My area is overrun by crows. They don't seem to have a natural predator and it is rare to see the carcass of one. Several years ago, an upscale restaurant in a nearby town was plagued by them. Their customers would park their cars in the lot and then RUN to get inside because thousands of crows roosted in the trees above. All the cars would be covered in crow feces when they would come out after dinner. The town wouldn't allow the restaurant owners to cut down the trees where the birds roosted.
The town would not do anything to help so the restaurant owner obtained some carcasses and strung them up in some trees. The result? The crows moved on elsewhere within the town. (They probably moved to the mayor's front yard.) The same town later instituted an annual crow shoot and gave out prizes to the hunter who killed the most on that day.
I believe that the crow shoot caused the crows to migrate to my area, as we now have more than ever. They are noisy and disruptive and for those who are not familiar with crow behavior in a city... They are extremely large (compared with their country cousins), aggressive and not afraid of people. They seem to know the city trash schedule and literally tear apart all trash bags. They have even seemed to enlist the aid of the local squirrels. The squirrels tip over the trash cans and then they all have a feast. I frequently joke that if my kids were little that they wouldn't be allowed outside alone because one of these giants would carry them off.
If I had access to a couple of crow carcasses, I would probably try stringing them up to keep their brethren away. If I lived in the country and found a coyote carcass on the side of the road, I would string it up if I felt that it would help to keep my flock safe. It is preferable to losing sleep, livestock and money to a predator.