Gene frequency and fitness change in an age‐structured population

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Abstract
Fisher's system of reproductive value weighting, whereby each age group is assigned a weight purported to measure its contribution to the ancestry of future generations, was suggested by him as a way of ironing out irregularities in the change of population numbers when the age structure is not in equilibrium. This has been extended to Mendelian populations for two models. In Model I, the reproductive value of each genotype is computed from a table of age-specific survival and reproduction rates, and the genic values are computed by averaging these genotypic values. The total reproductive value of an allele and of the population always increase at a rate equal to the reproductive value-weighted average fitness regardless of age structure. This has the disadvantage that the total reproductive value is not equal to the census numbers, when age-stability has been reached. In Model II, this difficulty is surmounted, but the formula is no longer exact. The reproductive value of an allele for a specific age x is measured from the average death and reproductive rates of individuals of age x carrying that allele. An expression is given for the rate of change of reproductive value of an allele or of the population. In many circumstances this changes nearly uniformly, regardless of irregularities of age structure, and goes over to the census numbers as age stability is approached. The special difficulties in populations with separate sexes are discussed and a formula for rate of change of mean reproductive value, analogous to Fisher's Fundamental Theorem of Natural Selection, is given.