An inclusive fitness model for the evolutionary advantage of sibmating

Summary We construct an inclusive fitness model of the relative selective advantage of sibmating and outbreeding behaviour, under the assumption that inbred offspring pay a fitness penalty. We are particularly interested in the question of whether such inbreeding depression is enough to generate a stable phenotypic polymorphism, with both kinds of breeding observed. The model predicts that, under diploidy, such a polymorphism is never found, but under haplodiploidy, it exists for a narrow range of parameter values. The inclusive fitness argument is technically interesting because care must be taken with reproductive values. We also present a corrected version of a one-locus genetic model for sibmating and find that the inclusive fitness and genetic models give identical results when selection is weak.     

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.     

Protected polymorphism and evolutionary stability in pleiotropic models with trait-specific dominance

When alleles have pleiotropic effects on a number of quantitative traits, the degree of dominance between a pair of alleles can be different for each trait. Such trait-specific dominance has been studied previously in models for the maintenance of genetic variation by antagonistic effects of an allele on two fitness components. By generalizing these models to an arbitrary number of fitness components or other phenotypic traits with different degrees of dominance, I show that genetic polymorphism is generally impossible without antagonistic fitness effects of different traits and without trait-specific dominance. I also investigate dominance and pleiotropy from a more long-term evolutionary perspective, allowing for the study of general ecological scenarios, and I discuss the effects of trait-specific dominance on evolutionary stability criteria. When selection is mainly directional and only trait-specific dominance and antagonism cause the emergence of polymorphism, then these polymorphisms can be overtaken by single mutants again, such that they are probably short-lived on an evolutionary time scale. Near evolutionarily singular points where directional selection is absent, trait-specific dominance and overdominance facilitate the emergence of polymorphism and cause evolutionary divergence in some cases. An important outcome of these models is that trait-specific dominance allows for the emergence of genetic polymorphisms without a selective disadvantage for heterozygotes. This removes the scope for the evolution of assortative mate choice and affects dominance modification. Sympatric speciation by disruptive ecological selection requires this heterozygote disadvantage in order to evolve, and therefore it becomes less plausible if the emergence of genetic polymorphism usually occurs via trait-specific dominance and antagonistic effects.     

Parental sex discrimination and intralocus sexual conflict

Intralocus sexual conflict occurs when populations segregate for alleles with opposing fitness consequences in the two sexes. This form of selection is known to be capable of maintaining genetic and fitness variation in nature, the extent of which is sensitive to the underlying genetics. We present a one-locus model of a haploid maternal effect that has sexually antagonistic consequences for offspring. The evolutionary dynamics of these maternal effects are distinct from those of haploid direct effects under sexual antagonism because the relevant genes are expressed only in females. Despite this, we find the same opportunity for sexually antagonistic polymorphism at the maternal effect locus as at a direct effect locus. Thus, sexually antagonistic maternal effects may underlie some natural genetic variation. The model we present permits alternative interpretations of how the genes are expressed and how the fitness variation is assigned, which invites a theoretical comparison to models of both imprinted genes and sex allocation.     

Hamilton goes empirical: estimation of inclusive fitness from life-history data

Hamilton's theory of kin selection is one of the most important advances in evolutionary biology since Darwin. Central to the kin-selection theory is the concept of inclusive fitness. However, despite the importance of inclusive fitness in evolutionary theory, empirical estimation of inclusive fitness has remained an elusive task. Using the concept of individual fitness, I present a method for estimating inclusive fitness and its components for diploid organisms with age-structured life histories. The method presented here: (i) allows empirical estimation of inclusive fitness from life-history data; (ii) simultaneously considers all components of fitness, including timing and magnitude of reproduction; (iii) is consistent with Hamilton's definition of inclusive fitness; and (iv) adequately addresses shortcomings of existing methods of estimating inclusive fitness. I also demonstrate the application of this new method for testing Hamilton's rule.     

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.     

Joint phenotypes,evolutionary conflict and the fundamental theorem of natural selection

Multiple organisms can sometimes affect a common phenotype. For example, the portion of a leaf eaten by an insect is a joint phenotype of the plant and insect and the amount of food obtained by an offspring can be a joint trait with its mother. Here, I describe the evolution of joint phenotypes in quantitative genetic terms. A joint phenotype for multiple species evolves as the sum of additive genetic variances in each species, weighted by the selection on each species. Selective conflict between the interactants occurs when selection takes opposite signs on the joint phenotype. The mean fitness of a population changes not just through its own genetic variance but also through the genetic variance for its fitness that resides in other species, an update of Fisher''s fundamental theorem of natural selection. Some similar results, using inclusive fitness, apply to within-species interactions. The models provide a framework for understanding evolutionary conflicts at all levels.     

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.     

