A country builds its empire by attacking its neighbours and taking over both their land and their people. The larger the empire becomes, the more resources they have available to take further territory by force. However the further they expand, the harder they find it to keep expanding due to the geographical and organisational logistics involved. Additionally the empire finds it harder to maintain social cooperation among its people, its asabiya.
An agrarian population can exceed its normal carrying capacity by the formation of a state that helps improve land productivity. As the population and state grow, it becomes harder for state revenue to stay ahead of expenditure. Eventually the state runs out of money and collapses, with a large drop in population numbers.