Social evolution and genetic interactions in the short and long term

The coalescent in finite populations with arbitrary, fixed structure

The coalescent is a stochastic process representing ancestral lineages in a population undergoing neutral genetic drift. Originally defined for a well-mixed population, the coalescent has been adapted in various ways to accommodate spatial, age, and class structure, along with other features of real-world populations. To further extend the range of population structures to which coalescent theory applies, we formulate a coalescent process for a broad class of neutral drift models with arbitrary - but fixed - spatial, age, sex, and class structure, haploid or diploid genetics, and any fixed mating pattern. Here, the coalescent is represented as a random sequence of mappings [Formula: see text] from a finite set G to itself. The set G represents the "sites" (in individuals, in particular locations and/or classes) at which these alleles can live. The state of the coalescent, Ct:G→G, maps each site g∈G to the site containing g's ancestor, t time-steps into the past. Using this representation, we define and analyze coalescence time, coalescence branch length, mutations prior to coalescence, and stationary probabilities of identity-by-descent and identity-by-state. For low mutation, we provide a recipe for computing identity-by-descent and identity-by-state probabilities via the coalescent. Applying our results to a diploid population with arbitrary sex ratio r, we find that measures of genetic dissimilarity, among any set of sites, are scaled by 4r(1-r) relative to the even sex ratio case.

The evolution of environmentally mediated social interactions and posthumous spite under isolation by distance

Many social interactions happen indirectly via modifications of the environment, e.g. through the secretion of functional compounds or the depletion of renewable resources. Here, we derive the selection gradient on a quantitative trait affecting dynamical environmental variables that feed back on reproduction and survival in a finite patch-structured population subject to isolation by distance. Our analysis shows that the selection gradient depends on how a focal individual influences the fitness of all future individuals in the population through modifications of the environmental variables they experience, weighted by the neutral relatedness between recipients and the focal. The evolutionarily relevant trait-driven environmental modifications are formalized as the extended phenotypic effects of an individual, quantifying how a trait change in an individual in the present affects the environmental variables in all patches at all future times. When the trait affects reproduction and survival through a payoff function, the selection gradient can be expressed in terms of extended phenotypic effects weighted by scaled relatedness. We show how to compute extended phenotypic effects, relatedness, and scaled relatedness using Fourier analysis, which allow us to investigate a broad class of environmentally mediated social interactions in a tractable way. We use our approach to study the evolution of a trait controlling the costly production of some lasting commons (e.g. a common-pool resource or a toxic compound) that can diffuse in space and persist in time. We show that indiscriminate posthumous spite readily evolves in this scenario. More generally, whether selection favours environmentally mediated altruism or spite is determined by the spatial correlation between an individual's lineage and the commons originating from its patch. The sign of this correlation depends on interactions between dispersal patterns and the commons' renewal dynamics. More broadly, we suggest that selection can favour a wide range of social behaviours when these have carry-over effects in space and time.

Strategy evolution on higher-order networks

Cooperation is key to prosperity in human societies. Population structure is well understood as a catalyst for cooperation, where research has focused on pairwise interactions. But cooperative behaviors are not simply dyadic, and they often involve coordinated behavior in larger groups. Here we develop a framework to study the evolution of behavioral strategies in higher-order population structures, which include pairwise and multi-way interactions. We provide an analytical treatment of when cooperation will be favored by higher-order interactions, accounting for arbitrary spatial heterogeneity and nonlinear rewards for cooperation in larger groups. Our results indicate that higher-order interactions can act to promote the evolution of cooperation across a broad range of networks, in public goods games. Higher-order interactions consistently provide an advantage for cooperation when interaction hyper-networks feature multiple conjoined communities. Our analysis provides a systematic account of how higher-order interactions modulate the evolution of prosocial traits.

Nonlinear social evolution and the emergence of collective action

Organisms from microbes to humans engage in a variety of social behaviors, which affect fitness in complex, often nonlinear ways. The question of how these behaviors evolve has consequences ranging from antibiotic resistance to human origins. However, evolution with nonlinear social interactions is challenging to model mathematically, especially in combination with spatial, group, and/or kin assortment. We derive a mathematical condition for natural selection with synergistic interactions among any number of individuals. This result applies to populations with arbitrary (but fixed) spatial or network structure, group subdivision, and/or mating patterns. In this condition, nonlinear fitness effects are ascribed to collectives, and weighted by a new measure of collective relatedness. For weak selection, this condition can be systematically evaluated by computing branch lengths of ancestral trees. We apply this condition to pairwise games between diploid relatives, and to dilemmas of collective help or harm among siblings and on spatial networks. Our work provides a rigorous basis for extending the notion of "actor", in the study of social evolution, from individuals to collectives.

