Birds don't eat. Cage those suckers.
I tend to have one pasture area[1] that I keep birds on, and probably could put the occasional single grazer on as well (unless they're 'fighty', and tend to snap at my goslings/whatever), and this is where I put my nestboxes as well. As I get more creatures, I'm learning to
- Move them on from pasture to pasture as the grazed grass/whatever dissappears,
- Individually move the bad-tempered ones onto pastures of their own, to which I might later add another creature to see if their mood has improved[2]
But so far I haven't had more than a couple of years of significant creature density per fort in the versions of DF in which these features exist. And this new mode of working certainly prompts one to butcher animals one doesn't want, where previously I'd have just tethered/caged/let roam free whole hoards of creatures (culling the less suitable off, in an attempt to promote better breeding). Now one has to in order to maintain order and, to a definitely non-zero degree, player sanity!
[1] Even though they don't need it, but it's usually my "starter pasture" from before I expand.
[2] Not enough experience to know if this is the case, but I'll give 'em a chance. Haven't had any squabbling-deaths, yet, or even anything more than a "knock over" incident. Certainly nothing like the suffocating and beheading incidents already mentioned in this thread.