Where I used to live in the outer sticks of GA there was a huge problem with dogs killing livestock. The owners would just let them run loose during the day and they'd form into packs killing chickens, goats and cattle. One friend of a friend came home and found a pack of about 20 dogs in his calf pen along with the heads and a few other parts of two calves. He went into the house get his AR-15 and killed all but two of them. Some of the "neighbors" whose "pets" they were complained to the sheriff (those who were responsible for the most dogs.) I think the one "neighbor" that actually made it to court owned about a dozen of them.
Bottom line was it was treated as a civil matter. the judge ordered him to pay $10 each for the dead dogs, and ordered the dog owners to pay $1000 each for the dead calves. Naturally the trash who owned the multiple dogs couldn't pay, and had nothing against which to place a lien. Funny how they could afford to harbor a pack of dogs.