Sociobiology and the structural stability of behavior patterns

A structural stability approach to population-genetic systems and to dynamic evolutionary games is attempted in order to examine the theoretical significance of sociobiological selection models. A criterion of weak selection is derived that is not restricted to differential reproduction in polymorphic systems but describes possible directions of evolutionary change in time scales governed by genetic mutation rates. The criterion applies to the problems of how the initial mutational basis of an adaptive trait may be established and how this may happen, for analogous traits, independently in different species. Two basic sociobiological concepts are reconsidered with reference to the criterion. It is shown that W. D. Hamilton's condition of increases in inclusive fitness due to altruistic interactions among kin expresses the structural instability of populations against the evolution of altruistic behavior. Using the dynamic approach to evolutionary game theory, it is demonstrated that if a behavioral phenotype is an evolutionarily stable strategy, it is structurally stable against perturbations of the fitness payoffs, provided selection is weak. These results are applied to material problems of the evolution of animal social behavior.     

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.     

Evolutionary Stability for Interactions among Kin under Quantitative Inheritance

The theory of evolutionarily stable strategies (ESS) predicts the long-term evolutionary outcome of frequency-dependent selection by making a number of simplifying assumptions about the genetic basis of inheritance. I use a symmetrized multilocus model of quantitative inheritance without mutation to analyze the results of interactions between pairs of related individuals and compare the equilibria to those found by ESS analysis. It is assumed that the fitness changes due to interactions can be approximated by the exponential of a quadratic surface. The major results are the following. (1) The evolutionarily stable phenotypes found by ESS analysis are always equilibria of the model studied here. (2) When relatives interact, one of the two conditions for stability of equilibria differs between the two models; this can be accounted for by positing that the inclusive fitness function for quantitative characters is slightly different from the inclusive fitness function for characters determined by a single locus. (3) The inclusion of environmental variance will in general change the equilibrium phenotype, but the equilibria of ESS analysis are changed to the same extent by environmental variance. (4) A class of genetically polymorphic equilibria occur, which in the present model are always unstable. These results expand the range of conditions under which one can validly predict the evolution of pairwise interactions using ESS analysis.     

Inclusive Fitness from Multitype Branching Processes

I use multitype branching processes to study genetic models for the evolution of social behaviour, i.e. behaviours that, when acted out, affect the success of the actor’s neighbours. Here, I suppose an individual bearing a mutant copy of a gene influences the reproductive success of a neighbour by altering its own competitive ability. Approximations based on assumptions about the rareness of the mutant allele and the strength of selection allow me to formulate statements concerning the probability of mutant extinction in terms of inclusive fitness. Inclusive fitness is an idea well known to biologists and can be thought of as a sum of an individual’s fitness and the fitness of each of its relatives, weighted by some measure of genetic relatedness. Previous work has led to some confusion surrounding the definition of the inclusive-fitness effect of a mutant allele when individuals carrying that allele experience demographic conditions that fluctuate randomly. In this paper, I emphasise the link between inclusive fitness and the probability of mutant extinction. I recover standard results for populations of constant size, and I show that inclusive fitness can be used to determine the short-term fate of mutants in the face of stochastic demographic fluctuations. Overall, then, I provide a connection between certain inclusive-fitness-based approaches routinely applied in theoretical studies of social evolution.     

EVOLUTION OF THE NUMBER OF SEXES

In sexually reproducing isogamous organisms, gametes (or diploid cells in ciliates) are classified into two or more groups called sexes, and mating occurs only between cells of different sexes. We have studied the evolutionary stability of the number of sexes maintained in a population by examining population-genetic models. For models in which the diploid genome determines the sex of conjugal cells, a one-locus system with three alleles of pecking-order dominance is assumed. Unlike traditional bisexual models, the genetic dynamics then depend on a rule, called mating kinetics, which determines the proportion of matings between each pair of sexes for given proportions of cells of the three sexes. The evolutionary consequences greatly depend on the mating kinetics assumed. Of the four mating kinetics examined, two give a large advantage to rare sexes whose cells quickly find heterosexual partners, which implies an evolutionary increase in the number of sexes. In contrast, the other two mating kinetics, in which gametes wait for suitable mates without being eliminated from the gamete pool during this waiting period, produce neutrally stable dynamics with curves or a surface of equilibria. Then random drift or differential fitness among sexes would result in the loss of sex alleles until only two remain in the population. This suggests a turnover of sexes; a new sex invades and replaces resident sexes after temporary coexistence. Similar results are obtained in models with haploid sex-determination and with autogamy. These two processes, however, may help to maintain many sexes indirectly by preventing the accumulation of recessive lethal mutations on sex chromosomes. The relationship of these models to models of self-sterility factors in plants and sex factors in honeybees is discussed. To summarize, the number of sexes should increase when conjugal cells must find mates during a limited period of time, otherwise a two-sex system should evolve. We conclude that there may be more isogamous species with three or more sexes than are currently known.     

There is no fitness but fitness,and the lineage is its bearer

Inclusive fitness has been the cornerstone of social evolution theory for more than a half-century and has matured as a mathematical theory in the past 20 years. Yet surprisingly for a theory so central to an entire field, some of its connections to evolutionary theory more broadly remain contentious or underappreciated. In this paper, we aim to emphasize the connection between inclusive fitness and modern evolutionary theory through the following fact: inclusive fitness is simply classical Darwinian fitness, averaged over social, environmental and demographic states that members of a gene lineage experience. Therefore, inclusive fitness is neither a generalization of classical fitness, nor does it belong exclusively to the individual. Rather, the lineage perspective emphasizes that evolutionary success is determined by the effect of selection on all biological and environmental contexts that a lineage may experience. We argue that this understanding of inclusive fitness based on gene lineages provides the most illuminating and accurate picture and avoids pitfalls in interpretation and empirical applications of inclusive fitness theory.     

