Defining fitness in an uncertain world

Paul Crewe, Richard Gratwick, Alan Grafen*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)1059-1099
Number of pages41
JournalJournal of Mathematical Biology
Issue number5
Early online date29 Jul 2017
Publication statusPublished - 1 Apr 2018


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


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