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 0,1,.... 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(iA)(j) 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(iA)(j). Then V(ij)( t+1)- V(ij)( t) is equal to m(ij) V(ij)( t), where m(ij) is the Malthusian parameter for A(iA)(j). Furthermore, if the mean and variance at time t of the m(ij)'s, weighted by their corresponding reproductive values, are respectively ( t) and Sigma(m)(2)( t), then m( t+1)-m( t)=Sigma(m)(2)( t)/(1+m( t)).