samples/texts/2274792/page_5.md

$$x''{sjk1} = \frac{1}{2} \left( \tau{sjk} x'{fjk1} + \sum_a \sum_b x'{makb} \sum_d \tau_{sjd} x'{fjd1} + \sum_g \sum_h x'{mjgh} \sum_m \tau_{smk} x'{fmk1} \right. \ \left. + \sum_n x'{njkn} \sum_p \sum_q \tau_{spq} x'_{fpq1} \right) \quad (C.2)$$

Individuals innovate a trait with a probability of $\nu$, increasing the fraction of encultured individuals, Equation C.4, and decreasing the fraction of unencultured individuals, Equation C.5.

$$x_{sk1}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}=x_{sjk1}^{\mathrm{\prime }\mathrm{\prime }}+\nu x_{sjk0}^{\mathrm{\prime }\mathrm{\prime }}(C.4)x\text{'}\text{'}\text{'}\_\{sk1\}\; =\; x\text{'}\text{'}\_\{sjk1\}\; +\; \backslash nu\; x\text{'}\text{'}\_\{sjk0\}\; \backslash quad\; (C.4)$$

$$x_{sjk0}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}=x_{sjk0}^{\mathrm{\prime }\mathrm{\prime }}(1-\nu )(C.5)x\text{'}\text{'}\text{'}\_\{sjk0\}\; =\; x\text{'}\text{'}\_\{sjk0\}(1\; -\; \backslash nu)\; \backslash quad\; (C.5)$$

As in the main text, I ran numeric simulations of this system for various combinations of teaching costs, $t$, and reproductive benefits of the cultural trait, $b$ and an innovation rate of $r = 0.005$. As in the main text, for each parameter combination, I ran the simulation starting with four initial allele frequencies. In each frequency one allele started at 5% of the population and the rest were evenly distributed among the rest of the population. I started the frequency of the cultural trait at zero in all four initial conditions and ran the simulations until they either converged to a shared equilibrium or ran for $10^7$ generations.

The results of the simulation are shown in Figure C.1