At the onset of the ancient Egyptian civilization the world population was somewhat more than 15 million people, i.e. the current population of the Netherlands only or fifth of the current of population of Egypt today. Ancient Egyptians had therefore enough natural resources to create a civilization on their own and within their boundaries. So, peak population had passed very long time ago!
http://www.vaughns-1-pagers.com/history/world-population-growth.htm