Linked selection and the evolution of altruism in family-structured populations

Much research on the evolution of altruism via kin selection, group selection, and reciprocity focuses on the role of a single locus or quantitative trait. Very few studies have explored how linked selection, or selection at loci neighboring an altruism locus, impacts the evolution of altruism. While linked selection can decrease the efficacy of selection at neighboring loci, it might have other effects including promoting selection for altruism by increasing relatedness in regions of low recombination. Here, we used population genetic simulations to study how negative selection at linked loci, or background selection, affects the evolution of altruism. When altruism occurs between full siblings, we found that background selection interfered with selection on the altruistic allele, increasing its fixation probability when the altruistic allele was disfavored and reducing its fixation when the allele was favored. In other words, background selection has the same effect on altruistic genes in family-structured populations as it does on other, nonsocial, genes. This contrasts with prior research showing that linked selective sweeps can favor the evolution of cooperation, and we discuss possibilities for resolving these contrasting results.

Symmetry in models of natural selection

Symmetry arguments are frequently used-often implicitly-in mathematical modelling of natural selection. Symmetry simplifies the analysis of models and reduces the number of distinct population states to be considered. Here, I introduce a formal definition of symmetry in mathematical models of natural selection. This definition applies to a broad class of models that satisfy a minimal set of assumptions, using a framework developed in previous works. In this framework, population structure is represented by a set of sites at which alleles can live, and transitions occur via replacement of some alleles by copies of others. A symmetry is defined as a permutation of sites that preserves probabilities of replacement and mutation. The symmetries of a given selection process form a group, which acts on population states in a way that preserves the Markov chain representing selection. Applying classical results on group actions, I formally characterize the use of symmetry to reduce the states of this Markov chain, and obtain bounds on the number of states in the reduced chain.

Extending eco-evolutionary theory with oligomorphic dynamics

Understanding the interplay between ecological processes and the evolutionary dynamics of quantitative traits in natural systems remains a major challenge. Two main theoretical frameworks are used to address this question, adaptive dynamics and quantitative genetics, both of which have strengths and limitations and are often used by distinct research communities to address different questions. In order to make progress, new theoretical developments are needed that integrate these approaches and strengthen the link to empirical data. Here, we discuss a novel theoretical framework that bridges the gap between quantitative genetics and adaptive dynamics approaches. 'Oligomorphic dynamics' can be used to analyse eco-evolutionary dynamics across different time scales and extends quantitative genetics theory to account for multimodal trait distributions, the dynamical nature of genetic variance, the potential for disruptive selection due to ecological feedbacks, and the non-normal or skewed trait distributions encountered in nature. Oligomorphic dynamics explicitly takes into account the effect of environmental feedback, such as frequency- and density-dependent selection, on the dynamics of multi-modal trait distributions and we argue it has the potential to facilitate a much tighter integration between eco-evolutionary theory and empirical data.

Evolutionarily stable strategy analysis and its links to demography and genetics through invasion fitness

Evolutionarily stable strategy (ESS) analysis pioneered by Maynard Smith and Price took off in part because it often does not require explicit assumptions about the genetics and demography of a population in contrast to population genetic models. Though this simplicity is useful, it obscures the degree to which ESS analysis applies to populations with more realistic genetics and demography: for example, how does ESS analysis handle complexities such as kin selection, group selection and variable environments when phenotypes are affected by multiple genes? In this paper, I review the history of the ESS concept and show how early uncertainty about the method lead to important mathematical theory linking ESS analysis to general population genetic models. I use this theory to emphasize the link between ESS analysis and the concept of invasion fitness. I give examples of how invasion fitness can measure kin selection, group selection and the evolution of linked modifier genes in response to variable environments. The ESSs in these examples depend crucially on demographic and genetic parameters, which highlights how ESS analysis will continue to be an important tool in understanding evolutionary patterns as new models address the increasing abundance of genetic and long-term demographic data in natural populations. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

Evolutionary game theory and the adaptive dynamics approach: adaptation where individuals interact

Evolutionary game theory and the adaptive dynamics approach have made invaluable contributions to understanding how gradual evolution leads to adaptation when individuals interact. Here, we review some of the basic tools that have come out of these contributions to model the evolution of quantitative traits in complex populations. We collect together mathematical expressions that describe directional and disruptive selection in class- and group-structured populations in terms of individual fitness, with the aims of bridging different models and interpreting selection. In particular, our review of disruptive selection suggests there are two main paths that can lead to diversity: (i) when individual fitness increases more than linearly with trait expression; (ii) when trait expression simultaneously increases the probability that an individual is in a certain context (e.g. a given age, sex, habitat, size or social environment) and fitness in that context. We provide various examples of these and more broadly argue that population structure lays the ground for the emergence of polymorphism with unique characteristics. Beyond this, we hope that the descriptions of selection we present here help see the tight links among fundamental branches of evolutionary biology, from life history to social evolution through evolutionary ecology, and thus favour further their integration. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

