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 $x_{f11} = x_{m11} = 1$.
128(1+b−2t)6(1+b)2λ6(2(1+b−2t)λ−1)((1+b−2t)λ−1)(2(1+b−2t)λ−(1+b−2t)−1)(4(1+b)(1+b−2t)λ2−2(1+b)(2+2b−3t)λ+b(1+b−t))(8(1+b)(1+b−2t)λ3+4(1+b)(1+b−2t)(2+b−2t)λ2−2(1+b)(b−t)(1+b−2t)λ+b(1+b−t))=0
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 $x_{f21} = x_{m20} = 1$.
256(1+b−t)5λ5(4(1+bt)λ2−2(2+2b−3t)λ−b(1+b−t))((4(1+b−t)λ2−2(2+b−t)λ−b))(2(1+b−t)λ−1)((1+b−t)λ−1)(8(1+b−t)λ3−4(2+b−t)λ2−2(b2−2bt+4b−2t)λ+b)=0
