The equations here are cool, but they start to yield some pretty large numbers when the empire size begins to grow. So I'm going to try an alternative system for a while and see how it feels.
This is just a simple logistic function centered at $I_0 = 400$, with a maximum value of 480 persons per 1hex: a very high rural density indeed for the development we're talking about. \[d(I) = {480 \over 1 + \exp\left(- k \left(I - I_0\right)\right)}\]. $k$ is set to 0.016, so that $d(5) = 1$. The population of a whole 20hex is given as $400 \cdot d(I)$.
Since this doesn't rely on the urbanization rate (which is now derived rather than set) to determine population, we can now get population numbers for hexes with no cities. The hinterlands.
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