Optimal investment in sons and daughters when parents do not know the sex of their offspring

Optimal parental investment usually differs depending on the sex of the offspring. However, parents in most organisms cannot discriminate the sex of their young until those young are energetically independent. In a species with physical male–male competition, males are often larger and usually develop sexual ornaments, so male offspring are often more costly to produce. However, Onthophagus dung beetles (Coleoptera; Scarabaeidae) are highly dimorphic in secondary sexual characters, but sexually monomorphic in body size, despite strong male–male competition for mates. We demonstrate that because parents provide all resources required by their offspring before adulthood, O. atripennis exhibits no sexual size dimorphism irrespective of sexual selection pressure favoring sexual dimorphism. By constructing a graphic model with three fitness curves (for sons, daughters, and expected fitness return for parents), we demonstrate that natural selection favors parents that provide both sons and daughters with the optimal amount of investment for sons, which is far greater than that for daughters. This is because the cost of producing small sons, that are unable to compete for mates, is far greater than the cost of producing daughters that are larger than necessary. This theoretical prediction can explain sexual dimorphism without sexual size dimorphism, widely observed in species with crucial parental care such as dung beetles and leaf-rolling beetles, and may provide an insight into the enigmatic relationship between sexual size dimorphism and sexual dimorphism.