders cannot drop down on you.

I position myself below the middle floating platform, on the right side. If you look closely, you can see that the bottom of the middle floating platform slants up to the right on the right side. I wait for the riders to come down to the bottom level, then I flap up into this small slant, keeping my head against the platform. Since there is no way for them to come down on top of you and when you are stationed in the upwards slant on the underside of the platform you are higher then opponents approaching from the left, you should win all duels in this space. So long as you are able to flap quickly, the worst that will happen is a tie in a duel from your right, since they can approach you at the same height.

This will keep you from losing lives until the middle floating platform eventually disappears. By that time, if you have saved all your lives, it should be fairly easy for you to make it to wave 10.

Edit: Another tip that has helped me get much farther - do not try to kill opponents with the front of your mount, use your butt. If you turn your back to your opponent, the AI does not come at you as fast or go as high. This allows you to flap up above them much more easily.