Mate limitation and sex ratio evolution

Sex ratio evolution has been one of the most successful areas of evolutionary theory. Pioneered by Düsing and Fisher under panmixia, and later extended by Hamilton to cover local mate competition (LMC), these models often assume, either implicitly or explicitly, that all females are fertilized. Here, we examine the effects of relaxing this assumption, under both panmictic and LMC models with diploid genetics. We revisit the question of the mathematical relationship between sex ratio and probability of fertilization, and use these results to model sex ratio evolution under risk of incomplete fertilization. We find that (i) under panmixia, mate limitation has no effect on the evolutionarily stable strategy (ESS) sex allocation; (ii) under LMC, mate limitation can make sex allocation less female-biased than under complete fertilization; (iii) contrary to what is occasionally stated, a significant fraction of daughters can remain unfertilized at the ESS in LMC with mate limitation; (iv) with a commonly used mating function, the fraction of unfertilized daughters can be quite large, and (v) with more realistic fertilization functions, the deviation becomes smaller. The models are presented in three equivalent forms: individual selection, kin selection and group selection. This serves as an example of the equivalence of the methods, while each approach has their own advantages. We discuss possible extensions of the model to haplodiploidy.


1) Data for Fig. 1: Empirical sex ratios for one foundress conditions
. Mean sex ratio (proportion male) produced by single foundresses. Data are from publications presented in West et al. (2005;Appendix ) as focusing on the role of foundress number under Local Mate Competition. Ploidy: HD = haplodiploid; PA = Pseudoarrhenotokous. When multiple treatments existed at single foundress conditions, the group sex ratio were averaged.

Reference
Species Ploidy

2) Derivation of the mating function
Assume that each male can successfully engage in matings on a given patch, in the time available for matings. If males mate indiscriminately, so that each female can be mated multiple times, and a male does not avoid re-mating with the same female, then: i) The total number of matings is , where l is the number of total number of individuals in the patch (i.e. all the offspring of both sexes of all foundresses).
ii) Because there are (1 − ) females, the probability of a given female being mated in a given mating event is 1 (1− ) , and the probability of not being mated in this event is iii) The probability of a given female not being mated in any of the matings is

vii)
Assuming that l is large, the same function can be found with a similar derivation even if a given pair never mates twice, but both males and females can mate multiply.

4) Stability analysis of equation (8)
To be an ESS (Eshel et al. 1997), the candidate trait value must satisfy the criterion 2 2 � = = * < 0 and to be convergence stable (Eshel et al. 1997), the criterion In both cases it is easy to see that the components in square brackets are positive when * ≥ 0, ≥ 0 and ≥ 1; therefore equation (8) is an ESS and convergence stable.

5) Stability analysis of equation (9)
With the mating function 2 ( ) we are restricted to numerical solutions for the equilibria.
Therefore we also take a partly numerical approach to stability analysis.
First we derive Next we numerically solve * for any combination of a and n from equation (9), plug these values of a, n and * into the two equations above and check their sign for a range of parameter values we are interested in. In both cases, all combinations in the range 1 ≤ ≤ 1000 and 1 ≤ ≤ 1000 resulted in negative values for both criteria. Therefore, the results with the mating function 2 ( ) are evolutionarily stable and convergence stable in this (very large) parameter range.

6) Derivation and stability analysis of equation (10)
The function 3 ( ) = min( 1− , 1) is continuous, but not differentiable at = 1 1+ , as can be seen in this example with a=2: The piecewise nature of this function complicates the analysis in some ways, and the resulting ESS is also piecewise defined.
Firstly, we know from classic LMC theory that when all females are fertilized, the ESS is * = −1 2 (Hamilton 1967;West 2009). This will be the case if To check this, we need only compute for the left side of = * = 1 1+ . This is because the right side is in the regime of standard LMC. We assume −1 2 < 1 1+ , and therefore in this region we already know that selection and the derivative are negative.
For the left side we use 3 ( ) = 1− and find = , which can be shown to be positive when < For convergence stability, we must show that � = similarly changes sign from + to -as y passes * = 1 1+ . This shows that if the population is perturbed from the ESS, it will return to it, and hence * = 1 1+ is convergence stable.
Again, with similar justification as above, we need only check the left side; on the right side, the derivative is known to be negative due to the stability of the standard LMC result.
For the left side we find that Therefore equation (7) in the main text is evolutionarily stable and convergence stable.