Inclusive fitness in finite populations—effects of heterogeneity and synergy

I review recent results concerning the relationship between the inclusive fitness (IF) effect and standard measures of allele fitness in a finite‐population, with attention to the effect of heterogeneity in population structure and nonadditive fitness effects. In both cases, existing theoretical work is somewhat technical and I try to provide a more transparent account. In a heterogeneous population it is known that inclusive fitness will generally fail to incorporate the effects of selection on the distribution of alleles among states unless a reproductive‐value weighting is used. But even given that, recent work shows that under certain updating rules, the IF effect can fail to be equivalent to standard measures such as fixation probability. In terms of synergistic fitness effects, I review the result that in the finite population model, the IF effect can be calculated using only “additive” relatedness coefficients so that computational difficulties found in the infinite‐population model do not arise. In my own work, there is an interaction here in that my 2012 work on synergy with Maciejewski made an assumption about inclusive fitness that my 2014 work on heterogeneity with Tarnita showed to be wrong. I include (Appendix C) a corrected argument for the 2012 result.     

Relatedness in spatially structured populations with empty sites: An approach based on spatial moment equations

Taking into account the interplay between spatial ecological dynamics and selection is a major challenge in evolutionary ecology. Although inclusive fitness theory has proven to be a very useful tool to unravel the interactions between spatial genetic structuring and selection, applications of the theory usually rely on simplifying demographic assumptions. In this paper, I attempt to bridge the gap between spatial demographic models and kin selection models by providing a method to compute approximations for relatedness coefficients in a spatial model with empty sites. Using spatial moment equations, I provide an approximation of nearest-neighbour relatedness on random regular networks, and show that this approximation performs much better than the ordinary pair approximation. I discuss the connection between the relatedness coefficients I define and those used in population genetics, and sketch some potential extensions of the theory.     

Relatedness with different interaction configurations

In an inclusive fitness model of social behaviour, a key concept is that of the relatedness between two interactants. This is typically calculated with reference to a “focal” actor taken to be representative of all actors, but when there are different interaction configurations, relatedness must be constructed as an average over all such configurations. We provide an example of such a calculation in an island model with local reproduction but global mortality, leading to variable island size and hence variable numbers of individual interactions. We find that the analysis of this example significantly sharpens our understanding of relatedness. As an application, we obtain a version of Hamilton's rule for a tag-based model of altruism in a randomly mixed population. For large populations, the selective advantage of altruism is enhanced by low (but not too low) tag mutation rates and large numbers of tags. For moderate population sizes and moderate numbers of tags, we find a window of tag mutation rates with critical benefit/cost ratios of between 1 and 3.     

A theoretical basis for measures of kin selection in subdivided populations: finite populations and localized dispersal

The analysis of kin selection in subdivided populations has been hampered by the lack of well‐defined measures of genealogical relatedness in the presence of localized dispersal. Furthermore, the usual arguments underlying the definition of game‐theoretical measures of inclusive fitness are not exact under localized dispersal. We define such measures to give the first‐order effects of selection on the probability of fixation of an allele. The derived measures of kin selection and relatedness are valid in finite populations and under localized dispersal. For the infinite island model, the resulting measure of kin selection is equivalent to a previously used measure. In other cases its definition is based on definitions of relatedness which are different from the usual ones. To illustrate the approach, we reanalyse a model with localized dispersal. We consider sex ratio evolution under sex‐specific dispersal behaviour, and the results confirm the earlier conclusion that the sex ratio is biased towards the sex with the dispersal rate closer to the optimal dispersal rate in the absence of sex‐specific dispersal behaviour.     

A quantitative genetic model of two-policy games between relatives

Equations are derived for the change per generation of the population mean of the probability that an individual adopts a policy 1 as opposed to a policy 2 in a behavioral interaction between two diploid individuals of the same generation in which two policies are possible. The probability is assumed to be a quantitative genetic trait determined by many additively acting genes of small effects and an independent environmental component. Equations are derived for the case that interactions occur at random between all members of the population and also for the case that interactions occur between relatives of the same average degree of relatedness. It is assumed that each group of relatives and the number of such groups is sufficiently large. For a quantitative genetic trait with the additional assumption of unlinked loci the latter equation can be heuristically derived from the first by substituting the corresponding inclusive fitness effects. When per locus selection coefficients are small and linkage equilibrium holds, the average degree of relatedness can be equated approximately with Wright's coefficient of relationship. Thus, the quantitative genetic model provides a genetic basis for the inclusive fitness approach toward games between relatives. By contrast, in a monogenic system with major gene effects we obtain substantially different results which contradict those obtained by the inclusive fitness approach in game theory. Applications are made to the hawk-dove game, and the simple and iterated forms of the prisoner's dilemma.     

