Hamilton's inclusive fitness in finite-structured populations

Hamilton''s formulation of inclusive fitness has been with us for 50 years. During the first 20 of those years attention was largely focused on the evolutionary trajectories of different behaviours, but over the past 20 years interest has been growing in the effect of population structure on the evolution of behaviour and that is our focus here. We discuss the evolutionary journey of the inclusive-fitness effect over this epoch, nurtured as it was in an essentially homogeneous environment (that of ‘transitive’ structures) having to adapt in different ways to meet the expectations of heterogeneous structures. We pay particular attention to the way in which the theory has managed to adapt the original constructs of relatedness and reproductive value to provide a formulation of inclusive fitness that captures a precise measure of allele-frequency change in finite-structured populations.     

Evolutionary dynamics in structured populations

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces.     

Optimization of inclusive fitness

The first fully explicit argument is given that broadly supports a widespread belief among whole-organism biologists that natural selection tends to lead to organisms acting as if maximizing their inclusive fitness. The use of optimization programs permits a clear statement of what this belief should be understood to mean, in contradistinction to the common mathematical presumption that it should be formalized as some kind of Lyapunov or even potential function. The argument reveals new details and uncovers latent assumptions. A very general genetic architecture is allowed, and there is arbitrary uncertainty. However, frequency dependence of fitnesses is not permitted. The logic of inclusive fitness immediately draws together various kinds of intra-genomic conflict, and the concept of 'p-family' is introduced. Inclusive fitness is thus incorporated into the formal Darwinism project, which aims to link the mathematics of motion (difference and differential equations) used to describe gene frequency trajectories with the mathematics of optimization used to describe purpose and design. Important questions remain to be answered in the fundamental theory of inclusive fitness.     

An inclusive fitness analysis of altruism on a cyclical network

A recent model studies the evolution of cooperation on a network, and concludes with a result connecting the benefits and costs of interactions and the number of neighbours. Here, an inclusive fitness analysis is conducted of the only case solved analytically, of a cycle, and the identical result is obtained. This brings the result within a biologically familiar framework. It is notable that the benefits and costs in the inclusive fitness framework need to be derived, and are not the benefits and costs that are the parameters in the original model. The relatedness is a quadratic function of position in a cycle of size N: an individual is related by 1 to itself, by (N - 5)/(N + 1) to an immediate neighbour, and by very close to -1/2 to the most distant individuals. The inclusive fitness analysis explains hitherto puzzling features of the results.     

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.     

Inclusive fitness analysis on mathematical groups

Recent work on the evolution of behaviour is set in a structured population, providing a systematic way to describe gene flow and behavioural interactions. To obtain analytical results one needs a structure with considerable regularity. Our results apply to such "homogeneous" structures (e.g., lattices, cycles, and island models). This regularity has been formally described by a "node-transitivity" condition but in mathematics, such internal symmetry is powerfully described by the theory of mathematical groups. Here, this theory provides elegant direct arguments for a more general version of a number of existing results. Our main result is that in large "group-structured" populations, primary fitness effects on others play no role in the evolution of the behaviour. The competitive effects of such a trait cancel the primary effects, and the inclusive fitness effect is given by the direct effect of the actor on its own fitness. This result is conditional on a number of assumptions such as (1) whether generations overlap, (2) whether offspring dispersal is symmetric, (3) whether the trait affects fecundity or survival, and (4) whether the underlying group is abelian. We formulate a number of results of this type in finite and infinite populations for both Moran and Wright-Fisher demographies.     

Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits

Summary We present models of adaptive change in continuous traits for the following situations: (1) adaptation of a single trait within a single population in which the fitness of a given individual depends on the population's mean trait value as well as its own trait value; (2) adaptation of two (or more) traits within a single population; (3) adaptation in two or more interacting species. We analyse a dynamic model of these adaptive scenarios in which the rate of change of the mean trait value is an increasing function of the fitness gradient (i.e. the rate of increase of individual fitness with the individual's trait value). Such models have been employed in evolutionary game theory and are often appropriate both for the evolution of quantitative genetic traits and for the behavioural adjustment of phenotypically plastic traits. The dynamics of the adaptation of several different ecologically important traits can result in characters that minimize individual fitness and can preclude evolution towards characters that maximize individual fitness. We discuss biological circumstances that are likely to produce such adaptive failures for situations involving foraging, predator avoidance, competition and coevolution. The results argue for greater attention to dynamical stability in models of the evolution of continuous traits.     

