How good is the first order approximation? What does the data tell us about the shape of demic expansion?
In looking at the Dodecad K10 data, there are signs that more than one "shape" of demic expansion is at work. In some cases, you see hard boundaries between peoples living several hundred miles from each other. The separation between the Armenians and Georgians comes to mind, with each confined to their respective nearby valleys. In other cases, you see an entirely different diffusion shape, with a very soft and gradual diffusion of peoples, such as that seen between Tuscans and Northern Italians or between Syrians, Jordanians and Egyptians.
The three models of diffusion can be graphed:
The First Order Model(FOM) is a straight line approximation to the decaying exponential. The gradual diffusion model can be described by an unbounded exponential decay. I'll call that the Unbounded Diffusion Model(UDM). The third model is bounded, with a leading edge that lies somewhere between the Unbounded Diffusion Model and the First Order Model. That's the Bounded Diffusion Model(BDM).
For purposes of estimating dates for when the leading edge of a population component reached a particular area, I'll give the FOM date and the UDM date. The UDM edge will be set at the 2% proportion level so that the UDM date will be twice that of the FOM date, as pictured in the above graph. That's arbitrary, but makes for a nice rule of thumb.