Relatedness,Conflict, and the Evolution of Eusociality

The evolution of sterile worker castes in eusocial insects was a major problem in evolutionary theory until Hamilton developed a method called inclusive fitness. He used it to show that sterile castes could evolve via kin selection, in which a gene for altruistic sterility is favored when the altruism sufficiently benefits relatives carrying the gene. Inclusive fitness theory is well supported empirically and has been applied to many other areas, but a recent paper argued that the general method of inclusive fitness was wrong and advocated an alternative population genetic method. The claim of these authors was bolstered by a new model of the evolution of eusociality with novel conclusions that appeared to overturn some major results from inclusive fitness. Here we report an expanded examination of this kind of model for the evolution of eusociality and show that all three of its apparently novel conclusions are essentially false. Contrary to their claims, genetic relatedness is important and causal, workers are agents that can evolve to be in conflict with the queen, and eusociality is not so difficult to evolve. The misleading conclusions all resulted not from incorrect math but from overgeneralizing from narrow assumptions or parameter values. For example, all of their models implicitly assumed high relatedness, but modifying the model to allow lower relatedness shows that relatedness is essential and causal in the evolution of eusociality. Their modeling strategy, properly applied, actually confirms major insights of inclusive fitness studies of kin selection. This broad agreement of different models shows that social evolution theory, rather than being in turmoil, is supported by multiple theoretical approaches. It also suggests that extensive prior work using inclusive fitness, from microbial interactions to human evolution, should be considered robust unless shown otherwise.     

Mating structure and nestmate relatedness in a communal bee, Andrena jacobi (Hymenoptera, Andrenidae), using microsatellites

Complex eusocial insect societies are generally matrifilial, suggesting kin selection has been of importance in their development. For simpler social systems, factors favouring their existence, in particular kin selection, have rarely been studied. Communal nesting is one of these simple social organizations, and is found in a diversity of insect species. To examine whether kin selection may play a role in the evolution and maintenance of communality, we estimated genetic relatedness of nestmate females of the facultatively communal bee, Andrena jacobi . Microsatellite loci were developed for this species and used to analyse individuals from two populations. Loci were variable, they were in heterozygote deficit and showed positive inbreeding coefficients. This may arise from nonrandom mating; previous observations (Paxton & Tengö 1996) indicate that a large proportion of females mate intranidally with nestmate males in their natal nests before first emerging. Nestmate relatedness was low, no different from zero for all loci in one population and for three of four loci in the other population. The large number of nestmates sharing a common nest (up to 594) may explain the low relatedness estimates, although relatedness was also independent of the number of females sharing a nest. Lack of inclusive fitness payoffs could constrain social evolution in this communal species.     

Colony-level sex ratio selection in the eusocial Hymenoptera

We present an inclusive fitness model on worker-controlled sex investments in eusocial Hymenoptera which expands the existing theory for random mating populations as formulated by Trivers and Hare (1976) and Benford (1978). We assume that relatedness asymmetry is variable among colonies — owing to multiple mating, worker reproduction and polygyny — and that workers are able to assess the relatedness asymmetry in their own colony. A simple marginal value argument shows that “assessing” workers maximize their inclusive fitness by specializing on the production of the sex to which they are relatively more related than the average worker in the population is related to that sex. The model confirms our earlier verbal argument on this matter (Boomsma and Grafen, 1990) and gives further quantitative predictions of the optimal sex ratio of relatedness-asymmetry classes for both infinite and finite, random mating populations. It is shown that in large populations all but one of the relatedness-asymmetry classes should specialize on the production of one sex only. The remaining, balancing class is selected to compensate any bias induced by the other class(es) such that the population sex ratio reflects the relatedness asymmetry of that balancing class. In the absence of worker-reproduction, the sex ratio compensation by the balancing-class is generally close to 100%, unless the population is very small. In the Discussion we address explicitly the likelihood of our relatedness-assessment hypothesis and other assumptions made in the model. The relationship of our model with previous theory on sex allocation in eusocial Hymenoptera is worked out in the Appendix.     

