Id almost be willing to bet its raccoons, they love to eat the heads of birds because of the protien in the brains. A couple years ago i had a coon wipe out almost 20 assorted ducklings while i was gone for a couple days. I live out in the wide open where the nearest wooded area or creek is about 3 miles away, at first i thought a hawk did it as i had an open top on the pen. One morning while going to work i went to check on the remaining ducklings and there was a coon in the tree in the pen eating the head on a duckling, i ran in the house to get the gun but it had taken off in the mean time. I spent a week trapping the area and ended up with THREE raccoons. I have only trapped one raccoon since then. I keep a pretty active trapping program around the house now and it seemed like things have stopped.
Sounds like you need smaller wire, i assume you have 1" wire?