How hunter-gatherers slowed farmers' advance.

A very technical paper yet one that addresses how farmers could not advance so fast in Europe:

Neus Isern and Joaquim Fort, Anisotropic dispersion, space competition and the slowdown of the Neolithic transition. New Journal of Physics, 2010. Open access.

We can see from equation (9) that if the Mesolithic (indigenous) population density M increases along the direction y, then the probability of Neolithic invaders to jump forward (θ = π/2) is minimum and the probability to jump backwards (θ = 3π/2) is maximum.

Where y is the main vector of Neolithic expansion, drawn along the Morava-Danube axis in SE-NW direction.

So there is a forager population density value for which farmers are more likely to fall back than continue advancing... interesting.

From equation (22), we see that if the Mesolithic population density increases with y, then the front speed decreases because of two effects: (i) the higher the gradient of the reduced Mesolithic density m, the higher the correction on the front speed; (ii) the speed also changes if there is less available space for the Neolithic population, i.e. for lower values of s = (1–m(y)) (if s = 1, this second effect disappears).

And there is an interesting issue of what happens if Epipaleolithic ("Mesolithic") population density increases through the same vector as that of Neolithic expansion (as it was the case in Europe without doubt)?

Fig. 3 (legend below)

Figure 3. Curves: relative Neolithic front speed predicted by a model with the dispersion and growth processes dependent on the presence of Mesolithic populations, equation (25). Symbols: observed front speeds calculated from archaeological data [9].

Interesting: archaeology seems to support m4 of all estimated curves.

The value of m4 is described earlier:

Where:

A₁ = 0.999/1300, B₁ = 0; A₂ = –0.1 = –B₂ = A₃ = B₃, τ₂ = –ln(10.99)/1300 = –τ₃; A₄ = 0.99, B₄ = 42, τ₄ = 1/0.007.

Just for the record.

They conclude:

Comparing the results from equation (25) to those from archaeological data in figure 3, we see that, even though none of the four test functions reproduce exactly the behavior of the archaeological data (which is not surprising for such a complex phenomenon), they do give a good approximation to the general behavior (especially m₄). Thus, a simple physical model can explain qualitatively the decrease in the front speed during the Neolithic expansion range in Europe. Therefore, physical models are useful to explain not only the average Neolithic front speed [8] , but also its gradual slowdown in space.

Press release: Eurekalert (found at AiE)