I read an article once that talked about coyotes and how they have been incredibly resistant to extirpation despite human's best efforts. We have been able to get rid of bison, wolves, elk, and whatever else we have tried (or did accidentally), but despite shooting, trapping, poison and whatever else, we have never been able to get rid of coyotes. It was a pretty interesting perspective and something I had never thought about before reading it.
I'm sure you can do it on a very local level, but they are a pretty unique animal and a real survivor.