"The south" designates Egypt, as it does in Daniel 8:9. "The king of the south," therefore, is the king of Egypt. Ptolemy I was "one of Alexander the Great’s favorite generals.... After Alexander’s death in 323 B.C., he seized Egypt as his share of the divided empire. He assumed the title of king of Egypt" (World Book Encyclopedia). And so Ptolemy became "the family name of the last dynasty of ... ancient Egypt. They ruled from 323 to 30 B.C." (Ibid.)
"The north" is the area north of Palestine which was controlled by the Seleucid kings in the division of Alexander's empire.
"He that cometh against him" in verse 16 is Rome.
"The king" introduced in verse 36 is the papacy.