Possible Forms of the Cultural Transmission Term

The demic models above can be extended by including cultural transmission. Then Fisher's speed, Eq. (5.1) is generalized into (Fort 2012)

where C is the intensity of cultural transmission (defined as the number of hunter-gatherers converted into farmers per farmer during his/her lifetime, in the leading edge of the front, i.e.

Equation (5.2) and other models with cultural transmission take into account that hunter-gatherers can learn agriculture not only from incoming farmers, but also from converted hunter-gatherers, i.e. former hunter-gatherers that have (partially) become farmers (as well as their descendants).

An integro-difference cohabitation model with cultural transmission leads to a more complicated equation than Eq. (5.2), and generalizes the integro-difference model summarized in the previous section (Fort 2012).

Both demic-cultural models (i.e., Eq. (5.2) and the integro-difference cohabita­tion model) are based on cultural transmission theory (Cavalli-Sforza and Feldman 1981), which shows that the number of hunter-gatherers converted into farmers per farmer during his/her lifetime is (Fort 2012)

where P_N and P_P are the population densities of Neolithic farmers and Mesolithic hunter-gatherers, respectively, and f and y are cultural transmission parameters. In the leading edge of the front (P_N«0), Eq. (5.3) becomes

with C = f Y.

A comparison to other approaches is of interest here. In Ecology a widely used model is based on Lotka-Volterra equations, which assume that the interaction between two populations (AP_N) is proportional to their population densities (Murray 2003),

where k is a constant. This model has the problem that AP_N/P_N to if P_P to, which seems inappropriate in cultural transmission, for the following reason. Assume that a farmer converts, e.g., 5 hunter-gatherers during his lifetime (AP_n/P_n = 5) if there are P_P = 10 hunter-gatherers per unit area. Then Eq. (5.5) predicts that he/she will convert AP_N/P_N = 50 hunter-gatherers if there are Pp = 100 hunter-gatherers per unit area, &P_N/P_N = 500 hunter-gatherers if there are P_P = 1000 hunter-gatherers per unit area, etc. Contrary to this, intuitively we expect that there should be a maximum in the number of hunter-gatherers that a famer can convert during his/her lifetime, i.e. that &P_N/P_N should have a finite limit if P_P to. This saturation effect is indeed predicted by Eq. (5.3), as shown by Eq. (5.4). Thus we think that Eq. (5.3) is more reasonable than the Lotka-Volterra interaction, Eq. (5.5).

This point has important consequences because for Eq. (5.3) the wave-of-advance speed is independent of the carrying capacity of hunter-gatherers, P_Pmax (see, e.g., Eq. 5.2). In contrast, for the Lotka-Volterra interaction the wave-of-advance speed does depend on P_Pmax. For example, if Fisher's model is generalized by including the Lotka-Volterra interaction, the front speed is (Minedez et al. 1999) (see also Murray 2003 for a similar model)

The point is that, in contrast to Eq.

Finally, some language competition models use population fractions (rather than population densities) and interaction terms with non-linear powers of P_N and P_P (Abrams and Strogatz 2003). We first consider the linear case. In one such model, Eq. (5.5) above is replaced by (Isern and Fort 2014)

with n a constant. Equation (5.7) is a special case of Eq. (5.3), thus the wave-of-advance speed is independent of P_{P max} also in this model (Isern et al. 2014). It can be argued that the complete model in Isern et al. (2014) is useful for modern populations but not for the Neolithic transition, because it assumes the same carrying capacity for both populations. But a model that allows for different carrying capacities (Fort and Perez-Losada 2012) also leads, in the linear case, to an equation with the form of Eq. (5.7). In conclusion, some models originally devised to describe language competition also lead to the conclusion we have stressed above, namely that the wave-of-advance speed is independent of PP max.

For completeness, in the non-linear case the following two limitations of the language-competition models discussed in the previous paragraph (Abrams and Strogatz 2003; Isern et al. 2014; Fort and Perez-Losada 2012) should be noted in the context of the Neolithic transition.

(i) In the non-linear case, Eq. (5.7) above is generalized into (Isern et al.

with a > 1 and p > 1 (Abrams and Strogatz 2003). Thus AP_n 0 if P_p to, i.e. AP_n/P_n does not have a finite, non-vanishing limit (except in the linear case a = p = 1, see Eq. (5.6). Alternatively, for the Abrams-Strogatz model in Ref. (Fort and Perez-Losada 2012), namely

where o < 1 is called the status of language N and a > 1 is the resistance to language change, we obtain a negative limit for AP_n/P_n if P_P to, which is counterintuitive (Isern et al. 2014) (except again in the linear case, a = 1). The main point here is that neither of both non-linear models displays the saturation effect discussed above.

(ii) Whereas Eq. (5.3) was derived from cultural transmission theory, the non-linear models introduced to describe language competition (Abrams and Strogatz 2003; Isern et al. 2014; Fort and Perez-Losada 2012) (Eqs. 5.8 and 5.9) were not.

The non-linear models given by Eqs. (5.8) and (5.9) compare favorably to observed data in non-spatial linguistic systems (Abrams and Strogatz 2003; Isern et al. 2014), and may be applicable to other modern instances of cultural trans­mission. Perhaps the effects of mass-media, schools, etc. in modern societies avoid the saturation effect discussed above. Such effects are not included in the cultural transmission theory leading to Eq. (5.3) (Fort 2012).

In any case, due to reasons (i) and (ii) above, for the Neolithic transition we prefer not to apply language-competition non-linear models, Eqs. (5.8) and (5.9), neither the Lotka-Volterra interaction, Eq. (5.5). Instead, we apply cultural trans­mission theory, Eq. (5.3) (or its frequency-dependent generalizations, which take into account the conformist effect but lead to the same conclusions (Fort 2012)).

We stress that the conclusion that the wave-of-advance speed is independent of the hunter-gatherer population density P_{P max} follows from cultural transmission theory, and is ultimately due to the fact that there should be a maximum number of hunter-gatherers converted to agriculture per farmer (or converted hunter-gatherer) during his/her lifetime (this is the saturation effect discussed above).

5.4