An inclusive fitness model for the sex ratio in a partially sibmating population with inbreeding cost

Summary We construct an inclusive fitness model to find the evolutionarily stable sex ratios in a partially sibmating diploid or haplodiploid population. We assume a constant rate of sibmating with inbred offspring incurring a fitness penalty which, under haplodiploidy, is only suffered by females. We construct a one-locus genetic model for the same problem and observe that when selection is weak it gives the same numerical results as the inclusive fitness model.     

Evolutionarily stable mixed mating in a variety of genetic systems

Many species display a mixture of close inbreeding and outbreeding which is referred to as mixed mating. For selfing species, models predict that such mixed mating systems can be stable. Conversely, models considering separate sex species have not been able to explain mixed mating systems. This failure may be a result of the unrealistic assumption that recurrent inbreeding does not increase the inbreeding coefficient. Here we show that mixed mating is expected in separate sex systems when recurrent inbreeding is taken into account. A female that allows her brother to sibmate with her gives an extra mating opportunity to said brother. This kin selective advantage should be strongest in genetic systems where the male is more related to the female. In support of this idea, we find that inbreeding evolves most easily in selfers, followed by diploid sibmating, followed by haplodiploid sibmating. Consideration of published values for the regression of fitness on inbreeding coefficient suggests that many species fall in a range where some selfing/sibmating is optimal.     

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.     

From inclusive fitness to fixation probability in homogeneous structured populations

The methods of inclusive fitness provide a powerful analysis of the action of selection on social behaviour. The key component of this analysis is the concept of relatedness R. In infinite populations, a standard method of calculating relatedness coefficients is through coefficients of consanguinity using the notion of genetic identity by descent. In this paper, we show that this approach can also be made to work in finite populations and we assume here that the population has a homogeneous structure, such as an island model. We demonstrate that, under the assumption that genetic effects are small and additive, the resulting formulation of inclusive fitness is equivalent to other significant measures of selection in finite populations, including the change in average allele frequency and fixation probability. The results are illustrated for a model of the evolution of cooperation in a finite island population.     

Sib‐mating in the ant Plagiolepis pygmaea: adaptative inbreeding?

Multiple functional queens in a colony (polygyny) and multiple mating by queens (polyandry) in social insects challenge kin selection, because they dilute inclusive fitness benefits from helping. Colonies of the ant Plagiolepis pygmaea brash contain several hundreds of multiply mated queens. Yet, within‐colony relatedness remains unexpectedly high. This stems from low male dispersal, extensive mating among relatives and adoption of young queens in the natal colony. We investigated whether inbreeding results from workers expelling foreign males, and/or from preferential mating between related partners. Our data show that workers actively repel unrelated males entering their colony, and that queens preferentially mate with related males. These results are consistent with inclusive fitness being a driving force for inbreeding: by preventing outbreeding, workers reduce erosion of relatedness within colonies due to polygyny and polyandry. That virgin queens mate preferentially with related males could result from a long history of inbreeding, which is expected to reduce depression in species with regular sibmating.     

Invasion of sibmating genes in diploid and haplodiploid populations

Summary Existing genetic models of the evolution of sibmating behaviour in diploids incorporate inbreeding depression in terms of reduced fecundity of consanguineous mating pairs rather than reduced survival or fecundity of the progeny of such matings. Here we derive a model to correct this deficiency and extend the model to haplodiploids where differential effects of inbreeding in males and females is a crucial consideration. Our analyses indicate that sibmating can readily evolve in both diploids and haplodiploids in which male mating costs and inbreeding depression are reasonably low, provided there is some mechanism to permit sibmating such as siblings being reared in nests or other forms of aggregation. Our analyses also indicate that once sibmating invades, it typically will go to fixation, although sib-/randommating polymorphisms can persist in both diploids and haplodiploids if male mating costs are close to zero and inbreeding depression reduces survival by around one-third. The conditions favouring sibmating are slightly more restrictive in haplodiploids than in diploids. In light of this we may ask why we see intense sibmating in many haplodiploids such as parasitic wasps, fig wasps, ants, bark beetles and mites, and only rarely in diploid animals. The common factor could be certain kinds of aggregation behaviour that are a prerequisite for sibmating in the absence of kin recognition. Another possibility is that inbreeding depression is likely to be more severe in diploids than in haplodiploids because deleterious recessives are purged from haplodiploid populations when expressed by haploid males. Thus, lower levels of inbreeding depression might be one important reason why sibmating appears to arise more frequently in haplodiploids than diploids. Phylogenetic analysis of groups, such as bark beetles and mites, exhibiting both diploid and haplodiploid populations may be useful in elucidating the relative importance of gregarious behaviour and haplodiploidy in facilitating sibmating systems.     

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.     

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.     

Much ado about nothing: Nowak et al.'s charge against inclusive fitness theory

In a recent article, Nowak et al. claim that the mathematical basis of inclusive fitness theory does not stand to scrunity and to have found an alternative explanation for eusociality. We show that these claims are based on false premises, many of which have been exposed more than 25 years ago, such as misrepresentations of the basic components of inclusive fitness and fallacious distinctions between individual fitness and inclusive fitness. Moreover, some limitations ascribed to inclusive fitness are actually limitations of current evolutionary theory, for which Nowak et al. propose no new solution. Likewise, their assertedly 'common sense' empirical alternative to estimating inclusive fitness is not applicable in cases of interest. Finally, their eusociality model merely confirms the importance of all the components of inclusive fitness. We conclude by discussing how rhetorical devices and editorial practices can impede scientific endeavours.     

