Time required for gene frequency change in a deterministic model of gene-culture coevolution, with special reference to the lactose absorption problem

Abstract

The time required for gene frequency change under natural selection in a deterministic model of gene-culture coevolution is investigated. A discrete generations model is formulated, and its continuous time approximation is derived. In passing to the continuous time limit, it is assumed that the frequency of the culturally transmitted trait does not change under oblique (between generations) transmission. The system of ordinary differential equations thus obtained are solved, and the dependence on the parameters of horizontal (within generations) transmission and natural selection is examined. The time required is found to be substantially longer when the determination of a phenotypic difference subject to natural selection is partly cultural rather than completely genetic. The predictions are relevant to the possibility of the coevolution of lactose absorbers and milk drinkers in some human populations. Alternative hypotheses are briefly discussed in the light of the theoretical results.