Conformity and anti-conformity in a finite population

Conformist and anti-conformist cultural transmission have been studied both empirically, in several species, and theoretically, with population genetic models. Building upon standard, infinite-population models (IPMs) of conformity, we introduce finite-population models (FPMs) and study them via simulation and a diffusion approximation. In previous IPMs of conformity, offspring observe the variants of n adult role models, where n is often three. Numerical simulations show that while the short-term behavior of the FPM with n=3 role models is well approximated by the IPM, stable polymorphic equilibria of the IPM become effective equilibria of the FPM at which the variation persists prior to fixation or loss, and which produce plateaus in curves for fixation probabilities and expected times to absorption. In the FPM with n=5 role models, the population may switch between two effective equilibria, which is not possible in the IPM, or may cycle between frequencies that are not effective equilibria, which is possible in the IPM. In all observed cases of 'equilibrium switching' and 'cycling' in the FPM, model parameters exceed O(1/N), required for the diffusion approximation, resulting in an over-estimation of the actual times to absorption. However, in those cases with n=5 role models that have one effective equilibrium and stable fixation states, even if conformity coefficients exceed O(1/N), the diffusion approximation matches closely the numerical simulations of the FPM. This suggests that the robustness of the diffusion approximation depends not only on the magnitudes of coefficients, but also on the qualitative behavior of the conformity model.

Hamilton's rule, the evolution of behavior rules and the wizardry of control theory

This paper formalizes selection on a quantitative trait affecting the evolution of behavior (or development) rules through which individuals act and react with their surroundings. Combining Hamilton's marginal rule for selection on scalar traits and concepts from optimal control theory, a necessary first-order condition for the evolutionary stability of the trait in a group-structured population is derived. The model, which is of intermediate level of complexity, fills a gap between the formalization of selection on evolving traits that are directly conceived as actions (no phenotypic plasticity) and selection on evolving traits that are conceived as strategies or function valued actions (complete phenotypic plasticity). By conceptualizing individuals as open deterministic dynamical systems expressing incomplete phenotypic plasticity, the model captures selection on a large class of phenotypic expression mechanisms, including developmental pathways and learning under life-history trade-offs. As an illustration of the results, a first-order condition for the evolutionary stability of behavior response rules from the social evolution literature is re-derived, strengthened, and generalized. All results of the paper also generalize directly to selection on multidimensional quantitative traits affecting behavior rule evolution, thereby covering neural and gene network evolution.

Evaluating the structure-coefficient theorem of evolutionary game theory

In order to accommodate the empirical fact that population structures are rarely simple, modern studies of evolutionary dynamics allow for complicated and highly heterogeneous spatial structures. As a result, one of the most difficult obstacles lies in making analytical deductions, either qualitative or quantitative, about the long-term outcomes of evolution. The "structure-coefficient" theorem is a well-known approach to this problem for mutation-selection processes under weak selection, but a general method of evaluating the terms it comprises is lacking. Here, we provide such a method for populations of fixed (but arbitrary) size and structure, using easily interpretable demographic measures. This method encompasses a large family of evolutionary update mechanisms and extends the theorem to allow for asymmetric contests to provide a better understanding of the mutation-selection balance under more realistic circumstances. We apply the method to study social goods produced and distributed among individuals in spatially heterogeneous populations, where asymmetric interactions emerge naturally and the outcome of selection varies dramatically, depending on the nature of the social good, the spatial topology, and the frequency with which mutations arise.

Evolution of warfare by resource raiding favours polymorphism in belligerence and bravery

From protists to primates, intergroup aggression and warfare over resources have been observed in several taxa whose populations typically consist of groups connected by limited genetic mixing. Here, we model the coevolution between four traits relevant to this setting: (i) investment into common-pool resource production within groups (helping); (ii) proclivity to raid other groups to appropriate their resources (belligerence); and investments into (iii) defense and (iv) offense of group contests (defensive and offensive bravery). We show that when traits coevolve, the population often experiences disruptive selection favouring two morphs: 'Hawks', who express high levels of both belligerence and offensive bravery; and 'Doves', who express neither. This social polymorphism involves further among-traits associations when the fitness costs of helping and bravery interact. In particular, if helping is antagonistic with both forms of bravery, coevolution leads to the coexistence of individuals that either: (i) do not participate into common-pool resource production but only in its defense and appropriation (Scrounger Hawks) or (ii) only invest into common pool resource production (Producer Doves). Provided groups are not randomly mixed, these findings are robust to several modelling assumptions. This suggests that inter-group aggression is a potent mechanism in favouring within-group social diversity and behavioural syndromes. This article is part of the theme issue 'Intergroup conflict across taxa'.

