Shoot all you want, as long as there's an abundant supply of food, they'll likely out-breed any culling you can muster. The best way to deal with predators getting at livestock is to either deny them the food source (higher fences, or other barriers), or employ guard animals like llamas (nasty, and they bite, kick and spit). It certainly seems to be more satisfying to people to shoot the coyotes, though. Culling inevitably seems to lead to a cyclical problem, and could even wind up with such reduced populations (if overwhelming enough to wipe them out) that they become protected. Usually, the best way to knock down the coyote population is to do a lot of rabbit hunting, and if you feel the need, shoot a few coyotes. If you shoot lots of coyotes, you'll make for a big breeding year or two for rabbits. The two populations are tied, so when you see lots of rabbits, figure on lots of coyotes next year, unless you eat allot of bunny instead of letting the 'yotes have them.
