Sex ratio theory for facultative parthenogens: from fortuitously optimal stick insects to the origin of haplodiploidy in Hymenoptera

Sex ratio theory usually assumes obligate sex; rare exceptions with facultative sex typically consider idiosyncratic cases of cyclic parthenogens. Here, we construct a general theoretical framework for facultative parthenogens. We show that facultative parthenogenesis selects for female-biased sex ratios by elevating the class reproductive value of females. The degree of this bias depends on the future rate of parthenogenesis. This complicates calculations for cyclic parthenogens, but in stable environments (with stable rates of parthenogenesis), the optimal sex ratio can result automatically from constraints caused by preexisting sex chromosomes: if sexually produced offspring retain unbiased sex ratios while parthenogenetically produced offspring are female (example: stick insects), optimality is achieved for any rate of parthenogenesis. Conversely, in birds and haplodiploids, parthenogenesis produces males, resulting in suboptimal sex ratios. Nevertheless, male-producing parthenogenesis can invade and reach an equilibrium frequency, if the reproductive value of parthenogenetically produced brood is compromised by less than 50%. We argue that this condition is not met in birds due to inviable WW and homozygous ZZ offspring. For haplodiploids, on the other hand, our work resurrects a somewhat forgotten idea by Bull (1981) that haplodiploidy in Hymenoptera evolved from a diplodiploid ancestor with complementary sex determination.