The fundamental theorem of natural selection in Ewens' sense. (Case of many loci)
Fisher's Fundamental Theorem of Natural Selection (FTNS) states that ``the rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time'' but ``the rigour of the demonstration requires that the terms employed should be used strictly as defined''. \textit{W. J. Ewens} [ibid. 36, No. 2, 167-180 (1989; Zbl 0702.92012)]\ claims that the FTNS should not be limited to continuous-time models and interprets the FTNS in the case of discrete nonoverlapping generations as follows: a partial change in the mean fitness of a population is exactly equal to the ratio of the additive genetic variance in fitness over the mean fitness. The partial change in the mean fitness is obtained by replacing the exact genotypic values of fitness by the associated additive genotypic values and by keeping these values constant from one generation to the next. Some of the main arguments in support of Ewens' interpretation include the following: the FTNS should be exact and it should hold without any assumption of random mating. A further contribution of Ewens (1989) is to show that the FTNS in its new interpretation holds in the case of fitness determined at many loci. But, in Eq. (26) therein, which characterizes the additive (average) allelic effects on fitness in this case, a term was inadvertently omitted. This equation is corrected in a follow-up paper of \textit{W. J. Ewens} [ibid. 42, No. 3, 333-346 (1992; Zbl 0768.92018)]. But a term is still missing in this corrected equation. We give the complete equation characterizing the additive allelic effects on fitness determined at many loci without the assumption of random mating. Then we confirm that the FTNS in Ewens' (1989) sense is fully valid in this general case. This is pertinent since ``the new interpretation leans heavily on the concept of the average effect of an allele'' (Ewens, 1992).
Fisher's fundamental theorem of natural selection
scientific article; zbMATH DE number 854732