| # Appendix B Stability of the monomorphic equilibria in the disparate benefits model | |
| There are six monomorphic equilibria in the disparate benefits model in the absence of innovation. These can be found by fixing each allele in turn and determining if cultural transmission loses or fixes the cultural in both sexes. The monomorphic equilibria are as follows: | |
| When $b_f > \mu$ and $b_m > \mu$, there is an equilibrium where the $B_1$ allele is fixed and cultural trait is fixed in both sexes. | |
| When $b_f > \mu$, there is a monomorphic equilibrium where the $B_2$ allele is fixed and the cultural trait is fixed in females but lost in males. | |
| When any of the alleles ($B_1, B_2, B_3$, or $B_4$) are fixed, there is a monomorphic equilibrium where the cultural trait is lost in both sexes. | |
| To determine the stability of each equilibrium, I calculated the characteristic equation for the system given by Equations 7 and 8 evaluated at the equilibrium. I then evaluated the eigenvalues numerically, where required, by substituting the values for $b_f, b_m$ and $\mu$ over the range explored in the main text. The results of this analysis are described below and shown in Figure B.1. Figure B.1 shows a close correspondence between the results of the stability analysis and the numerical simulation from the main text. | |
| There is one stable monomorphic equilibrium, which occurs over the range shown in A. In this equilibrium the $A_1$ allele is fixed and the cultural trait is fixed in both sexes. This range corresponds to the results of the numeric simulation where the $A_1$ allele and the cultural trait goes to fixation as shown in B and C. |