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.
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).
|Fig. 3 (legend below)
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.
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  , but also its gradual slowdown in space.