Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
10223732 | Theoretical Population Biology | 2018 | 10 Pages |
Abstract
A finite-population, discrete-generation model of cultural evolution is described, in which multiple discrete traits are transmitted independently. In this model, each newborn may inherit a trait from multiple cultural parents. Transmission fails with a positive probability unlike in population genetics. An ancestral process simulating the cultural genealogy of a sample of individuals is derived for this model. This ancestral process, denoted by Mâ, is shown to be dual to a process M+ describing the change in the frequency of a trait. The age-frequency spectrum is defined as a two-dimensional array whose (i,k) element is the expected number of distinct cultural traits introduced k generations ago and now carried by i individuals in a sample of a particular size n. Numerical calculations reveal that the age-frequency spectrum and related metrics undergo a critical transition from a phase with a moderate number of young, rare traits to a phase with numerous very old, common traits when the expected number of cultural parents per individual exceeds one. It is shown that M+ and Mâ converge to branching or deterministic processes, depending on the way population size tends to infinity, and these limiting processes bear some duality relationships. The critical behavior of the original processes M+ and Mâ is explained in terms of a phase transition of the branching processes. Using the results of the limiting processes in combination, we derive analytical formulae that well approximate the age-frequency spectrum and also other metrics.
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Authors
Yutaka Kobayashi, Joe Yuichiro Wakano, Hisashi Ohtsuki,