Fisher’s fundamental theorem of inclusive fitness and the change in fitness due to natural selection when conspecifics interact

Competition and cooperation is fundamental to evolution by natural selection, both in animals and plants. Here, I investigate the consequences of such interactions for response in fitness due to natural selection. I provide quantitative genetic expressions for heritable variance and response in fitness due to natural selection when conspecifics interact. Results show that interactions among conspecifics generate extra heritable variance in fitness, and that interacting with kin is the key to evolutionary success because it translates the extra heritable variance into response in fitness. This work also unifies Fisher’s fundamental theorem of natural selection (FTNS) and Hamilton’s inclusive fitness (IF). The FTNS implies that natural selection maximizes fitness, whereas Hamilton proposed maximization of IF. This work shows that the FTNS describes the increase in IF, rather than direct fitness, at a rate equal to the additive genetic variance in fitness. Thus, Hamilton’s IF and Fisher’s FTNS both describe the maximization of IF.     

Group selection and kin selection: formally equivalent approaches

Inclusive fitness theory, summarised in Hamilton's rule, is a dominant explanation for the evolution of social behaviour. A parallel thread of evolutionary theory holds that selection between groups is also a candidate explanation for social evolution. The mathematical equivalence of these two approaches has long been known. Several recent papers, however, have objected that inclusive fitness theory is unable to deal with strong selection or with non-additive fitness effects, and concluded that the group selection framework is more general, or even that the two are not equivalent after all. Yet, these same problems have already been identified and resolved in the literature. Here, I survey these contemporary objections, and examine them in the light of current understanding of inclusive fitness theory.     

The stationary distribution of a continuously varying strategy in a class-structured population under mutation-selection-drift balance

Many traits and/or strategies expressed by organisms are quantitative phenotypes. Because populations are of finite size and genomes are subject to mutations, these continuously varying phenotypes are under the joint pressure of mutation, natural selection and random genetic drift. This article derives the stationary distribution for such a phenotype under a mutation-selection-drift balance in a class-structured population allowing for demographically varying class sizes and/or changing environmental conditions. The salient feature of the stationary distribution is that it can be entirely characterized in terms of the average size of the gene pool and Hamilton's inclusive fitness effect. The exploration of the phenotypic space varies exponentially with the cumulative inclusive fitness effect over state space, which determines an adaptive landscape. The peaks of the landscapes are those phenotypes that are candidate evolutionary stable strategies and can be determined by standard phenotypic selection gradient methods (e.g. evolutionary game theory, kin selection theory, adaptive dynamics). The curvature of the stationary distribution provides a measure of the stability by convergence of candidate evolutionary stable strategies, and it is evaluated explicitly for two biological scenarios: first, a coordination game, which illustrates that, for a multipeaked adaptive landscape, stochastically stable strategies can be singled out by letting the size of the gene pool grow large; second, a sex-allocation game for diploids and haplo-diploids, which suggests that the equilibrium sex ratio follows a Beta distribution with parameters depending on the features of the genetic system.     

Inclusive fitness models with two sexes

Much recent work has focused on the transition from G. R. Price's (1970, Nature 227, 520-521) formula for allele frequency change to an inclusive fitness condition for the selective advantage of a certain behaviour. In case there is any kind of asymmetry between the sexes, the analysis must keep track of the two sexes separately, and this leads to a number of different forms of the expression for inclusive fitness. In this paper I gather these forms together and note the assumptions needed to make each valid. I also show how inclusive fitness should be formulated when the behaviour of the actor is controlled by another individual. I illustrate the inclusive fitness approach with a sex allocation example in a haplodiploid population with a local breeding structure.     

The validity and value of inclusive fitness theory

Social evolution is a central topic in evolutionary biology, with the evolution of eusociality (societies with altruistic, non-reproductive helpers) representing a long-standing evolutionary conundrum. Recent critiques have questioned the validity of the leading theory for explaining social evolution and eusociality, namely inclusive fitness (kin selection) theory. I review recent and past literature to argue that these critiques do not succeed. Inclusive fitness theory has added fundamental insights to natural selection theory. These are the realization that selection on a gene for social behaviour depends on its effects on co-bearers, the explanation of social behaviours as unalike as altruism and selfishness using the same underlying parameters, and the explanation of within-group conflict in terms of non-coinciding inclusive fitness optima. A proposed alternative theory for eusocial evolution assumes mistakenly that workers' interests are subordinate to the queen's, contains no new elements and fails to make novel predictions. The haplodiploidy hypothesis has yet to be rigorously tested and positive relatedness within diploid eusocial societies supports inclusive fitness theory. The theory has made unique, falsifiable predictions that have been confirmed, and its evidence base is extensive and robust. Hence, inclusive fitness theory deserves to keep its position as the leading theory for social evolution.     

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.