| ## B.1 Stability of the $B_1$ equilibrium with culture | |
| To determine the stability of this equilibrium, I calculated the characteristic equation for the system given by Equations 7 and 8 evaluated where $y_{f11} = y_{m11} = 1$. | |
| $$ \frac{\lambda^8}{32(1+b_m-\mu)^3(1+b_f-\mu)^4} \left(2(1+b_m-\mu)\lambda-1\right) \left((1+b_f-\mu)\lambda-1\right) $$ | |
| $$ \left(2(1+b_f-\mu)\lambda-1\right) \left(2(1+b_m-\mu)(1+b_f-\mu)\lambda - (2+b_m+b_f-2\mu)\right) $$ | |
| $$ \left(4(1+b_m-\mu)(1+b_f-\mu)\lambda^2 - 2(1+b_m-\mu)(2+b_f-\mu)\lambda + (b_m-\mu)\right) = 0 $$ | |
| Explicit expressions for the eigenvalues can be found, though the eigenvalues resulting from the cubic factor of the characteristic polynomial are difficult to evaluate symbolically. Therefore, I evaluated the eigenvalues numerically by substituting the values for $b_f - \mu$, and $b_m - \mu$ explored in the main text. Figure B.1 shows a close correspondence between the region where this equilibrium is stable and where $B_1$ and the cultural trait go to fixation in the numerical simulations in the main text. | |
| ## B.2 Stability of the $B_2$ equilibrium with culture in females | |
| To determine the stability of this equilibrium, I calculated the characteristic equation for the system given by Equations 7 and 8 evaluated where $y_{f21} = y_{m20} = 1$. | |
| $$ \frac{\lambda^8}{32(1+b_f-\mu)^4} \left(2\lambda - (2+b_m-\mu)\right) \left((1+b_f-\mu)\lambda - 1\right) $$ | |
| $$ \left(-4(1+b_f-\mu)\lambda^2 + 2(1+(1+b_m-\mu)(1+b_f-\mu))\lambda - (b_m-\mu)\right) $$ | |
| $$ \left(2(1+b_f-\mu)\lambda - (2+b_f-\mu)\right) \left(2(1+b_f-\mu)\lambda - 1\right) = 0 $$ | |
| I evaluated the resulting eigenvalues numerically by substituting the values of $b_f - \mu$ and |