samples/texts/2274792/page_20.md

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 $y_{f11} = y_{m11} = 1$.

$$\frac{{\lambda }^8}{32(1+b_m-\mu {)}^3(1+b_f-\mu {)}^4}(2(1+b_m-\mu )\lambda -1)((1+b_f-\mu )\lambda -1)\backslash frac\{\backslash lambda^8\}\{32(1+b\_m-\backslash mu)^3(1+b\_f-\backslash mu)^4\}\; \backslash left(2(1+b\_m-\backslash mu)\backslash lambda-1\backslash right)\; \backslash left((1+b\_f-\backslash mu)\backslash lambda-1\backslash right)$$

$$(2(1+b_f-\mu )\lambda -1)(2(1+b_m-\mu )(1+b_f-\mu )\lambda -(2+b_m+b_f-2\mu ))\backslash left(2(1+b\_f-\backslash mu)\backslash lambda-1\backslash right)\; \backslash left(2(1+b\_m-\backslash mu)(1+b\_f-\backslash mu)\backslash lambda\; -\; (2+b\_m+b\_f-2\backslash mu)\backslash right)$$

$$(4(1+b_m-\mu )(1+b_f-\mu ){\lambda }^2-2(1+b_m-\mu )(2+b_f-\mu )\lambda +(b_m-\mu ))=0\backslash left(4(1+b\_m-\backslash mu)(1+b\_f-\backslash mu)\backslash lambda^2\; -\; 2(1+b\_m-\backslash mu)(2+b\_f-\backslash mu)\backslash lambda\; +\; (b\_m-\backslash mu)\backslash right)\; =\; 0$$

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 $y_{f21} = y_{m20} = 1$.

$$\frac{{\lambda }^8}{32(1+b_f-\mu {)}^4}(2\lambda -(2+b_m-\mu ))((1+b_f-\mu )\lambda -1)\backslash frac\{\backslash lambda^8\}\{32(1+b\_f-\backslash mu)^4\}\; \backslash left(2\backslash lambda\; -\; (2+b\_m-\backslash mu)\backslash right)\; \backslash left((1+b\_f-\backslash mu)\backslash lambda\; -\; 1\backslash right)$$

$$(-4(1+b_f-\mu ){\lambda }^2+2(1+(1+b_m-\mu )(1+b_f-\mu ))\lambda -(b_m-\mu ))\backslash left(-4(1+b\_f-\backslash mu)\backslash lambda^2\; +\; 2(1+(1+b\_m-\backslash mu)(1+b\_f-\backslash mu))\backslash lambda\; -\; (b\_m-\backslash mu)\backslash right)$$

$$(2(1+b_f-\mu )\lambda -(2+b_f-\mu ))(2(1+b_f-\mu )\lambda -1)=0\backslash left(2(1+b\_f-\backslash mu)\backslash lambda\; -\; (2+b\_f-\backslash mu)\backslash right)\; \backslash left(2(1+b\_f-\backslash mu)\backslash lambda\; -\; 1\backslash right)\; =\; 0$$

I evaluated the resulting eigenvalues numerically by substituting the values of $b_f - \mu$ and