a depicts the temporal trajectory for a set of 100 cities in a non-stochastic, non-linear environment, described by Eq. (8). Despite generalizing the demographic dynamics in Fig. 1, this results in a fixed urban hierarchy at late times (vertical red line), regardless of initial conditions. However, when migration and vital rates become stochastic, the demographic dynamics no longer has a static solution at long times, b. Instead, cities constantly change their relative sizes (rank) over time. After an initial transient, the population structure fluctuates close to Zipf's law. The insets show the population structure (red) at the point in time when the non-stochastic evolution (a) becomes approximately stationary, with Zipf's law (dashed blue line) shown for reference.