Weak Selection and the Separation of Eco-evo Time Scales using Perturbation Analysis

We show that under the assumption of weak frequency-dependent selection a wide class of population dynamical models can be analysed using perturbation theory. The inner solution corresponds to the ecological dynamics, where to zeroth order, the genotype frequencies remain constant. The outer solution provides the evolutionary dynamics and corresponds, to zeroth order, to a generalisation of the replicator equation. We apply this method to a model of public goods dynamics and construct, using matched asymptotic expansions, a composite solution valid for all times. We also analyse a Lotka-Volterra model of predator competition and show that to zeroth order the fraction of wild-type predators follows a replicator equation with a constant selection coefficient given by the predator death rate. For both models, we investigate how the error between approximate solutions and the solution to the full model depend on the order of the approximation and show using numerical comparison, for [Formula: see text] and 2, that the error scales according to [Formula: see text], where [Formula: see text] is the strength of selection and k is the order of the approximation.

Cooperative interactions among females can lead to even more extraordinary sex ratios

Hamilton's local mate competition theory provided an explanation for extraordinary female-biased sex ratios in a range of organisms. When mating takes place locally, in structured populations, a female-biased sex ratio is favored to reduce competition between related males, and to provide more mates for males. However, there are a number of wasp species in which the sex ratios appear to more female biased than predicted by Hamilton's theory. It has been hypothesized that the additional female bias in these wasp species results from cooperative interactions between females. We investigated theoretically the extent to which cooperation between related females can interact with local mate competition to favor even more female-biased sex ratios. We found that (i) cooperation between females can lead to sex ratios that are more female biased than predicted by local competition theory alone, and (ii) sex ratios can be more female biased when the cooperation occurs from offspring to mothers before dispersal, rather than cooperation between siblings after dispersal. Our models formally confirm the verbal predictions made in previous experimental studies, which could be applied to a range of organisms. Specifically, cooperation can help explain sex ratio biases in Sclerodermus and Melittobia wasps, although quantitative comparisons between predictions and data suggest that some additional factors may be operating.

Fixation probabilities in evolutionary dynamics under weak selection

In evolutionary dynamics, a key measure of a mutant trait's success is the probability that it takes over the population given some initial mutant-appearance distribution. This "fixation probability" is difficult to compute in general, as it depends on the mutation's effect on the organism as well as the population's spatial structure, mating patterns, and other factors. In this study, we consider weak selection, which means that the mutation's effect on the organism is small. We obtain a weak-selection perturbation expansion of a mutant's fixation probability, from an arbitrary initial configuration of mutant and resident types. Our results apply to a broad class of stochastic evolutionary models, in which the size and spatial structure are arbitrary (but fixed). The problem of whether selection favors a given trait is thereby reduced from exponential to polynomial complexity in the population size, when selection is weak. We conclude by applying these methods to obtain new results for evolutionary dynamics on graphs.

Hamilton's rule, gradual evolution, and the optimal (feedback) control of phenotypically plastic traits

Most traits expressed by organisms, such as gene expression profiles, developmental trajectories, behavioural sequences and reaction norms are function-valued traits (colloquially "phenotypically plastic traits"), since they vary across an individual's age and in response to various internal and/or external factors (state variables). Furthermore, most organisms live in populations subject to limited genetic mixing and are thus likely to interact with their relatives. We here formalise selection on genetically determined function-valued traits of individuals interacting in a group-structured population, by deriving the marginal version of Hamilton's rule for function-valued traits. This rule simultaneously gives a condition for the invasion of an initially rare mutant function-valued trait and its ultimate fixation in the population (invasion thus implies substitution). Hamilton's rule thus underlies the gradual evolution of function-valued traits and gives rise to necessary first-order conditions for their uninvadability (evolutionary stability). We develop a novel analysis using optimal control theory and differential game theory, to simultaneously characterise and compare the first-order conditions of (i) open-loop traits - functions of time (or age) only, and (ii) closed-loop (state-feedback) traits - functions of both time and state variables. We show that closed-loop traits can be represented as the simpler open-loop traits when individuals do not interact or when they interact with clonal relatives. Our analysis delineates the role of state-dependence and interdependence between individuals for trait evolution, which has implications to both life-history theory and social evolution.

