Cities I: History II: Fall and Decline

The basic city growth model is \[\Delta P = B + D + I + E\], where $\Delta P$ is population change, $B$ is births, $D$ is deaths, $I$ is immigration, and $E$ is emigration.

I handle $B$ and $D$ with the logistic equation \[B - D = r P \left(1 - {P \over K}\right)\], where $K$ is the carrying capacity of the city. I want some randomness here also, so I add a bit of random walk. I also add some random walk to $r$ (the maximum rate of increase) and $K$, to account for changing conditions in the city's management, infrastructure, and what-have-you.

This leaves $I$ and $E$, which is a bit trickier. I read a lot of papers on this, which mostly deal with models for modern cities, where people often live in totally different cities. Since permanent immigration is due to many varied non-deterministic reasons, these terms often refer to the flow of workers between cities (who aren't necessarily permanent residents). These phenomena are pretty well studied, but that's not helpful to me.

So I figured that a certain percentage of people from a city would want to move someplace better. The bigger the difference in desirability ($\delta_j - \delta_i$), the more people who would try to move there, as long as the host could bear them. The number of migrants is then defined as \[M = {\delta_j - \delta_i \over 10\cdot \textrm{max}\left(\delta_j, \delta_i\right)} P_i\], where $i$ is the source and $j$ is the destination. This captures reverse flow, so I don't have to worry about which one is nicer. Since $\delta$ is already partially a function of $P$, the dense tend to get denser, to a point.

To make things more interesting, there's also a ~5% chance of a "disaster" striking (or perhaps a series of such) which decrease the population even further.

Here's a city undergoing these processes:

All of this assumes near-perfect growth, not to mention a maximum $K$ that's probably optimistic (around 100k or so, but with magic, who knows?). But more importantly, there is no war here. And that, my friends, will not do.