The evolution of altruism: game theory in multilevel selection and inclusive fitness

Although the prisoner's dilemma (PD) has been used extensively to study reciprocal altruism, here we show that the n-player prisoner's dilemma (NPD) is also central to two other prominent theories of the evolution of altruism: inclusive fitness and multilevel selection. An NPD model captures the essential factors for the evolution of altruism directly in its parameters and integrates important aspects of these two theories such as Hamilton's rule, Simpson's paradox, and the Price covariance equation. The model also suggests a simple interpretation of the Price selection decomposition and an alternative decomposition that is symmetrical and complementary to it. In some situations this alternative shows the temporal changes in within- and between-group selection more clearly than the Price equation. In addition, we provide a new perspective on strong vs. weak altruism by identifying their different underlying game structures (based on absolute fitness) and showing how their evolutionary dynamics are nevertheless similar under selection (based on relative fitness). In contrast to conventional wisdom, the model shows that both strong and weak altruism can evolve in periodically formed random groups of non-conditional strategies if groups are multigenerational. An integrative approach based on the NPD helps unify different perspectives on the evolution of altruism.     

Extrapair paternity, inclusive fitness, and within-group benefits of helping in western bluebirds

In central coastal California, USA, 3–16% of western bluebird ( Sialia mexicana ) pairs have adult male helpers at the nest. Demographic data on a colour-ringed population over a 13-year period indicate that helpers gain a small indirect fitness benefit through increases in the number of young fledged from nests of close kin. A small proportion of adult helpers (16%) that were able to breed and help simultaneously had higher annual inclusive fitness than males that only bred. These males comprised such a minor proportion of helpers that the mean fitness of helpers was still lower than the mean fitness of independent breeders. We used DNA fingerprinting to determine whether extrapair fertilizations alter within-group benefits enough to tip the balance in favour of helping behaviour. Overall, 19% of 207 offspring were sired by males other than their social father and extrapair fertilizations occurred in 45% of 51 nests. Intraspecific brood parasitism was rare so that mean mother-nestling relatedness approximated the expected value of 0.5. Extrapair paternity reduced putative father-offspring relatedness to 0.38. Mean helper-nestling relatedness was 0.41 for helpers assisting one or both parents and 0.28 for helpers aiding their brothers. Helpers rarely sired offspring in the nests at which they helped. Helping was not conditional on paternity and helpers were not significantly more closely related to offspring in their parents' nests than to offspring in their own nests. Although helpers may derive extracurricular benefits if helping increases their own or their father's opportunities for extrapair fertilizations, within-nest inclusive fitness benefits of helping do not compensate males for failing to breed. Breeding failure and constraints on breeding are the most likely explanations for why most helpers help.     

A genetic affinity analysis of human populations

Genetic affinity of human populations based on allele frequency data was studied from two viewpoints. (1) The effect of the number of polymorphic loci on the reconstruction of a phylogenetic tree of human populations was empirically investigated. Genetic affinity trees were constructed based on data for 1–12 polymorphic loci, by using the neighbor-joining method. Geographical clustering of populations gradually appeared when the number of loci was increased. A new classification and terminology of higher order human population clusters is proposed based on these and other studies. (2) A new method of estimating the absolute divergence time of two populations is proposed, which is based on a diffusion equation that describes random genetic drift.     

Probabilistic participation in public goods games

Voluntary participation in public goods games (PGGs) has turned out to be a simple but effective mechanism for promoting cooperation under full anonymity. Voluntary participation allows individuals to adopt a risk-aversion strategy, termed loner. A loner refuses to participate in unpromising public enterprises and instead relies on a small but fixed pay-off. This system leads to a cyclic dominance of three pure strategies, cooperators, defectors and loners, but at the same time, there remain two considerable restrictions: the addition of loners cannot stabilize the dynamics and the time average pay-off for each strategy remains equal to the pay-off of loners. Here, we introduce probabilistic participation in PGGs from the standpoint of diversification of risk, namely simple mixed strategies with loners, and prove the existence of a dynamical regime in which the restrictions ono longer hold. Considering two kinds of mixed strategies associated with participants (cooperators or defectors) and non-participants (loners), we can recover all basic evolutionary dynamics of the two strategies: dominance; coexistence; bistability; and neutrality, as special cases depending on pairs of probabilities. Of special interest is that the expected pay-off of each mixed strategy exceeds the pay-off of loners at some interior equilibrium in the coexistence region.     

The joint allele frequency spectrum of multiple populations: a coalescent theory approach

The allele frequency spectrum is a series of statistics that describe genetic polymorphism, and is commonly used for inferring population genetic parameters and detecting natural selection. Population genetic theory on the allele frequency spectrum for a single population has been well studied using both coalescent theory and diffusion equations. Recently, the theory was extended to the joint allele frequency spectrum (JAFS) for three populations using diffusion equations and was shown to be very useful in inferring human demographic history. In this paper, I show that the JAFS can be analytically derived with coalescent theory for a basic model of two isolated populations and then extended to multiple populations and various complex scenarios, such as those involving population growth and bottleneck, migration, and positive selection. Simulation study is used to demonstrate the accuracy and applicability of the theoretical model. The coalescent theory-based approach for the JAFS can characterize the demographic history with comprehensive statistical models as the diffusion approach does, and in addition gains several novel advantages: the computational complexity of calculating the JAFS with coalescent theory is reduced, and thus it is feasible to analytically obtain the JAFS for multiple populations; the hitchhiking effect can be efficiently modeled in coalescent theory, enabling the development of methodologies for detecting selection via multi-population polymorphism data. As an alternative to the diffusion approximation approach, the coalescent theory for the JAFS also provides a foundation for population genetic inference with the advent of large-scale genomic polymorphism data.     