THE EFFECT OF SIBMATING ON THE INFECTION DYNAMICS OF MALE-KILLING BACTERIA

Male-killing (MK) bacteria are vertically transmitted endosymbionts that selectively kill the male offspring of their hosts. Simple mathematical models describe the infection dynamics using two parameters, the bacterial transmission rate and a fitness compensation for siblings of killed males. These models cannot explain two phenomena that have been observed in nature: the persistence of extreme MK causing all-female broods, and the coexistence of two different strains of MK bacteria in the same host population. In the present study, we extend the simple MK models and investigate theoretically the effects of sibmating on the infection dynamics. We demonstrate analytically that, in general, sibmating reduces MK prevalence, and can even cause its extinction. As a special case of this finding, we show that sibmating allows a stable coexistence between no infection and extreme MK. Furthermore, we performed computer simulations and showed that, depending on male mating capacity, a stable coexistence of two strains is possible if sibmating occurs but is below a threshold. The results suggest that sibmating might be an important factor for understanding the infection dynamics of MK bacteria.     

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.     

The generation and maintenance of genetic variation by frequency-dependent selection: constructing polymorphisms under the pairwise interaction model

Frequency-dependent selection remains the most commonly invoked heuristic explanation for the maintenance of genetic variation. For polymorphism to exist, new alleles must be both generated and maintained in the population. Here we use a construction approach to model frequency-dependent selection with mutation under the pairwise interaction model. The pairwise interaction model is a general model of frequency-dependent selection at the genotypic level. We find that frequency-dependent selection is able to generate a large number of alleles at a single locus. The construction process generates multiallelic polymorphisms with a wide range of allele-frequency distributions and genotypic fitness relationships. Levels of polymorphism and mean fitness are uncoupled, so constructed polymorphisms remain permanently invasible to new mutants; thus the model never settles down to an equilibrium state. Analysis of constructed fitness sets reveals signatures of heterozygote advantage and positive frequency dependence.     

A comparative study of egg recognition signature mixtures in Formica ants

Processing of information from the environment, such as assessing group membership in social contexts, is a major determinant of inclusive fitness. For social insects, recognizing brood origin is crucial for inclusive fitness in many contexts, such as social parasitism and kin conflicts within colonies. Whether a recognition signature is informative in kin conflicts depends on the extent of a genetic contribution into the cues. We investigated colony‐ and matriline‐specific variation in egg surface hydrocarbons in seven species of Formica ants. We show that chemical variance is distributed similarly to genetic variation, suggesting a significant genetic contribution to eggs odors in the genus. Significant among matriline components, and significant correlations between chemical and genetic similarity among individuals also indicate kin informative egg odors in several species. We suggest that egg odor surface variation could play a large role in within colony conflicts, and that a comparative method can reveal novel insight into communication of identity.     

Inclusive fitness arguments in genetic models of behaviour

My purpose here is to provide a coherent account of inclusive fitness techniques, accessible to a mathematically literate graduate student in evolutionary biology, and to relate these to standard one-locus genetic models. I begin in Sect. 2 with a general formulation of evolutionary stability; in Sect. 3 and Sect. 4 I interpret the basic stability conditions within genetic and inclusive fitness models. In Sect. 5 I extend these concepts to the case of a class-structured population, and in Sect. 6 I illustrate these notions with a sex ratio example. In Sect. 7 I give a proof of the result that under additive gene action and weak selection, an inclusive fitness argument is able to verify an important stability condition (2.5) for one-locus genetic models. Most of these results have been published.     

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 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.     

Emergence of long‐term balanced polymorphism under cyclic selection of spatially variable magnitude

A fundamental question in evolutionary biology is what promotes genetic variation at nonneutral loci, a major precursor to adaptation in changing environments. In particular, balanced polymorphism under realistic evolutionary models of temporally varying environments in finite natural populations remains to be demonstrated. Here, we propose a novel mechanism of balancing selection under temporally varying fitnesses. Using forward‐in‐time computer simulations and mathematical analysis, we show that cyclic selection that spatially varies in magnitude, such as along an environmental gradient, can lead to elevated levels of nonneutral genetic polymorphism in finite populations. Balanced polymorphism is more likely with an increase in gene flow, magnitude and period of fitness oscillations, and spatial heterogeneity. This polymorphism‐promoting effect is robust to small systematic fitness differences between competing alleles or to random environmental perturbation. Furthermore, we demonstrate analytically that protected polymorphism arises as spatially heterogeneous cyclic fitness oscillations generate a type of storage effect that leads to negative frequency dependent selection. Our findings imply that spatially variable cyclic environments can promote elevated levels of nonneutral genetic variation in natural populations.     

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.     

Polymorphism and differential selection for the sexes.

Various genetic models with different fitnesses for the sexes are investigated. Only a limited set of fitness values will result in a stable polymorphism, and the rate of approach to these equilibrium frequencies is extremely slow. These results indicate that there are problems associated with the interpretation of some human genetic traits by such models.     

Kinship reinforces cooperative predator inspection in a cichlid fish

Kin selection theory predicts that cooperation is facilitated between genetic relatives, as by cooperating with kin an individual might increase its inclusive fitness. Although numerous theoretical papers support Hamilton's inclusive fitness theory, experimental evidence is still underrepresented, in particular in noncooperative breeders. Cooperative predator inspection is one of the most intriguing antipredator strategies, as it implies high costs on inspectors. During an inspection event, one or more individuals leave the safety of a group and approach a potential predator to gather information about the current predation risk. We investigated the effect of genetic relatedness on cooperative predator inspection in juveniles of the cichlid fish Pelvicachromis taeniatus, a species in which juveniles live in shoals under natural conditions. We show that relatedness significantly influenced predator inspection behaviour with kin dyads being significantly more cooperative. Thus, our results indicate a higher disposition for cooperative antipredator behaviour among kin as predicted by kin selection theory.