Ancient Egypt was actually in decline by the time the Western civilization rose. Ancient Egypt was concentrated along the lower end of the Nile River where the modern country of Egypt is currently located. Egyptian civilization came together around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh.
The history of ancient Egypt occurred in a series of stable Kingdoms, separated by periods of relative instability known as Intermediate Periods. Egypt reached the pinnacle of its power during the New Kingdom, in the Ramesside period, after which it entered a period of slow decline.
In the aftermath of Alexander the Great's death, one of his generals, Ptolemy Soter, established himself as the new ruler of Egypt. The Ptolemaic Dynasty ruled Egypt until 30 BC, when it fell to the Roman Empire and became a Roman province.
Ancient Egypt held together for an impressive number of years. The West today, still has not managed to match the number of living years as Ancient Egypt did.