Genetic variation within the Merino sheep breed: analysis of closely related populations using microsatellites

Genetic relationships among six populations of Merino sheep were investigated using microsatellites. The history of the six populations is relatively well documented, with all being derived from the Spanish Merino breed within the last 400 years. Genetic variation was highest amongst the Spanish and Portuguese populations, although the preservation of genetic diversity within the other populations was high. By a variety of different statistical tests the French Mutton, German Mutton and New Zealand Merino populations could be differentiated from each other and the Iberian Merinos, indicating that microsatellites are able to track relatively recent changes in the population structure of sheep breeds. The dendrograms constructed on the basis of microsatellite allelic frequencies showed that populations that have shared selection criteria (meat vs. wool) tend to cluster together.     

The evolution of juvenile-adult interactions in populations structured in age and space

We study the evolution of a spatially structured population with two age classes using spatial moment equations. In the model, adults can either help juveniles by increasing their survival, or adopt a cannibalistic behaviour and consume juveniles. While cannibalism is the sole evolutionary outcome when the population is well-mixed, both cannibalism and parental care can be evolutionarily stable if the population is viscous. Our analysis allows us to make two main technical points. First, we present a method to define invasion fitness in class-structured viscous populations, which allows us to apply adaptive dynamics methodology. Second, we show that ordinary pair approximation introduces an important quantitative bias in the evolutionary model, even on random networks. We propose a correction to the ordinary pair approximation that yields quantitative accuracy, and discuss how the bias associated with this approach is precisely what allows us to identify subtle aspects associated with the evolutionary dynamics of spatially structured populations.     

Cross entropy minimization in uninvadable states of complex populations

Selection is often. viewed as a process that maximizes the average fitness of a population. However, there are often constraints even on the phenotypic level which may prevent fitness optimization. Consequently, in evolutionary game theory, models of frequency dependent selection are investigated, which focus on equilibrium states that are characterized by stability (or uninvadability) rather than by optimality. The aim of this article is to show that nevertheless there is a biologically meaningful quantity, namely cross (fitness) entropy, which is optimized during the course of evolution: a dynamical model adapted to evolutionary games is presented which has the property that relative entropy decreases monotonically, if the state of a (complex) population is close to an uninvadable state. This result may be interpreted as if evolution has an order stabilizing effect.     

Effectiveness of sexual selection in preventing fitness deterioration in bulb mite populations under relaxed natural selection

Under the 'good genes' mechanism of sexual selection (SS), females benefit from mate choice indirectly: their offspring inherit genes of the preferred, high quality fathers. Recent models assume that the genetic variance for male quality is maintained by deleterious mutations. Consequently, SS can be predicted to remove deleterious mutations from populations. We tested this prediction by relaxing selection in populations of the bulb mite, thus increasing their rate of accumulation of deleterious mutation. SS, allowed to operate in half of these populations, did not prevent the fitness decline observed in the other half of the relaxed selection lines. After 11 generations of relaxed selection, female fecundity in lines in which males were allowed to compete for females declined compared with control populations by similar amount as in monogamous lines (17.5 and 14.5%, respectively), whereas other fitness components (viability, longevity, male reproductive success) did not differ significantly between both types of lines and control populations.     

Decompositions of Price’s formula in an inhomogeneous population structure

The central tool for the study of allele frequency change due to selection is the remarkably simple but powerful formula of Price [Nature 227 (1970) 520] . Here, I provide what might be called a structural analysis of this formula. The formula essentially accumulates the average allele frequency change over many instances of a fitness‐determining interaction, but there are different ways of organizing this average and these lead to quite different computational algorithms. I present three of these: an analysis by population state, an analysis by recipient and an analysis by actor. A comparison of these can lead to a heightened understanding of the different factors behind selective allele frequency change. In particular, I pay attention to the effects of structural inhomogeneity on reproductive value (RV) and emphasize that Price’s formula measures RV‐weighted allele frequency change. I examine in detail a simple example as a crucial way of cementing the different theoretical pathways. My aim was to produce a simple transparent presentation and therefore I work with a simple population structure and have omitted a number of technical details that are found elsewhere.