Fundamental theorem of natural selection in biocultural populations

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I derive a general equation for the evolution of mean biological fitness in large, randomly mating populations in which the phenotype is subject to both biological and cultural transmission and in which fitness and the effects of cultural transmission are genotype-dependent. The equation is f m ̄ dt = 2 Cov(m, m′), where d m ̄ dt is the instantaneous rate of change in the mean biological fitness of the population and Cov( m, m′) is the covariance between offspring and parental fitnesses. This formulation differs from Fisher's fundamental theorem in at least two important respects. First, because covariances can be either positive or negative, mean fitness can decline over evolutionary time, in contrast to Fisher's result that fitness always increases. Second, since Cov( m, m′) is a phenotypic rather than a genotypic covariance, natural selection can bring about changes in mean fitness even in the absence of additive genetic variance for fitness. Comparison of simple models of vertical (from parents to offspring) and oblique (from members of the parental generation to offspring) cultural transmission indicates that in the former case, mean fitness can evolve in the absence of genotypic variation for both fitness and the effects of cultural transmission, while in the latter case, both are required for evolution to occur.