Allow me again to be repetitive, this time with the help of our good friend Bob Plamondon:
"... I rarely hear people mention this, but predators are smarter and more observant than people give them credit for. Now that I've killed two crows, at least a hundred are avoiding my farm. I've seen the same effect with four-footed predators: when the farmers and the federal trapper are on their toes about livestock-eaters, the predators not only get the message, they pass it on to their young, and a balance is struck.
Practically all the predators are eating wildlife rather than livestock, and this means that both predators and livestock get to have a normal lifespan.
But if you don't kill any predators, their caution fades. After a couple of generations, the mothers stop teaching farm-avoidance to their young, and then the clueless young predators kill a lot of livestock before inevitably being killed themselves. Which is a bad deal all around. To misquote the song, "Mamas, don't let your babies grow up to be livestock eaters!"