Measures of Relatedness

Measures of genetic relatedness are essential to models of evolution by kin selection and determinations of inclusive fitness. Under a kin selection paradigm, individuals are expected to distribute actions influencing the fitness of relatives based on the relatedness of these relatives. In addition, it is necessary to have an accurate measure of relatedness to estimate heritability (h²) of phenotypic characters and to predict the efficacy of selection. Relatedness is often defined as the genotypic correlation between individuals. Assessed on the basis of common ancestry, relatedness can only be determined sensu strictu from pedigree analysis. Recent methodological and statistical advances allow the estimation of relatedness from allele frequency data. Many coefficients of relatedness can be found in the literature; I review and evaluate these, with emphasis on situations for which each is appropriate.     

Contrasting population genetic structure for workers and queens in the putatively unicolonial ant Formica exsecta

The theory of inclusive fitness provides a powerful explanation for reproductive altruism in social insects, whereby workers gain inclusive fitness benefit by rearing the brood of related queens. Some ant species, however, have unicolonial population structures where multiple nests, each containing numerous queens, are interconnected and individuals move freely between nests. In such cases, nestmate relatedness values may often be indistinguishable from zero, which is problematic for inclusive fitness-based explanations of reproductive altruism. We conducted a detailed population genetic study in the polygynous ant Formica exsecta, which has been suggested to form unicolonial populations in its native habitat. Analyses based on adult workers indeed confirmed a genetic structuring consistent with a unicolonial population structure. However, at the population level the genetic structuring inferred from worker pupae was not consistent with a unicolonial population structure, but rather suggested a multicolonial population structure of extended family-based nests. These contrasting patterns suggest limited queen dispersal and free adult worker dispersal. That workers indeed disperse as adults was confirmed by mark-recapture measures showing consistent worker movement between nests. Together, these findings describe a new form of social organization, which possibly also characterizes other ant species forming unicolonial populations in their native habitats. Moreover, the genetic analyses also revealed that while worker nestmate relatedness was indistinguishable from zero at a small geographical scale, it was significantly positive at the population level. This highlights the need to consider the relevant geographical scale when investigating the role of inclusive fitness as a selective force maintaining reproductive altruism.     

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.     

Invasion fitness,inclusive fitness,and reproductive numbers in heterogeneous populations

How should fitness be measured to determine which phenotype or “strategy” is uninvadable when evolution occurs in a group‐structured population subject to local demographic and environmental heterogeneity? Several fitness measures, such as basic reproductive number, lifetime dispersal success of a local lineage, or inclusive fitness have been proposed to address this question, but the relationships between them and their generality remains unclear. Here, we ascertain uninvadability (all mutant strategies always go extinct) in terms of the asymptotic per capita number of mutant copies produced by a mutant lineage arising as a single copy in a resident population (“invasion fitness”). We show that from invasion fitness uninvadability is equivalently characterized by at least three conceptually distinct fitness measures: (i) lineage fitness, giving the average individual fitness of a randomly sampled mutant lineage member; (ii) inclusive fitness, giving a reproductive value weighted average of the direct fitness costs and relatedness weighted indirect fitness benefits accruing to a randomly sampled mutant lineage member; and (iii) basic reproductive number (and variations thereof) giving lifetime success of a lineage in a single group, and which is an invasion fitness proxy. Our analysis connects approaches that have been deemed different, generalizes the exact version of inclusive fitness to class‐structured populations, and provides a biological interpretation of natural selection on a mutant allele under arbitrary strength of selection.     