The components of directional and disruptive selection in heterogeneous group-structured populations

We derive how directional and disruptive selection operate on scalar traits in a heterogeneous group-structured population for a general class of models. In particular, we assume that each group in the population can be in one of a finite number of states, where states can affect group size and/or other environmental variables, at a given time. Using up to second-order perturbation expansions of the invasion fitness of a mutant allele, we derive expressions for the directional and disruptive selection coefficients, which are sufficient to classify the singular strategies of adaptive dynamics. These expressions include first- and second-order perturbations of individual fitness (expected number of settled offspring produced by an individual, possibly including self through survival); the first-order perturbation of the stationary distribution of mutants (derived here explicitly for the first time); the first-order perturbation of pairwise relatedness; and reproductive values, pairwise and three-way relatedness, and stationary distribution of mutants, each evaluated under neutrality. We introduce the concept of individual k-fitness (defined as the expected number of settled offspring of an individual for which k-1 randomly chosen neighbors are lineage members) and show its usefulness for calculating relatedness and its perturbation. We then demonstrate that the directional and disruptive selection coefficients can be expressed in terms individual k-fitnesses with k=1,2,3 only. This representation has two important benefits. First, it allows for a significant reduction in the dimensions of the system of equations describing the mutant dynamics that needs to be solved to evaluate explicitly the two selection coefficients. Second, it leads to a biologically meaningful interpretation of their components. As an application of our methodology, we analyze directional and disruptive selection in a lottery model with either hard or soft selection and show that many previous results about selection in group-structured populations can be reproduced as special cases of our model.

Deconstructing Evolutionary Game Theory: Coevolution of Social Behaviors with Their Evolutionary Setting

Evolution of social behaviors is one of the most fascinating and active fields of evolutionary biology. During the past half century, social evolution theory developed into a mature field with powerful tools to understand the dynamics of social traits such as cooperation under a wide range of conditions. In this article, I argue that the next stage in the development of social evolution theory should consider the evolution of the setting in which social behaviors evolve. To that end, I propose a conceptual map of the components that make up the evolutionary setting of social behaviors, review existing work that considers the evolution of each component, and discuss potential future directions. The theoretical work reviewed here illustrates how unexpected dynamics can happen when the setting of social evolution itself is evolving, such as cooperation sometimes being self-limiting. I argue that a theory of how the setting of social evolution itself evolves will lead to a deeper understanding of when cooperation and other social behaviors evolve and diversify.

An evolutionary quantitative genetics model for phenotypic (co)variances under limited dispersal, with an application to socially synergistic traits

Darwinian evolution consists of the gradual transformation of heritable traits due to natural selection and the input of random variation by mutation. Here, we use a quantitative genetics approach to investigate the coevolution of multiple quantitative traits under selection, mutation, and limited dispersal. We track the dynamics of trait means and of variance-covariances between traits that experience frequency-dependent selection. Assuming a multivariate-normal trait distribution, we recover classical dynamics of quantitative genetics, as well as stability and evolutionary branching conditions of invasion analyses, except that due to limited dispersal, selection depends on indirect fitness effects and relatedness. In particular, correlational selection that associates different traits within-individuals depends on the fitness effects of such associations between-individuals. We find that these kin selection effects can be as relevant as pleiotropy for the evolution of correlation between traits. We illustrate this with an example of the coevolution of two social traits whose association within-individuals is costly but synergistically beneficial between-individuals. As dispersal becomes limited and relatedness increases, associations between-traits between-individuals become increasingly targeted by correlational selection. Consequently, the trait distribution goes from being bimodal with a negative correlation under panmixia to unimodal with a positive correlation under limited dispersal.

A mathematical formalism for natural selection with arbitrary spatial and genetic structure

We define a general class of models representing natural selection between two alleles. The population size and spatial structure are arbitrary, but fixed. Genetics can be haploid, diploid, or otherwise; reproduction can be asexual or sexual. Biological events (e.g. births, deaths, mating, dispersal) depend in arbitrary fashion on the current population state. Our formalism is based on the idea of genetic sites. Each genetic site resides at a particular locus and houses a single allele. Each individual contains a number of sites equal to its ploidy (one for haploids, two for diploids, etc.). Selection occurs via replacement events, in which alleles in some sites are replaced by copies of others. Replacement events depend stochastically on the population state, leading to a Markov chain representation of natural selection. Within this formalism, we define reproductive value, fitness, neutral drift, and fixation probability, and prove relationships among them. We identify four criteria for evaluating which allele is selected and show that these become equivalent in the limit of low mutation. We then formalize the method of weak selection. The power of our formalism is illustrated with applications to evolutionary games on graphs and to selection in a haplodiploid population.

