Rita, I have seen squirrels do that with not ripe fruit.

Probably squirrels. Possums have large teeth and would easily take huge chunks or run off with the whole thing. Possums might also damage limbs. When they get up in our fig and dwarf mulberry tree they always cause damage.

Squirrels and rats often take one bite of a fruit and leave it. Rats usually leave a deep hole in the fruit, especially with citrus.