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The recently elucidated definition of fitness employed by Fisher in his fundamental theorem of natural selection is combined with reproductive values as appropriately defined in the context of both random environments and continuing fluctuations in the distribution over classes in a class-structured population. We obtain astonishingly simple results, generalisations of the Price Equation and the fundamental theorem, that show natural selection acting only through the arithmetic expectation of fitness over all uncertainties, in contrast to previous studies with fluctuating demography, in which natural selection looks rather complicated. Furthermore, our setting permits each class to have its characteristic ploidy, thus covering haploidy, diploidy and haplodiploidy at the same time; and allows arbitrary classes, including continuous variables such as condition. The simplicity is achieved by focussing just on the effects of natural selection on genotype frequencies: while other causes are present in the model, and the effect of natural selection is assessed in their presence, these causes will have their own further effects on genoytpe frequencies that are not assessed here. Also, Fisher's uses of reproductive value are shown to have two ambivalences, and a new axiomatic foundation for reproductive value is endorsed. The results continue the formal darwinism project, and extend support for the individual-as-maximising-agent analogy to finite populations with random environments and fluctuating class-distributions. The model may also lead to improved ways to measure fitness in real populations.

Original publication




Journal article


J Math Biol

Publication Date





1059 - 1099


Class-distribution stochasticity, Formal darwinism, Fundamental theorem of natural selection, Price equation, Reproductive value, Stochastic environment