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.
Highlights
The individual-as-maximising-agent analogy is a central principle in many subdisciplines of biology, but in other sub-disciplines is regarded as inaccurate and misleading
Mathematical population genetics stays very close to the Mendelian machinery (e.g. Ewens 2004), with the work of Charlesworth and colleagues going furthest towards Darwinian ideas; adaptive dynamics makes necessary assumptions to provide an ecologically useful representation of the effects of natural selection, including extinction and speciation; other literatures such as that of Taylor and colleagues (e.g. Taylor 1990, 1996; Taylor et al 2007) and that of Rousset and colleagues (e.g. Rousset 2004; Lehmann et al 2006), aim to make reasonable genetic assumptions to solve important modelling problems suggested by behavioural ecology, and within their frameworks to provide general results about inclusive fitness
The formal darwinism project (Grafen 1999, 2002, 2006a, b; Batty et al 2014) aims to justify the individualas-maximising agent analogy in terms of population genetics, but eschews dynamic sufficiency in order to obtain results that apply to a wider range of genetical systems
Summary
The individual-as-maximising-agent analogy is a central principle in many subdisciplines of biology, but in other sub-disciplines is regarded as inaccurate and misleading We study this question in structured populations with stochastic demography, that is, with continuing fluctuations in class distribution. This formula is Fisher’s definition transferred to a discrete-time setting (Grafen 2015a, eqn 39) This relativising of Williams’ reproductive value to the class-predicted reproductive value provides a concept of fitness with a number of technically convenient properties, that together permit the generalisations of the Price Equation and fundamental theorem in which fitness appears only as an arithmetic expectation over relevant uncertainties. The concept of fitness has further significant properties suitable for a maximand in the ‘individual-as-maximising-agent’ analogy It is not a single ‘tombstone’ measure for an individual, but is defined in each year, and in each year is based entirely on future reproduction. We hope this introduction provides enough motivation for further reading
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