Host-parasite coevolution and the stability of genetic kin recognition.

Crozier's paradox suggests that genetic kin recognition will not be evolutionarily stable. The problem is that more common tags (markers) are more likely to be recognized and helped. This causes common tags to increase in frequency, eliminating the genetic variability that is required for genetic kin recognition. Two potential solutions to this problem have been suggested: host-parasite coevolution and multiple social encounters. We show that the host-parasite coevolution hypothesis does not work as commonly assumed. Host-parasite coevolution only stabilizes kin recognition at a parasite resistance locus if parasites adapt rapidly to hosts and cause intermediate or high levels of damage (virulence). Additionally, when kin recognition is stabilized at a parasite resistance locus, this can have an additional cost of making hosts more susceptible to parasites. However, we show that if the genetic architecture is allowed to evolve, meaning natural selection can choose the recognition locus, genetic kin recognition is more likely to be stable. The reason for this is that host-parasite coevolution can maintain tag diversity at another (neutral) locus by genetic hitchhiking, allowing that other locus to be used for genetic kin recognition. These results suggest a way that host-parasite coevolution can resolve Crozier's paradox, without making hosts more susceptible to parasites. However, the opportunity for multiple social encounters may provide a more robust resolution of Crozier's paradox.

Multiple social encounters can eliminate Crozier's paradox and stabilise genetic kin recognition.

Crozier's paradox suggests that genetic kin recognition will not be evolutionarily stable. The problem is that more common tags (markers) are more likely to be recognised and helped. This causes common tags to increase in frequency, and hence eliminates the genetic variability that is required for genetic kin recognition. It has therefore been assumed that genetic kin recognition can only be stable if there is some other factor maintaining tag diversity, such as the advantage of rare alleles in host-parasite interactions. We show that allowing for multiple social encounters before each social interaction can eliminate Crozier's paradox, because it allows individuals with rare tags to find others with the same tag. We also show that rare tags are better indicators of relatedness, and hence better at helping individuals avoid interactions with non-cooperative cheats. Consequently, genetic kin recognition provides an advantage to rare tags that maintains tag diversity, and stabilises itself.

Extending the range of additivity in using inclusive fitness.

Inclusive fitness is a concept widely utilized by social biologists as the quantity organisms appear designed to maximize. However, inclusive fitness theory has long been criticized on the (uncontested) grounds that other quantities, such as offspring number, predict gene frequency changes accurately in a wider range of mathematical models. Here, we articulate a set of modeling assumptions that extend the range of scenarios in which inclusive fitness can be applied. We reanalyze recent formal analyses that searched for, but did not find, inclusive fitness maximization. We show (a) that previous models have not used Hamilton's definition of inclusive fitness, (b) a reinterpretation of Hamilton's definition that makes it usable in this context, and (c) that under the assumption of probabilistic mixing of phenotypes, inclusive fitness is indeed maximized in these models. We also show how to understand mathematically, and at an individual level, the definition of inclusive fitness, in an explicit population genetic model in which exact additivity is not assumed. We hope that in articulating these modeling assumptions and providing formal support for inclusive fitness maximization, we help bridge the gap between empiricists and theoreticians, which in some ways has been widening, demonstrating to mathematicians why biologists are content to use inclusive fitness, and offering one way to utilize inclusive fitness in general models of social behavior.

A simple completion of Fisher's fundamental theorem of natural selection.

Fisher's fundamental theorem of natural selection shows that the part of the rate of change of mean fitness that is due to natural selection equals the additive genetic variance in fitness. Fisher embedded this result in a model of total fitness, adding terms for deterioration of the environment and density dependence. Here, a quantitative genetic version of this neglected model is derived that relaxes its assumptions that the additive genetic variance in fitness and the rate of deterioration of the environment do not change over time, allows population size to vary, and includes an input of mutational variance. The resulting formula for total rate of change in mean fitness contains two terms more than Fisher's original, representing the effects of stabilizing selection, on the one hand, and of mutational variance, on the other, making clear for the first time that the fundamental theorem deals only with natural selection that is directional (as opposed to stabilizing) on the underlying traits. In this model, the total (rather than just the additive) genetic variance increases mean fitness. The unstructured population allows an explanation of Fisher's concept of fitness as simply birth rate minus mortality rate, and building up to the definition in structured populations.

