When Dumbledore died, Fawkes, too, left Hogwarts for good. Rowling said, “Dumbledore was a very great and irreplaceable man, and the loss of Fawkes (and the fact that he was ‘non-transferable’!) expresses this symbolically.”
“The Patronus often mutates to take the image of the love of one’s life (because they so often become the ‘happy thought’ that generates a Patronus),” said Rowling on James’ being a stag and Lily’s being a doe.