Inclusive fitness,asymmetric competition and kin selection in plants

The findings that some plants alter their competitive phenotype in response to genetic relatedness of its conspecific neighbour (and presumed competitor) has spurred an increasing interest in plant kin‐interactions. This phenotypic response suggests the ability to assess the genetic relatedness of conspecific competitors, proposing kin selection as a process that can influence plant competitive interactions. Kin selection can favour restrained competitive growth towards kin, if the fitness loss from reducing own growth is compensated by increased fitness in the related neighbour. This may lead to positive frequency dependency among related conspecifics with important ecological consequences for species assemblage and coexistence. However, kin selection in plants is still controversial. First, many studies documenting a plastic response to neighbour relatedness do not estimate fitness consequences of the individual that responds, and when estimated, fitness of individuals grown in competition with kin did not necessarily exceed that of individuals grown in non‐kin groups. Although higher fitness in kin groups could be consistent with kin selection, this could also arise from mechanisms like asymmetric competition in the non‐kin groups. Here we outline the main challenges for studying kin selection in plants taking genetic variation for competitive ability into account. We emphasize the need to measure inclusive fitness in order to assess whether kin selection occurs, and show under which circumstances kin selected responses can be expected. We also illustrate why direct fitness estimates of a focal plant, and group fitness estimates are not suitable for documenting kin selection. Importantly, natural selection occurs at the individual level and it is the inclusive fitness of an individual plant – not the mean fitness of the group – that can capture if a differential response to neighbour relatedness is favoured by kin selection.     

Assortative mating for relatedness in a large naturally occurring population of Drosophila melanogaster

New theoretical work on kin selection and inclusive fitness benefits predicts that individuals will sometimes choose close or intermediate relatives as mates to maximize their fitness. However, empirical examples supporting such predictions are rare. In this study, we look for such evidence in a natural population of Drosophila melanogaster. We compared mating and nonmating individuals to test whether mating was nonrandom with respect to relatedness. Consistent with optimal inbreeding, males were more closely related to their mate than to randomly sampled females. However, all individuals collected mating showed higher relatedness and males were not significantly more related to their mate than to other mating females. We also found a negative relationship between relatedness and fecundity. Our results are consistent with the hypothesis that inclusive fitness benefits may drive inbreeding tolerance despite direct costs to fitness; however, an experimental approach is needed to investigate the link between mate preference and relatedness.     

Evolution of stepping-stone dispersal rates

We present a general model of the evolution of dispersal in a population with any distribution of dispersal distance. We use this model to analyse evolutionarily stable (ES) dispersal rates for the classical island model of dispersal and for three different stepping-stone models. Using general techniques to compute relatedness coefficients in the different dispersal models which we consider, we find that the distribution of dispersal distance may affect the ES dispersal rate when the cost of dispersal is low. In this case the ES dispersal rate increases with the number of demes that can be reached by one dispersal event. However, for increasing cost the ES dispersal rate converges to a value independent of the distribution of dispersal distance. These results are in contrast to previous analyses of similar models. The effects of the size (number of demes) and shape (ratio between the width and the length) of the population on the evolution of dispersal are also studied. We find that larger and more elongated populations lead generally to higher ES dispersal rates. However, both of these effects can only be observed for extreme parameter values (i.e. for very small and very elongated populations). The direct fitness method and the analytical techniques used here to compute relatedness coefficients provide an efficient way to analyse ES strategies in subdivided populations.     

Parasite-mediated heterozygote advantage in an outbred songbird population

Coevolution with parasites is thought to maintain genetic diversity in host populations. However, while there are sound theoretical reasons to expect heterozygosity and parasite resistance to be related, this pattern has generally been shown only in inbred laboratory and island populations. This leaves doubt as to whether parasite-mediated selection for genetic diversity is in fact a general process. Here we show that haematozoan parasite load is linked to two complementary measures of microsatellite variability in an outbred population of mountain white-crowned sparrows (Zonotrichia leucophrys oriantha) for which we know that parasites reduce fitness. Moreover, each of the genetic measures predicts a subtly different aspect of parasitism. Microsatellite heterozygosity is related to an individual's risk of parasitism, and mean d2 (a broader, more long-term measure of parental relatedness) to the severity of infection among parasitized individuals.     