The Price equation and reproductive value.

The Price equation is widely recognized as capturing conceptually important properties of natural selection, and is often used to derive versions of Fisher's fundamental theorem of natural selection, the secondary theorems of natural selection and other significant results. However, class structure is not usually incorporated into these arguments. From the starting point of Fisher's original connection between fitness and reproductive value, a principled way of incorporating reproductive value and structured populations into the Price equation is explained, with its implications for precise meanings of (two distinct kinds of) reproductive value and of fitness. Once the Price equation applies to structured populations, then the other equations follow. The fundamental theorem itself has a special place among these equations, not only because it always incorporated class structure (and its method is followed for general class structures), but also because that is the result that justifies the important idea that these equations identify the effect of natural selection. The precise definitions of reproductive value and fitness have striking and unexpected features. However, a theoretical challenge emerges from the articulation of Fisher's structure: is it possible to retain the ecological properties of fitness as well as its evolutionary out-of-equilibrium properties? This article is part of the theme issue 'Fifty years of the Price equation'.

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

A nonmathematical exposition of the current status of the formal darwinism project is presented, linking it to the fundamental theorem of natural selection, which is regarded as Fisher's own 'formal darwinism project'. The purpose is to found organism-level thinking about design and adaptation, in short Darwinism, on what is known about the mechanics of genetic inheritance, in short Mendelism, and the project is to do so in as general a biological setting as possible. This view also makes sense of the name 'fundamental theorem of natural selection'.

Inclusive fitness is an indispensable approximation for understanding organismal design.

For some decades most biologists interested in design have agreed that natural selection leads to organisms acting as if they are maximizing a quantity known as "inclusive fitness." This maximization principle has been criticized on the (uncontested) grounds that other quantities, such as offspring number, predict gene frequency changes accurately in a wider range of mathematical models. Here, we adopt a resolution offered by Birch, who accepts the technical difficulties of establishing inclusive fitness maximization in a fully general model, while concluding that inclusive fitness is still useful as an organizing framework. We set out in more detail why inclusive fitness is such a practical and powerful framework, and provide verbal and conceptual arguments for why social biology would be more or less impossible without it. We aim to help mathematicians understand why social biologists are content to use inclusive fitness despite its theoretical weaknesses. Here, we also offer biologists practical advice for avoiding potential pitfalls.

A general model of biological signals, from cues to handicaps.

Organisms sometimes appear to use extravagant traits, or "handicaps", to signal their quality to an interested receiver. Before they were used as signals, many of these traits might have been selected to increase with individual quality for reasons apart from conveying information, allowing receivers to use the traits as "cues" of quality. However, current theory does not explain when and why cues of individual quality become exaggerated into costly handicaps. We address this here, using a game-theoretic model of adaptive signalling. Our model predicts that: (1) signals will honestly reflect signaler quality whenever there is a positive relationship between individual quality and the signalling trait's naturally selected, non-informational optimum; and (2) the slope of this relationship will determine the amount of costly signal exaggeration, with more exaggeration favored when the slope is more shallow. A shallow slope means that a lower quality male would pay only a small fitness cost to have the same trait value as a higher quality male, and this drives the exaggeration of signals as high-quality signalers are selected to distinguish themselves. Our model reveals a simple and potentially widespread mechanism for ensuring signal honesty and predicts a natural continuum of signalling strategies, from cost-free cues to costly handicaps.

The left hand side of the Fundamental Theorem of Natural Selection.

The fundamental theorem of natural selection is explained here in very simple terms, suitable for students. The biological significance of the left hand side - the rate of change in mean fitness due to changes in gene frequencies, which is also described as the rate of change due to natural selection - has been regarded since 1972 as problematic, but here a simple graph is used to show that Fisher's poor explanation was of a robust and simple intuition. Simple numerical examples show the theorem at work with fixed genotypic fitness under two different mating systems, with bland density dependence, and also with fitnesses determined by an evolutionary game. The content of the theorem has long been taken for granted by whole-organism evolutionary biologists, though in an imprecise way, even while mathematical population geneticists have been, in sequence, wrongly proving it false, wrongly proving it requires more assumptions than Fisher admitted, and accepting the truth of the theorem as Fisher proved it, but doubting its biological significance. An important emphasis on the instantaneous nature of natural selection, and of its measurement, emerges from the argument. Price's disappointments with the content of the theorem are directly confronted. The new explanation allows us to recognise the central place the theorem already occupies in evolutionary biology, and to begin to incorporate more fully the insights embedded in it.

