| ## A.1 Stability of the equilibrium where the $A_1$ allele and cultural trait are fixed in both sexes | |
| To determine the stability of this equilibrium, I found the characteristic equation for the system given by Equations 2 and 3 evaluated where $x_{f11} = x_{m11} = 1$. | |
| $$ | |
| \begin{aligned} | |
| & \frac{\lambda^6}{128(1+b-2t)^6(1+b)^2} (2(1+b-2t)\lambda - 1) ((1+b-2t)\lambda - 1) \\ | |
| & \quad (2(1+b-2t)\lambda - (1+b-2t)-1) (4(1+b)(1+b-2t)\lambda^2 - 2(1+b)(2+2b-3t)\lambda + b(1+b-t)) \\ | |
| & \quad (8(1+b)(1+b-2t)\lambda^3 + 4(1+b)(1+b-2t)(2+b-2t)\lambda^2 - 2(1+b)(b-t)(1+b-2t)\lambda + \\ | |
| & \qquad \qquad \qquad \qquad b(1+b-t)) = 0 | |
| \end{aligned} | |
| $$ | |
| I then evaluated the resulting eigenvalues numerically by substituting the values for $b$ and $t$ over the range explored in the main text. Figure A.1A shows the range of parameters where this equilibrium is stable. There is a close correspondence between this region and where the $A_1$ allele (Figure A.1B) and the cultural trait (Figure A.1C) go to fixation in the numerical simulations described in the main text. | |
| ## A.2 Stability of $A_2$ equilibrium with culture in females | |
| To determine the stability of this equilibrium, I calculated the characteristic equation for the system given by Equations 2 and 3 evaluated where $x_{f21} = x_{m20} = 1$. | |
| $$ | |
| \begin{aligned} | |
| & \frac{\lambda^5}{256(1+b-t)^5} \Biggl(4(1+bt)\lambda^2 - 2(2+2b-3t)\lambda - b(1+b-t)\Biggr) \\ | |
| & \quad \Biggl((4(1+b-t)\lambda^2 - 2(2+b-t)\lambda - b)\Biggr) \Biggl(2(1+b-t)\lambda - 1\Biggr) \Biggl((1+b-t)\lambda - 1\Biggr) \\ | |
| & \quad \Biggl(8(1+b-t)\lambda^3 - 4(2+b-t)\lambda^2 - 2(b^2 - 2bt + 4b - 2t)\lambda + b\Biggr) = 0 | |
| \end{aligned} | |
| $$ |