Well, I'm in Coral Springs (the north side near university and Sample) also and nothing has been molesting my three small papaya. Deer might do that but CS has none and I don't think butterflies eat papaya, at least not in my yard.
I would get some tent stakes and stake that plastic fence to the ground and cover the top with the same fence / net and tie wrap it to the sidewalls. Maybe at the very least a destroyed fence will give you a clue as to the size of the culprit.
Also if it's a herbivore, spreading some dried blood on it might keep whatever is from decimating the leaves and may leave it alone.