Defining fitness in an uncertain world.

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.

Biological fitness and the fundamental theorem of natural selection.

Fisher's fundamental theorem of natural selection is proved satisfactorily for the first time, resolving confusions in the literature about the nature of reproductive value and fitness. Reproductive value is defined following Fisher, without reference to genetic variation, and fitness is the proportional rate of increase in an individual's contribution to the demographic population size. The mean value of fitness is the same in each age class, and it also equals the population's Malthusian parameter. The statement and derivation are regarded as settled here, and so the general biological significance of the fundamental theorem can be debated. The main purpose of the theorem is to find a quantitative measure of the effect of natural selection in a Mendelian system, thus founding Darwinism on Mendelism and identifying the design criterion for biological adaptation, embodied in Fisher's ingenious definition of fitness. The relevance of the newly understood theorem to five current research areas is discussed.

Biological fitness and the Price Equation in class-structured populations.

Price׳s extended covariance selection mathematics is applied to class-structured populations with additional assumptions, to derive the 'genetic Price Equation with class structure'. Each individual belongs to a class, and there may be overlapping generations; the equation is genetic because the trait is restricted to an arbitrary weighted sum of allele frequencies. Two special cases are then considered, a demography-like case corresponding to Fisher's fundamental theorem of natural selection, and a sex-ratio-like case corresponding to Fisher's sex ratio argument: these differ in whether it is natural to assume that the per-capita or the total reproductive values of each class are maintained from the parental to the descendant population. These cases also match the two existing attempts to eliminate from the effects of natural selection those passive changes in allele frequencies that are caused by the class structure, and suggest improvements in one of them. In each case a more specialised Price Equation, and a 'fundamental theorem of natural selection', are proved, which hold out of class-structure equilibrium, showing that passive changes can be eliminated in more than one way, and hinting at the possibility of a more general formulation. Previous class-structured Price Equations and a 'fundamental theorem' are linked to these results. The power of Price's formal approach is vividly illustrated by this lucid conspectus of otherwise self-standing theories with confusing interconnections.

The evolution of index signals to avoid the cost of dishonesty.

Animals often convey useful information, despite a conflict of interest between the signaller and receiver. There are two major explanations for such 'honest' signalling, particularly when the size or intensity of signals reliably indicates the underlying quality of the signaller. Costly signalling theory (including the handicap principle) predicts that dishonest signals are too costly to fake, whereas the index hypothesis predicts that dishonest signals cannot be faked. Recent evidence of a highly conserved causal link between individual quality and signal growth appears to bolster the index hypothesis. However, it is not clear that this also diminishes costly signalling theory, as is often suggested. Here, by incorporating a mechanism of signal growth into costly signalling theory, we show that index signals can actually be favoured owing to the cost of dishonesty. We conclude that costly signalling theory provides the ultimate, adaptive rationale for honest signalling, whereas the index hypothesis describes one proximate (and potentially very general) mechanism for achieving honesty.

Total reproductive values for females and for males in sexual diploids are not equal.

A very simple mathematical exposition of reproductive value in an age- and sex-structured sexual diploid population employs reproductive value as the probability that a random gene in a distant generation traces its ancestry to a given individual or set of individuals today. Although the total reproductive values of all females and that of all males are not in general equal, but instead proportional to the average age of a new mother and a new father, respectively, Fisher׳s equal-investment conclusion for the sex ratio remains valid because the total reproductive value of age-zero females equals the total reproductive value of age-zero males. However, the conclusion is seen to require an extra assumption, namely stability of the age-distribution.

Foundations of a mathematical theory of darwinism.

This paper pursues the 'formal darwinism' project of Grafen, whose aim is to construct formal links between dynamics of gene frequencies and optimization programmes, in very abstract settings with general implications for biologically relevant situations. A major outcome is the definition, within wide assumptions, of the ubiquitous but problematic concept of 'fitness'. This paper is the first to present the project for mathematicians. Within the framework of overlapping generations in discrete time and no social interactions, the current model shows links between fitness maximization and gene frequency change in a class-structured population, with individual-level uncertainty but no uncertainty in the class projection operator, where individuals are permitted to observe and condition their behaviour on arbitrary parts of the uncertainty. The results hold with arbitrary numbers of loci and alleles, arbitrary dominance and epistasis, and make no assumptions about linkage, linkage disequilibrium or mating system. An explicit derivation is given of Fisher's Fundamental Theorem of Natural Selection in its full generality.

