Gene-culture waves of advance

Major genetic and cultural changes may have been coupled during hominid evolution. Since hominids have had a wide geographical distribution for about one million years, any mutant gene or cultural innovation that became established had to spread from its origin. A pair of nonlinear diffusion equations is derived which models the propagation of a mutant gene and a cultural innovation. Both are assumed to originate in the same locality along a linear habitat. The mutant gene and its allele are semidominant, and the two cultural choices are transmitted according to what I call the logistic attraction-repulsion model. The genes influence cultural choice, and the two interact to determine fitness. Of particular interest is the case in which mutant gene and cultural innovation are mutually dependent, neither being able to spread without the other. Each equation of the pair is similar in form to Fisher's equation, with a linear function of the other dependent variable replacing the constant coefficient in the reaction term. The partial differential equations are solved numerically to obtain the asymptotic speeds. Their form also suggests an heuristic argument which has proved useful, but I have been unable to obtain any analytic results. The waves of the system are shown to be of two types, synchronous and asynchronous. When genes and culture are mutually dependent, synchronous travelling waves can exist. However, their existence is dependent on initial conditions, and the speed of propagation is slow.