In this Appendix we provide the expressions for quantities referred to in the main text. We then derive these expressions, and provide proofs of our assertions.

We begin with the case where φ=0 (obligate asexuality). In this case it is straightforward to show that only the B_{1}B_{2} heterozygote is present at equilibrium, and thus, *W¯^*=1. For the remainder of the analysis presented in this Appendix, we assume φ > 0.

Let *x*_{1}, *x*_{2} and *x*_{3} represent, respectively, the frequencies of the genotypes B_{1}B_{1}, B_{1}B_{2} and B_{2}B_{2} among adults. Let us define *z* as follows:

Let *◯*_{1}, *◯*_{2} and *◯*_{3} represent, respectively, the equilibrium values of *x*_{1}, *x*_{2} and *x*_{3}, and let *z^* represent the quantity *◯*_{1}−*◯*_{3}. Using the assumption that *s*_{1} < *s*_{2}, we can write *z^* explicitly in terms of the parameters of the model as follows:

We can now write *◯*_{1}, *◯*_{2} and *◯*_{3} as

Let *p^* represent the equilibrium value of *p* (the frequency of B_{1}). We have

In the limit as φ goes to zero we have *z^*=0, and thus *z^*=½. To provide an estimate of *p^* when φ is small we note that, when φ << 1 we have (from eqn 2)

Thus, when φ is sufficiently small, *p^* is close to ½.

From the assumptions of the model, the equilibrium mean viability, *W¯^*, is given by

The assumptions also lead directly to the following expressions for *W¯^*_{S} and *W¯^*_{A}:

One can use eqns 2, 3, 4, 7, 8 and 9 to write *W¯^*, *W¯^*_{S} and 1*W¯^*_{A} as explicit functions of the parameters of the model, but the resulting expressions are lengthy, and not obviously useful.

To derive the foregoing quantities, let us begin by defining *q*=1 – *p*. We denote the average viability of sexually produced offspring during a particular generation as *W¯*_{S}, and the average viability for asexually produced offspring is *W¯*_{A}. The average viability of offspring (without respect to mode of reproduction) is simply *W¯*. From the assumptions of the model, we have

and *W¯*=(1−φ)*W¯*_{A}+φ*W¯*_{S}.

Let *x′*_{i} represent the value of *x*_{i} in the next generation. The assumptions of the model lead to the following expressions:

Equations 12–14 are not independent, as *x*_{1} + *x*_{2} + *x*_{3}=1. Thus, we need solve only two equations for two quantities. It is convenient to work with the variables *x*_{2} and *z*. We have

and from this we can write the following expressions:

where *z*′ is the value of *z* in the next generation.

Next, we search for *◯*_{2} and *z^* by finding values of *x*_{2} and *z* for which the left and right sides of the preceding two equations are equal. After some algebra, eqns 17 and 18 allow us to write the following expression for *◯*_{2}, in which φ does not appear:

We can use this, along with eqn 18, to get

This leads to a quadratic equation in *z^* which has two solutions. However, only one, which is given by eqn 2, corresponds to a valid solution, for which all three equilibrium genotype frequencies are non-negative.

To prove our assertions about the way that the equilibrium values of *W¯^*_{S}, *W¯^*_{A} and *W¯^* change with φ, we must first establish the fact that d*z^*/dφ>0. To do this, we use eqn 2 to get

By inspection, we can confirm that this is positive for all values of φ that satisfy 0 < φ < 1.

Next, we show that d*W¯^*_{S}/dφ>0. To begin, we note, using eqns 4 and 8, that

We know that d*z^*/dφ>0, and thus, from eqn 23, we see that [*s*_{2}−*s*_{1}−*z^*(*s*_{1}+*s*_{2})] reaches its minimum value when φ=1. When φ=1, we have (from eqn 2) *z^*=(*s*_{2}−*s*_{1})/(*s*_{1}+*s*_{2}), and thus, eqn 23 shows us that d*W¯^*_{S}/dφ=0 when φ=1. These observations imply that, for 0 < φ < 1, we have d*W¯^*_{S}/dφ>0.

To show that d*W¯^*_{A}/dφ<0, we substitute into eqn 9 from eqn 15 to get

It can be shown, using eqn 19 that (*s*_{2}−*s*_{1}+(*s*_{1}+*s*_{2}) (d*◯*_{2}/d*z^*))<0. Thus, since d*z^*/dφ>0 for 0 < φ < 1, we have d *W¯^*_{A}/dφ<0 for 0 < φ < 1.

Let us now prove that *W¯^*_{S}<*W¯^*_{A} always holds over the interval 0 < φ < 1. We have seen that d*W¯^*_{S}/dφ>0 and that d *W¯^*_{A}/dφ<0. Thus, we need only show that, in the limit as φ 1, we have *W¯^*_{S}<*W¯^*_{A}. In this limit, we have (from eqns 2, 3, 22 and 24)

Thus, in the same limit, we have

As this ratio is clearly in excess of unity, we have proved that *W¯^*_{A}>*W¯^*_{S} always holds for 0 < φ < 1.

Finally, we will show that d*W¯^*/dφ<0 within the interval 0 < φ < 1. Using eqns 3, 7 and 20, we can write *W¯^* in terms of *s*_{1}, *s*_{2} and *z^* as follows:

We know that, for 0 < φ < 1 we have d*z^*/dφ>0, and thus, since *s*_{1} < *s*_{2}, we have d*W¯^*/dφ<0.