Limitations of Fisher’s Model
Consider a population of preindustrial farmers, initially located in some region. Assume they can disperse into other regions that are also suitable for farming but initially empty of farmers.
The next generations of farmers will, in general, disperse away from their parents. Then Fisher's model predicts that a wave of advance (also called a front) of farmers will form and propagate with the following speed (Fisher 1937)
where aN is the initial reproduction rate of Neolithic farmers (which is easily related to their net fecundity and generation time) and DN is the diffusion coefficient of Neolithic farmers (which is easily related to the probability that farmers disperse away from their parents as a function of distance). Equation (5.1) is very useful. Ammerman and Cavalli-Sforza (Ammerman and Cavalli-Sforza 1973, 1984) used observed values for aN and DN into Eq. (5.1) and found that Fisher's model predicts a speed of about 1 km/y, i.e. similar to the observed one for the Neolithic transition in Europe. In recent years, Fisher's model has been refined (Fort et al. 2007). Note that Eq. (5.1) predicts that, for a given value of DN, the speed increases without bound (sF to) for increasing values of the initial reproduction rate (aN to). This is counterintuitive because, for a given value of DN, the dispersal behavior of the population is fixed. Thus individuals can disperse up to some maximum distance, Amas. Then we should expect that (no matter how large is aN) the speed sF should not be faster than smax = Amas /T, where T is the time interval between two subsequent migrations (mean age difference between parents and their children). An integro-difference cohabitation model solves this problem (Fort et al. 2007, 2008, 2012). Then Eq. (5.1) is replaced by a more complicated and accurate equation that takes into account a set of dispersal distances per generation and their respective probabilities. However Fisher's speed, Eq. (5.1), is very useful as a first approximation. It is even quite accurate for some pre-industrial farming populations. For example, for the Yanomano (Isern et al. 2008) Fisher's speed (1.22 km/y) yields an error of only 6 % relative to the integro-difference cohabitation model (1.30 km/y). In other cases, Fisher's speed is not so accurate. For example, for the Issocongos (Isern et al. 2008) Fisher's speed (0.56 km/y) yields an error of 30 % relative to the integro-difference cohabitation model (0.80 km/y).
5.3