Is there an expected relationship between parental expenditure and sex ratio of litters or broods?

Parents may be selected to adjust the sex ratio of their offspring when parental expenditure yields different fitness returns from sons and daughters. This prediction is clear when parents produce only one offspring per reproductive attempt, but more complicated when parental resources are shared by several offspring, and parents may potentially influence the resource allocation among offspring as well as their number and sex. Here we present an optimization model to make predictions on how total parental expenditure may relate to the number and sex of offspring at every rank position within the litter or brood as well as the sex ratio of the litter/brood for the case of a large population with stable Fisherian sex ratio. We show that selection for sex ratio adjustment should be stronger for offspring at higher-ranking positions, for which the amount of resources received can be more predictable. Also, the relationship between parental resources devoted to a litter/brood of a given size and the primary sex ratio (proportion of males) is not expected to be a monotonically increasing function but rather a J-shaped relationship, steeper for small litters/broods and more extreme sexual dimorphism. Parental expenditure relates to increased sex ratio only for small variations in parental expenditure and for a given brood/litter size. For variable litter/brood sizes, a general relationship between parental resources and litter/brood sex ratio is not expected, although in practice pooling litters or broods of different sizes may produce negative relationships between parental expenditure and sex ratio of the litter/brood.