The wave-of-advance model was introduced to describe the spread of advantageous genes in a population. It can be adapted to model the uptake of any advantageous technology through a population, such as the arrival of neolithic farmers in Europe, the domestication of the horse, and the development of the wheel, iron tools, political organization, or advanced weaponry.

Any trait that preexists alongside the advantageous one could be carried along with it, such as genetics or language, regardless of any intrinsic superiority. Decoupling of the advantageous trait from other "hitchhiking" traits depends on its adoption by the preexisting population.

Here, we adopt a similar wave-of-advance model based on food production on a heterogeneous landscape with multiple populations. Two key results arise from geographic inhomogeneity: the "subsistence boundary," land so poor that the wave of advance is halted, and the temporary "diffusion boundary" where the wave cannot move into poorer areas until its gradient becomes sufficiently large. At diffusion boundaries, farming technology may pass to indigenous people already in those poorer lands, allowing their population to grow and resist encroachment by farmers.

Ultimately, this adoption of technology leads to the halt in spread of the hitchhiking trait and establishment of a permanent "cultural boundary" between distinct cultures with equivalent technology.

If you don't have access to PNAS, you can download a preprint of the paper.

For a visual demonstration:

This page contains the test site applet for the Neolithic farming simulation in an environment with a Gaussian shaped hill in the middle.

The upper panels show the populations of Farmers, Hunter Gatherers and converts as the wave travels from left to right. Populations are suppressed in the central region.

Lower panels show the integrated fraction of the various populations (white). With the default settings, the wave of advance of the farmers outcompetes the hunter-gatherers, a few converts are formed, but rapidly overwhelmed. The wave is it is slowed by the hills, and Shortly after crossing the hills the converts become the dominant population, with a sharp cultural boundary between them and the original Farmers.

The timestep is hardcoded at 1 year, and the length unit is 1km. For fast moving waves these fixed scales lead to a numerical instability in the diffusion equation, which could be fixed by reducing the timestep (at the expense of slowing the applet)