How did the Battle of Actium lead to the fall of Ptolemaic Egypt?

The Battle of Actium, fought on September 2, 31 BCE, was the decisive turning point that led to the fall of Ptolemaic Egypt.12

Following the defeat of Mark Antony and Cleopatra VII by Octavian (later Augustus) at Actium, the morale of Antony’s forces collapsed, and his remaining military support disintegrated. Cleopatra and Antony fled back to Alexandria, where they could only await the arrival of the victorious Octavian.312

When Octavian captured Egypt in 30 BCE, both Antony and Cleopatra committed suicide. Octavian subsequently executed Cleopatra’s son, Ptolemy XV Caesar (Caesarion), to eliminate any rival claims to power. By annexing Egypt and placing it under his direct control, Octavian secured the kingdom's vast wealth to pay his veterans and established himself as the undisputed master of the Roman world, effectively ending the Ptolemaic dynasty.3425