Formalizing Darwinism and inclusive fitness theory.

Inclusive fitness maximization is a basic building block for biological contributions to any theory of the evolution of society. There is a view in mathematical population genetics that nothing is caused to be maximized in the process of natural selection, but this is explained as arising from a misunderstanding about the meaning of fitness maximization. Current theoretical work on inclusive fitness is discussed, with emphasis on the author's 'formal Darwinism project'. Generally, favourable conclusions are drawn about the validity of assuming fitness maximization, but the need for continuing work is emphasized, along with the possibility that substantive exceptions may be uncovered. The formal Darwinism project aims more ambitiously to represent in a formal mathematical framework the central point of Darwin's Origin of Species, that the mechanical processes of inheritance and reproduction can give rise to the appearance of design, and it is a fitting ambition in Darwin's bicentenary year to capture his most profound discovery in the lingua franca of science.

Natural selection of altruism in inelastic viscous homogeneous populations.

Biological explanations are given of three main uninterpreted theoretical results on the selection of altruism in inelastic viscous homogeneous populations, namely that non-overlapping generations hinder the evolution of altruism, fecundity effects are more conducive to altruism than survival effects, and one demographic regime (so-called death-birth) permits altruism whereas another (so-called birth-death) does not. The central idea is 'circles of compensation', which measure how far the effects of density dependence extend from a focal individual. Relatednesses can then be calculated that compensate for density dependence. There is very generally a 'balancing circle of compensation', at which the viscosity of the population slows up selection of altruism, but does not affect its direction, and this holds for altruism towards any individual, not just immediate neighbours. These explanations are possible because of recent advances in the theory of inclusive fitness on graphs. The assumption of node bitransitivity in that recent theory is relaxed to node transitivity and symmetry of the dispersal matrix, and new formulae show how to calculate relatedness from dispersal and vice versa.

Detecting kin selection at work using inclusive fitness.

A recent model shows that altruism can evolve with limited migration and variable group sizes, and the authors claim that kin selection cannot provide a sufficient explanation of their results. It is demonstrated, using a recent reformulation of Hamilton's original arguments, that the model falls squarely within the scope of inclusive fitness theory, which furthermore shows how to calculate inclusive fitness and the relevant relatedness. A distinction is drawn between inclusive fitness, which is a method of analysing social behaviour; and kin selection, a process that operates through genetic similarity brought about by common ancestry, but not by assortation by genotype or by direct assessment of genetic similarity. The recent model is analysed, and it turns out that kin selection provides a sufficient explanation to considerable quantitative accuracy, contrary to the authors' claims. A parallel analysis is possible and would be illuminating for all models of social behaviour in which individuals' effects on each other's offspring numbers combine additively.

A theory of Fisher's reproductive value.

The formal Darwinism project aims to provide a mathematically rigorous basis for optimisation thinking in relation to natural selection. This paper deals with the situation in which individuals in a population belong to classes, such as sexes, or size and/or age classes. Fisher introduced the concept of reproductive value into biology to help analyse evolutionary processes of populations divided into classes. Here a rigorously defined and very general structure justifies, and shows the unity of concept behind, Fisher's uses of reproductive value as measuring the significance for evolutionary processes of (i) an individual and (ii) a class; (iii) recursively, as calculable for a parent as a sum of its shares in the reproductive values of its offspring; and (iv) as an evolutionary maximand under natural selection. The maximand is the same for all parental classes, and is a weighted sum of offspring numbers, which implies that a tradeoff in one aspect of the phenotype can legitimately be studied separately from other aspects. The Price equation, measure theory, Markov theory and positive operators contribute to the framework, which is then applied to a number of examples, including a new and fully rigorous version of Fisher's sex ratio argument. Classes may be discrete (e.g. sex), continuous (e.g. weight at fledging) or multidimensional with discrete and continuous components (e.g. sex and weight at fledging and adult tarsus length).