Value of species and the evolution of conservation ethics

The theory of evolution by natural selection can help explain why people care about other species. Building upon recent insights that morality evolves to secure fitness advantages of cooperation, we propose that conservation ethics (moral beliefs, attitudes, intuitions and norms regarding other species) could be adaptations that support cooperation between humans and non-humans. We present eco-evolutionary cost-benefit models of conservation behaviours as interspecific cooperation (altruism towards members of other species). We find that an evolutionary rule identical in structure to Hamilton's rule (which explains altruistic behaviour towards related conspecifics) can explain altruistic behaviour towards members of other species. Natural selection will favour traits for selectively altering the success of members of other species (e.g. conserving them) in ways that maximize inclusive fitness return benefits. Conservation behaviours and the ethics that evolve to reinforce them will be sensitive to local ecological and socio-cultural conditions, so will assume different contours in different places. Difficulties accurately assessing costs and benefits provided by other species, time required to adapt to ecological and socio-cultural change and barriers to collective action could explain the apparent contradiction between the widespread existence of conservation ethics and patterns of biodiversity decline globally.

From the Price equation to the selection gradient in class-structured populations: a quasi-equilibrium route

Recent studies in theoretical evolutionary ecology have emphasised two approaches to modelling evolution. On the one hand, models based on a separation of time scales rely on the concept of invasion fitness. On the other hand, models based on the Price equation track the dynamics of a trait average, coupled with a description of ecological dynamics. The aim of this article is to show that, in class-structured populations, both approaches yield the same expression for the selection gradient under weak selection. Although the result is not new, I propose an alternative route to its derivation using the dynamics of scaled measures of between-class phenotypic differentiation. Under weak selection, these measures of phenotypic differentiation can be treated as fast variables compared to the trait mean, which allows for a quasi-equilibrium approximation. This suggests a different approach to calculating weak selection approximations of evolutionary dynamics, and clarifies the links between short- and long-term perspectives on evolution in structured populations.

Class Structure, Demography, and Selection: Reproductive-Value Weighting in Nonequilibrium, Polymorphic Populations

In natural populations, individuals of a given genotype may belong to different classes. Such classes can represent different age groups, developmental stages, or habitats. Class structure has important evolutionary consequences because the fitness of individuals with the same genetic background may vary depending on their class. As a result, demographic transitions between classes can cause fluctuations in the trait mean that need to be removed when estimating selection on a trait. Intrinsic differences between classes are classically taken into account by weighting individuals by class-specific reproductive values, defined as the relative contribution of individuals in a given class to the future of the population. These reproductive values are generally constant weights calculated from a constant projection matrix. Here, I show for large populations and clonal reproduction that reproductive values can be defined as time-dependent weights satisfying dynamical demographic equations that depend only on the average between-class transition rates over all genotypes. Using these time-dependent demographic reproductive values yields a simple Price equation where the nonselective effects of between-class transitions are removed from the dynamics of the trait. This generalizes previous theory to a large class of ecological scenarios, taking into account density dependence, ecological feedbacks, arbitrary strength of selection, and arbitrary trait distributions. I discuss the role of reproductive values for prospective and retrospective analyses of the dynamics of phenotypic traits.

Theoretical Approaches in Evolutionary Ecology: Environmental Feedback as a Unifying Perspective

Evolutionary biology and ecology have a strong theoretical underpinning, and this has fostered a variety of modeling approaches. A major challenge of this theoretical work has been to unravel the tangled feedback loop between ecology and evolution. This has prompted the development of two main classes of models. While quantitative genetics models jointly consider the ecological and evolutionary dynamics of a focal population, a separation of timescales between ecology and evolution is assumed by evolutionary game theory, adaptive dynamics, and inclusive fitness theory. As a result, theoretical evolutionary ecology tends to be divided among different schools of thought, with different toolboxes and motivations. My aim in this synthesis is to highlight the connections between these different approaches and clarify the current state of theory in evolutionary ecology. Central to this approach is to make explicit the dependence on environmental dynamics of the population and evolutionary dynamics, thereby materializing the eco-evolutionary feedback loop. This perspective sheds light on the interplay between environmental feedback and the timescales of ecological and evolutionary processes. I conclude by discussing some potential extensions and challenges to our current theoretical understanding of eco-evolutionary dynamics.

Antisocial rewarding in structured populations

Cooperation in collective action dilemmas usually breaks down in the absence of additional incentive mechanisms. This tragedy can be escaped if cooperators have the possibility to invest in reward funds that are shared exclusively among cooperators (prosocial rewarding). Yet, the presence of defectors who do not contribute to the public good but do reward themselves (antisocial rewarding) deters cooperation in the absence of additional countermeasures. A recent simulation study suggests that spatial structure is sufficient to prevent antisocial rewarding from deterring cooperation. Here we reinvestigate this issue assuming mixed strategies and weak selection on a game-theoretic model of social interactions, which we also validate using individual-based simulations. We show that increasing reward funds facilitates the maintenance of prosocial rewarding but prevents its invasion, and that spatial structure can sometimes select against the evolution of prosocial rewarding. Our results suggest that, even in spatially structured populations, additional mechanisms are required to prevent antisocial rewarding from deterring cooperation in public goods dilemmas.

