Theory of fads: Traveling-wave solution of evolutionary dynamics in a one-dimensional trait space

Abstract

We consider an infinite-sized population where an infinite number of traits compete simultaneously. The replicator equation with a diffusive term describes time evolution of the probability distribution over the traits due to selection and mutation on a mean-field level. We argue that this dynamics can be expressed as a variant of the Fisher equation with high-order correction terms. The equation has a traveling-wave solution, and the phase-space method shows how the wave shape depends on the correction. We compare this solution with empirical time-series data of given names in Quebec, treating it as a descriptive model for the observed patterns. Our model explains the reason that many names exhibit a similar pattern of the rise and fall as time goes by. At the same time, we have found that their dissimilarities are also statistically significant.