Abstract
Human evolution depends on gene-culture co-evolution: genetically determined behavior rules co-evolve with socially learned information. Social transmission of information can occur vertically from parent to child, horizontally from peer to peer, or obliquely from unrelated individuals of different generations. Models for the emergence of cumulative culture have in their majority studied gene-culture co-evolution under random interactions between individuals, which rules out vertical transmission. Yet, vertical transmission is thought to be a primary form of transmission in nature. In order to study gene-culture co-evolution within the family, we derive here the invasion fitness of a mutant allele that influence a deterministic environmental state variable (e.g., amount of knowledge or skill, stock of culture, or any other non-innate environmental feature) to which diploid carriers of the mutant are exposed in subsequent generations. Under the assumption that this environment has a single attractor point, we show that singular strategies and the concomitant environmental state they generate can be evaluated analytically, thereby making gene-culture co-evolution in the family with multigenerational effects mathematically tractable. We then apply our results to model the co-evolution between individual and social learning rules on one hand, and the amount of adaptive information these rules generate on the other. We find that vertical transmission generally favours the accumulation of adaptive information due to kin selection effects, but that when adaptive information is learned as efficiently between family members as between unrelated individuals, this effect is moderate in diploids. We also show that vertical transmission prevents evolutionary branching and therefore can have a qualitative impact on gene-culture co-evolutionary dynamics. More generally, our model is of relevance to study scenarios of niche construction and gene-culture co-evolution within the family, where modifications of the environment by individuals have long-lasting effects on genetic lines of descent.