How do you keep the birds from eating all your figs?

We built a cage(plastic mesh on a wood frame) around ours. It helped to a large extent but some birds managed to get under the mesh. Then they were too stupid to get back out and we had to open the door for them. Then they didn't have the sense to fly out, they kept flying around the cage so we had to go in the cage and chase them out. Also rabbits would chew through the mesh and climb up the tree(yes, rabbits are good climbers) and eat the figs. Don't bother with those nets they sell to d over your trees. The birds will get into them(and once again they don't know how to get out).