Supplemental material for:
Mothers teach daughters because daughters teach granddaughters: the evolution of sex-biased transmission
Matthew R. Zefferman
Appendix A Stability of the monomorphic equilibria in the uniparental teaching model
There are six monomorphic equilibria in the uniparental teaching 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 $b > 2t$ there is a monomorphic equilibrium where the $A_1$ allele is fixed and the cultural trait is fixed in both sexes.
When $b > t$ there is a monomorphic equilibrium where the $A_2$ allele is fixed and the cultural trait is fixed in females but lost in males.
When any of the alleles ($A_1$, $A_2$, $A_3$, or $A_4$) 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 2 and 3 evaluated at the equilibrium. I then evaluated the eigenvalues numerically, where required, by substituting the values for $b$, and $t$ over the ranges explored in the main text. The results of this analysis are described below and shown in Figure A.1. Figure A.1 shows a close correspondence between the results of the stability analysis and the numerical simulation from the main text.