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 . 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 A_iAj is denoted by m_ij+1=m_ji+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 V_ij(t) for genotype A_iAj. Then V_ij(t+1)–V_ij(t) is equal to m_ijV_ij(t), where m_ij is the Malthusian parameter for A_iAj. Furthermore, if the mean and variance at time t of the m_ijs, weighted by their corresponding reproductive values, are respectively (t) and _m²(t), then m¯(t+1)–m¯(t)=_m²(t)/(1+m¯(t)).