The general form of Hamilton's rule makes no predictions and cannot be tested empirically

Hamilton's rule asserts that a trait is favored by natural selection if the benefit to others, [Formula: see text], multiplied by relatedness, [Formula: see text], exceeds the cost to self, [Formula: see text] Specifically, Hamilton's rule states that the change in average trait value in a population is proportional to [Formula: see text] This rule is commonly believed to be a natural law making important predictions in biology, and its influence has spread from evolutionary biology to other fields including the social sciences. Whereas many feel that Hamilton's rule provides valuable intuition, there is disagreement even among experts as to how the quantities [Formula: see text], [Formula: see text], and [Formula: see text] should be defined for a given system. Here, we investigate a widely endorsed formulation of Hamilton's rule, which is said to be as general as natural selection itself. We show that, in this formulation, Hamilton's rule does not make predictions and cannot be tested empirically. It turns out that the parameters [Formula: see text] and [Formula: see text] depend on the change in average trait value and therefore cannot predict that change. In this formulation, which has been called "exact and general" by its proponents, Hamilton's rule can "predict" only the data that have already been given.

Altruistic aging: The evolutionary dynamics balancing longevity and evolvability

Altruism is typically associated with traits or behaviors that benefit the population as a whole, but are costly to the individual. We propose that, when the environment is rapidly changing, senescence (age-related deterioration) can be altruistic. According to numerical simulations of an agent-based model, while long-lived individuals can outcompete their short lived peers, populations composed of long-lived individuals are more likely to go extinct during periods of rapid environmental change. Moreover, as in many situations where other cooperative behavior arises, senescence can be stabilized in a structured population.

The Effectiveness of Reward and Punishment in Spatial Social Games

Punishment and reward are usually regarded as two potential mechanisms to explain the evolution of cooperation especially among multiple participators. However, the performance of these two scenarios in spatial environment needs to be discussed. To figure out this issue, we resort to the [Formula: see text]-player Iterated Snowdrift Dilemma (ISD) game and Iterated Prisoner’s Dilemma (IPD) game. More importantly, the evolution of punishment and reward in social network-structured populations has not been formally addressed. The numerical results show the equilibrium cooperation frequency can be influenced by cost-to-benefit ratio [Formula: see text], the punishment-to-benefit ratio [Formula: see text] and the reward-to-benefit ratio [Formula: see text]. And one intriguing observation is that under the same situation, the punishment is more effective than reward to the population. Then we further probe the effectiveness of neighborhood relationship to the cooperation, which is reflected by the random rewired probability [Formula: see text]. From the distribution of the four roles of the participator we can find that individuals can cooperate easily when they have close relationship. The results of this paper may be helpful to understand the cooperation in complex project or among industry–university–research cooperation project.

Fidelity of parent-offspring transmission and the evolution of social behavior in structured populations

The theoretical investigation of how spatial structure affects the evolution of social behavior has mostly been done under the assumption that parent-offspring strategy transmission is perfect, i.e., for genetically transmitted traits, that mutation is very weak or absent. Here, we investigate the evolution of social behavior in structured populations under arbitrary mutation probabilities. We consider populations of fixed size N, structured such that in the absence of selection, all individuals have the same probability of reproducing or dying (neutral reproductive values are the all same). Two types of individuals, A and B, corresponding to two types of social behavior, are competing; the fidelity of strategy transmission from parent to offspring is tuned by a parameter μ. Social interactions have a direct effect on individual fecundities. Under the assumption of small phenotypic differences (implying weak selection), we provide a formula for the expected frequency of type A individuals in the population, and deduce conditions for the long-term success of one strategy against another. We then illustrate our results with three common life-cycles (Wright-Fisher, Moran Birth-Death and Moran Death-Birth), and specific population structures (graph-structured populations). Qualitatively, we find that some life-cycles (Moran Birth-Death, Wright-Fisher) prevent the evolution of altruistic behavior, confirming previous results obtained with perfect strategy transmission. We also show that computing the expected frequency of altruists on a regular graph may require knowing more than just the graph's size and degree.

Ordering structured populations in multiplayer cooperation games

Spatial structure greatly affects the evolution of cooperation. While in two-player games the condition for cooperation to evolve depends on a single structure coefficient, in multiplayer games the condition might depend on several structure coefficients, making it difficult to compare different population structures. We propose a solution to this issue by introducing two simple ways of ordering population structures: the containment order and the volume order. If population structure is greater than population structure in the containment or the volume order, then can be considered a stronger promoter of cooperation. We provide conditions for establishing the containment order, give general results on the volume order, and illustrate our theory by comparing different models of spatial games and associated update rules. Our results hold for a large class of population structures and can be easily applied to specific cases once the structure coefficients have been calculated or estimated.

