Defining fitness in an uncertain world

Abstract

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.

The Price equation and reproductive value

“…1 of Queller [1] very usefully displays a variety of mathematical results that help biologists think about evolution and natural selection. The current paper argues that these results can all be extended very straightforwardly, using the equation of Price [26] and its generalization by Crewe et al [3], to allow class structure and environmental uncertainty, by adopting the structure of basic reproductive value, Williams's reproductive value, and fitness, as derived from Fisher's original fundamental theorem. Fisher's concept of fitness, therefore, has an important role to play in biologists' thinking about natural selection.…”

“…It is good news for the study of evolution that the structure of reproductive value and fitness survives density dependence and stochasticity, though there are practical implications, notably that class-based schedule of per capita reproductive values will not remain constant over time; specifically, it will depend on the environment. The main result to justify this claim in the current context is due to Crewe et al [3], and Lion [7] makes parallel points about the dynamic nature of reproductive value in the presence of environmental change and density dependence, in his own setting. There is, of course, a kind of stationarity involved, whose foundational centrality will be discussed now.…”

“…Fisher did not reach this conclusion, because he allowed proportionality, and specifically did not require equality, in the one-generation-to-the-next requirement that supplies the premiss of the current argument. (See Lemma 2 on page 1080 of [3], for a mathematical version of the second point. )…”

“…However, the definitions required to make these fundamental theorems work have a special claim on us, and biologists should use them wherever possible and teach them to their students. The purpose of the current paper is to present verbally important points that have emerged in highly mathematical papers-readers seeking a mathematical exposition are referred elsewhere [2][3][4][5], though some of the points below represent more recent advances in understanding.…”

Should we ask for more than consistency of Darwinism with Mendelism?

“…The formal darwinism project is about that relationship, especially the foundational property (Darwin, 1859) that natural selection is an improving process. For a more detailed account of many of the points here that is suitable for a biological audience, with full citations, the reader is referred elsewhere (Grafen, 2007;Batty et al, 2014;Grafen, 2015a;Crewe et al, 2018). The formal darwinism project has already been discussed in a mixed biological and philosophical context (see Grafen, 2014, and associated papers): the main changes since 2014 are the publication of much more rigorous and general mathematical papers (Batty et al, 2014;Crewe et al, 2018), and the greater appreciation of the connection to Fisher (1930)'s fundamental theorem (Grafen, 2015a,b).…”