The simplest model for an empire’s growth curve is simple logistic (drawn-out S), leading to a steady maximum area. During this steady state, the share of coordinators in the population increases at the expense of producers. Collapse follows sooner or later, even in the absence of other inbuilt defects. Actual growth–decline curves at times deviate widely from the simple model. While the size of top empires has increased, empire duration has not over five millennia. They have tended to last for 180 years at half their maximum size. When including lesser but still major states, duration drops to 100 years. Slow growth tends to lead to more durable states. The median ratio of duration and rise times is 2.5 (as compared to five for individual rats and humans), but variation is huge.

The size of states, including empires, is measured by two crude measures: their area and their population. Areas graphed over time represent the growth–decline curves of empires. These are the basic portraits of empires, showing at a glance how rapidly they formed, how large they became, and how long they lasted. Measuring areas in historical atlases supplies most data. Rome, Ottoman, and Manchu Qing offer examples of completed life cycles. Russia and the USA show curves still incomplete. The beginnings of these curves often approximate the “simple logistic” pattern typical of growth of bacteria colonies when they have plenty of food but space is limited. A table lists the following characteristic of 98 major empires during the last 5000 years: maximum size, duration at least at one-half of maximum size, and rise time from 20−80% of maximum size. Of these, 20 are Runner Empires (3000 to 600 BCE), 66 are Rider Empires (600 BCE to 1800 CE), and 12 are Engineer Empires (from 1800 on).