Total reproductive value of juvenile females is twice that of juvenile males under X-linkage and haplodiploidy

Abstract

Grafen (2014) has shown that, although the total reproductive value of females is not generally equal to that of males in an age-structured population under diploidy and autosomal inheritance, the total reproductive value of juvenile females is equal to that of juvenile males, provided there is a stable class distribution. It is the latter equality that is key to R.A. Fisher's famous explanation for equal investment into daughters and sons. Here, I simplify the derivation of Grafen's key result and extend the analysis to consider X-linkage and haplodiploid inheritance, i.e. scenarios in which a female receives one set of genes from her mother and one set from her father but where males receive genes only from their mother. I find that, although the total reproductive value of females need not be twice that of males, as is commonly supposed, the total reproductive value of juvenile females is twice that of juvenile males. This recovers the principle of equal maternal investment into daughters and sons in panmictic populations.