Speaker
Description
In cultural evolution, the frequency of a cultural trait often exhibits a nonlinear relationship with its frequency in the previous time step: a trait held by the majority tends to become increasingly common even in the absence of any advantage over the minority trait. This pattern has been attributed to conformist bias, which has been shown to generate a sigmoidal relationship between past and present frequencies. However, recent studies suggest that other mechanisms, including sampling biases, can produce similar patterns. Here, we identify a novel mechanism: fluctuations in the proportion of individuals allowed to switch traits. Consider the discrete-time dynamics of two cultural traits, A and B. Let $p_t$ denote the frequency of trait-A carriers at time $t$. At each time step, a fraction $k^+_t$ of trait-A carriers and a fraction $k^-_t$ of trait-B carriers update their traits based on their observations of the frequencies of the two traits. The values of $(k^+_t, k^-_t)$ change periodically with period $n$. For $n=2$, we analytically obtain the following results. Only the trivial equilibria, $p=0$ and $p=1$, can be locally stable. The sigmoidal relationship between $p_t$ and $p_{t-n}$ can arise when the changes in $p_t$ at two consecutive steps occur in opposite directions; in this case, an unstable internal equilibrium exists, and $p=0$ and $p=1$ are simultaneously locally stable. We confirmed analytically or numerically that these properties also hold for $n \geq 3$.
