The "Kingdoms of the South and the North" refer to the visions of Daniel the Prophet. He predicted that Alexander the Great's Empire would break up into four smaller empires "toward the four winds of Heaven."

After Alexander's death, the Greek Empire was indeed divided by his 4 generals into a Western (Macedonia-Greece), Northern (Northern-Turkey; Azerbaidjan), Eastern (Syria; Irak; Iran) & Southern Division (Lybia; Egypt-Israel)! [See Map below.] Daniel foresaw that out of one of these four factions the Antichrist is going to rise.
See Daniel 8 Tour

"Therefore the male goat (Greece) waxed very great: and when he was strong, the great horn (Alexander) was broken; and for it came up four notable ones toward the four winds of heaven. And out of one of them came forth a little horn (AC), which waxed exceeding great, toward the south, and toward the east, and toward the pleasant land. Daniel 8:8,9:

NOTE: The Kings of the West and the East don't come into the picture at all. The Bible concentrates only on the power struggles between the Northern and Southern Kings!

ACTUAL HISTORY
The history of the successive Kings of the South and the North are predicted and described in detail in Daniel 11:1-21, all the way from Alexander the Great in 333 B.C. until the Antichrist after 2000 AD! However the territories of the 2 dominions of the North and the South, have of course fluctuated a lot over the last 2.300 years. The 1st king of the Southern division was Ptolemeus and his descendents ruled it after him for 300 years until Cleopatra.

The Kingdom that then ruled the North--The Roman Empire--conquered her and her Egypt effectively taking over the role of the King of the North AND the South, until Byzantium split off from Rome! Byzantium was then conquered by the Ottoman Empire, which was the next King of the North! It also conquered the Kingdom of the South, Egypt & Palestine in 1512 AD, until 1918 when Egypt & Palestine were captured by the British. Then the British Empire became "King over the South", until their role was taken over by US-sponsored Jewry, Israel and Egypt, and finally the U.S itself with great concentration of military force in the Middle East until today!
And WHO did become "King of the North" today? Well, just look who is on the opposing side of the Balance of Power in the Middle East! …Greece? …Turkey? Irak? Hmmm! Look a little more North. Who has been the BIG Superpower of the North opposed to Israel & the U.S? Since 1917 Russia has not been as friendly to Israel as the U.S. has. Not just that, but the Bible predicts that the Antichrist will become the final King of the North, and invade Israel! Are you ready?

Daniel Chapter 11
3. "And a mighty king shall stand up (Alexander), that shall rule with great dominion, and do according to his will. 4. And when he shall stand up, his kingdom shall be broken, and shall be divided toward the four winds of heaven; and not to his posterity, nor according to his dominion which he ruled: for his kingdom shall be plucked up, even for others beside those." 5 "And the king of the south (Ptolemaeus) shall be strong, and one of his princes; and he shall be strong above him, and have dominion; his dominion shall be a great dominion."
(read the rest in Daniel 11:1-20 for yourself!
20. Then [in the Endtime] shall stand up in his [Northern] estate a raiser of taxes in the glory of the kingdom: but within few days he shall be destroyed, neither in anger, nor in battle......
AFTER THIS COMES VERSE 21, COMMONLY REFERRED TO AS THE FIRST DESCRIPTION OF THE ANTICHRIST IN DANIEL 11.

21 And in his estate shall stand up A VILE PERSON, to whom they shall not give the honour of the kingdom: but he shall come in peaceably, and obtain the kingdom by flatteries.