Malthusian parameters, reproductive values and change under selection in self fertilizing age-structured populations

A population, reproducing wholly by selfing, is assumed to be observed at times \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{0,1,\ldots}}$\end{document}. Individuals between x−1 and x units of age at time t are said to be in age class x at that time. The rate of increase in the long run of individuals of type AiAj is denoted by mij+1=mji+1. For each genotype there is also a set of reproductive values, corresponding to all age classes and genotypes of individuals having descendants of that genotype. Then, if the number of individuals of each sort of ancestor is multiplied by its reproductive value and the products are summed, the result is the total value, which is Vij(t) for genotype AiAj. Then Vij(t+1)−Vij(t) is equal to mijVij(t), where mij is the Malthusian parameter for AiAj. Furthermore, if the mean and variance at time t of the mij‘s, weighted by their corresponding reproductive values, are respectively (t) and Σm2(t), then m¯(t+1)−m¯(t)=Σm2(t)/(1+m¯(t)).