Evolutionary Games of Multiplayer Cooperation on Graphs

There has been much interest in studying evolutionary games in structured populations, often modeled as graphs. However, most analytical results so far have only been obtained for two-player or linear games, while the study of more complex multiplayer games has been usually tackled by computer simulations. Here we investigate evolutionary multiplayer games on graphs updated with a Moran death-Birth process. For cycles, we obtain an exact analytical condition for cooperation to be favored by natural selection, given in terms of the payoffs of the game and a set of structure coefficients. For regular graphs of degree three and larger, we estimate this condition using a combination of pair approximation and diffusion approximation. For a large class of cooperation games, our approximations suggest that graph-structured populations are stronger promoters of cooperation than populations lacking spatial structure. Computer simulations validate our analytical approximations for random regular graphs and cycles, but show systematic differences for graphs with many loops such as lattices. In particular, our simulation results show that these kinds of graphs can even lead to more stringent conditions for the evolution of cooperation than well-mixed populations. Overall, we provide evidence suggesting that the complexity arising from many-player interactions and spatial structure can be captured by pair approximation in the case of random graphs, but that it need to be handled with care for graphs with high clustering.

Cooperation can be defined as the act of providing fitness benefits to other individuals, often at a personal cost. When interactions occur mainly with neighbors, assortment of strategies can favor cooperation but local competition can undermine it. Previous research has shown that a single coefficient can capture this trade-off when cooperative interactions take place between two players. More complicated, but also more realistic, models of cooperative interactions involving multiple players instead require several such coefficients, making it difficult to assess the effects of population structure. Here, we obtain analytical approximations for the coefficients of multiplayer games in graph-structured populations. Computer simulations show that, for particular instances of multiplayer games, these approximate coefficients predict the condition for cooperation to be promoted in random graphs well, but fail to do so in graphs with more structure, such as lattices. Our work extends and generalizes established results on the evolution of cooperation on graphs, but also highlights the importance of explicitly taking into account higher-order statistical associations in order to assess the evolutionary dynamics of cooperation in spatially structured populations.

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.

Evolutionary Stability of Jointly Evolving Traits in Subdivided Populations

The evolutionary stability of quantitative traits depends on whether a population can resist invasion by any mutant. While uninvadability is well understood in well-mixed populations, it is much less so in subdivided populations when multiple traits evolve jointly. Here, we investigate whether a spatially subdivided population at a monomorphic equilibrium for multiple traits can withstand invasion by any mutant or is subject to diversifying selection. Our model also explores the correlations among traits arising from diversifying selection and how they depend on relatedness due to limited dispersal. We find that selection tends to favor a positive (negative) correlation between two traits when the selective effects of one trait on relatedness is positively (negatively) correlated to the indirect fitness effects of the other trait. We study the evolution of traits for which this matters: dispersal that decreases relatedness and helping that has positive indirect fitness effects. We find that when dispersal cost is low and the benefits of helping accelerate faster than its costs, selection leads to the coexistence of mobile defectors and sessile helpers. Otherwise, the population evolves to a monomorphic state with intermediate helping and dispersal. Overall, our results highlight the effects of population subdivision for evolutionary stability and correlations among traits.

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.

Evolutionary dynamics of collective action in spatially structured populations

Many models proposed to study the evolution of collective action rely on a formalism that represents social interactions as n-player games between individuals adopting discrete actions such as cooperate and defect. Despite the importance of spatial structure in biological collective action, the analysis of n-player games games in spatially structured populations has so far proved elusive. We address this problem by considering mixed strategies and by integrating discrete-action n-player games into the direct fitness approach of social evolution theory. This allows to conveniently identify convergence stable strategies and to capture the effect of population structure by a single structure coefficient, namely, the pairwise (scaled) relatedness among interacting individuals. As an application, we use our mathematical framework to investigate collective action problems associated with the provision of three different kinds of collective goods, paradigmatic of a vast array of helping traits in nature: "public goods" (both providers and shirkers can use the good, e.g., alarm calls), "club goods" (only providers can use the good, e.g., participation in collective hunting), and "charity goods" (only shirkers can use the good, e.g., altruistic sacrifice). We show that relatedness promotes the evolution of collective action in different ways depending on the kind of collective good and its economies of scale. Our findings highlight the importance of explicitly accounting for relatedness, the kind of collective good, and the economies of scale in theoretical and empirical studies of the evolution of collective action.