Time to extinction of a cultural trait in an overlapping generation model.

How long a newly emerging trait will stay in a population is a fundamental but rarely asked question in cultural evolution. To tackle this question, the distribution and mean of the time to extinction of a discrete cultural trait are derived for models with overlapping generations, in which trait transmission occurs from multiple role models to a single newborn and may fail with a certain probability. We explore two models. The first is a Moran-type model, which allows us to derive the exact analytical formula for the mean time to extinction of a trait in a finite population. The second is a branching process, which assumes an infinitely large population and allows us to derive approximate analytical formulae for the distribution and mean of the time to extinction in the first model under a large population size. We show that in the first model, the mean time to extinction apparently diverges (becomes so large that even numerical computation is impractical) under a certain parameter condition as the population size tends to infinity. Using the second model, we explain the underlying mechanism of the apparent divergence found in the first model and derive the mathematical condition for this divergence in terms of transmission efficiency and the number of role models per newborn. When this mathematical condition is satisfied in the second model, the probability of extinction is less than 1, and the mean extinction time does not exist. In addition, we find that in both models, the time to extinction of the trait becomes longer as the number of role models per individual increases and as cultural transmission becomes more efficient.