Mate choice for nonadditive genetic benefits and the maintenance of genetic diversity in song sparrows

The lek paradox asserts that strong directional selection via female choice should deplete additive genetic variation in fitness and consequently any benefit to females expressing the preference. Recently, we have provided a novel resolution to the paradox by showing that nonadditive genetic effects such as overdominance can be inherited from parent to offspring, and populations with females that express a mating preference for outbred males maintain higher genetic variation than populations with females that mate randomly. Here, we test our dynamic model using empirical data previously published from a small island population of song sparrows (Melospiza melodia). The model assumes that fitness and male trait expression display overdominance effects. The results demonstrate that female choice for outbred males mediated by directional selection on song repertoire size provides a heritable benefit to offspring through reduced inbreeding depression. Within the population, we estimate the heritability of the inbreeding coefficient to be 0.18 ± 0.08 (SD). Furthermore, we show that mate choice for outbred males increases fitness‐related genetic variation in the population by 12% and thereby reduces inbreeding depression by 1% per generation in typical years and upwards of 15% in severe years. Thus, mate choice may help to stave off population extinction in this and other small populations.     

Allelic Diversity and Its Implications for the Rate of Adaptation

Genetic variation is usually estimated empirically from statistics based on population gene frequencies, but alternative statistics based on allelic diversity (number of allelic types) can provide complementary information. There is a lack of knowledge, however, on the evolutionary implications attached to allelic-diversity measures, particularly in structured populations. In this article we simulated multiple scenarios of single and structured populations in which a quantitative trait subject to stabilizing selection is adapted to different fitness optima. By forcing a global change in the optima we evaluated which diversity variables are more strongly correlated with both short- and long-term adaptation to the new optima. We found that quantitative genetic variance components for the trait and gene-frequency-diversity measures are generally more strongly correlated with short-term response to selection, whereas allelic-diversity measures are more correlated with long-term and total response to selection. Thus, allelic-diversity variables are better predictors of long-term adaptation than gene-frequency variables. This observation is also extended to unlinked neutral markers as a result of the information they convey on the demographic population history. Diffusion approximations for the allelic-diversity measures in a finite island model under the infinite-allele neutral mutation model are also provided.     

The socially parasitic ant Polyergus mexicanus has host‐associated genetic population structure and related neighbouring colonies

The genetic structure of populations can be both a cause and a consequence of ecological interactions. For parasites, genetic structure may be a consequence of preferences for host species or of mating behaviour. Conversely, genetic structure can influence where conspecific interactions among parasites lay on a spectrum from cooperation to conflict. We used microsatellite loci to characterize the genetic structure of a population of the socially parasitic dulotic (aka “slave‐making”) ant (Polyergus mexicanus), which is known for its host‐specificity and conspecific aggression. First, we assessed whether the pattern of host species use by the parasite has influenced parasite population structure. We found that host species use was correlated with subpopulation structure, but this correlation was imperfect: some subpopulations used one host species nearly exclusively, while others used several. Second, we examined the viscosity of the parasite population by measuring the relatedness of pairs of neighbouring parasitic ant colonies at varying distances from each other. Although natural history observations of local dispersal by queens suggested the potential for viscosity, there was no strong correlation between relatedness and distance between colonies. However, 35% of colonies had a closely related neighbouring colony, indicating that kinship could potentially affect the nature of some interactions between colonies of this social parasite. Our findings confirm that ecological forces like host species selection can shape the genetic structure of parasite populations, and that such genetic structure has the potential to influence parasite‐parasite interactions in social parasites via inclusive fitness.     

A New Metric of Inclusive Fitness Predicts the Human Mortality Profile

Biological species have evolved characteristic patterns of age-specific mortality across their life spans. If these mortality profiles are shaped by natural selection they should reflect underlying variation in the fitness effect of mortality with age. Direct fitness models, however, do not accurately predict the mortality profiles of many species. For several species, including humans, mortality rates vary considerably before and after reproductive ages, during life-stages when no variation in direct fitness is possible. Variation in mortality rates at these ages may reflect indirect effects of natural selection acting through kin. To test this possibility we developed a new two-variable measure of inclusive fitness, which we term the extended genomic output or EGO. Using EGO, we estimate the inclusive fitness effect of mortality at different ages in a small hunter-gatherer population with a typical human mortality profile. EGO in this population predicts 90% of the variation in age-specific mortality. This result represents the first empirical measurement of inclusive fitness of a trait in any species. It shows that the pattern of human survival can largely be explained by variation in the inclusive fitness cost of mortality at different ages. More generally, our approach can be used to estimate the inclusive fitness of any trait or genotype from population data on birth dates and relatedness.