Evolution of dispersal under spatio-temporal heterogeneity

Theoretical studies over the past decades have revealed various factors that favor or disfavor the evolution of dispersal. Among these, environmental heterogeneity is one driving force that can impact dispersal traits, because dispersing individuals can obtain a fitness benefit through finding better environments. Despite this potential benefit, some previous works have shown that the existence of spatial heterogeneity hinders evolution of dispersal. On the other hand, temporal heterogeneity has been shown to promote dispersal through a bet-hedging mechanism. When they are combined in a patch-structured population in which the quality of each patch varies over time independently of the others, it has been shown that spatiotemporal heterogeneity can favor evolution of dispersal. When individuals can use patch quality information so that dispersal decision is conditional, the evolutionary outcome can be different since individuals have options to disperse more/less offspring from bad/good patches. In this paper, we generalize the model and results of previous studies. We find richer dynamics including bistable evolutionary dynamics when there is arrival bias towards high-productivity patches. Then we study the evolution of conditional dispersal strategy in this generalized model. We find a surprising result that no offspring will disperse from a patch whose productivity was low when these offspring were born. In addition to mathematical proofs, we also provide intuition behind this initially counter-intuitive result based on reproductive-value arguments. Dispersal from high-productivity patches can evolve, and its parameter dependence behaves similarly, but not identically, to the case of unconditional dispersal. Our results unveil an importance of whether or not individuals can use patch quality information in dispersal evolution.

Evolutionary stability of cooperation in indirect reciprocity under noisy and private assessment

Indirect reciprocity is a mechanism that explains large-scale cooperation in humans. In indirect reciprocity, individuals use reputations to choose whether or not to cooperate with a partner and update others' reputations. A major question is how the rules to choose their actions and the rules to update reputations evolve. In the public reputation case where all individuals share the evaluation of others, social norms called Simple Standing (SS) and Stern Judging (SJ) have been known to maintain cooperation. However, in the case of private assessment where individuals independently evaluate others, the mechanism of maintenance of cooperation is still largely unknown. This study theoretically shows for the first time that cooperation by indirect reciprocity can be evolutionarily stable under private assessment. Specifically, we find that SS can be stable, but SJ can never be. This is intuitive because SS can correct interpersonal discrepancies in reputations through its simplicity. On the other hand, SJ is too complicated to avoid an accumulation of errors, which leads to the collapse of cooperation. We conclude that moderate simplicity is a key to stable cooperation under the private assessment. Our result provides a theoretical basis for the evolution of human cooperation.

Ancestral social environments plus nonlinear benefits can explain cooperation in human societies

Human cooperation (paying a cost to benefit others) is puzzling from a Darwinian perspective, particularly in groups with strangers who cannot repay nor are family members. The beneficial effects of cooperation typically increase nonlinearly with the number of cooperators, e.g., increasing returns when cooperation is low and diminishing returns when cooperation is high. Such nonlinearity can allow cooperation between strangers to persist evolutionarily if a large enough proportion of the population are already cooperators. However, if a lone cooperator faces a conflict between the group's and its own interests (a social dilemma), that raises the question of how cooperation arose in the first place. We use a mathematically tractable evolutionary model to formalise a chronological narrative that has previously only been investigated verbally: given that ancient humans interacted mostly with family members (genetic homophily), cooperation evolved first by kin selection, and then persisted in situations with nonlinear benefits as homophily declined or even if interactions with strangers became the norm. The model also predicts the coexistence of cooperators and defectors observed in the human population (polymorphism), and may explain why cooperators in behavioural experiments prefer to condition their contribution on the contributions of others (conditional cooperation in public goods games).

Reputation structure in indirect reciprocity under noisy and private assessment

Evaluation relationships are pivotal for maintaining a cooperative society. A formation of the evaluation relationships has been discussed in terms of indirect reciprocity, by modeling dynamics of good or bad reputations among individuals. Recently, a situation that individuals independently evaluate others with errors (i.e., noisy and private reputation) is considered, where the reputation structure (from what proportion of individuals in the population each receives good reputations, defined as goodness here) becomes complex, and thus has been studied mainly with numerical simulations. The present study gives a theoretical analysis of such complex reputation structure. We formulate the time change of goodness of individuals caused by updates of reputations among individuals. By considering a large population, we derive dynamics of the frequency distribution of goodnesses. An equilibrium state of the dynamics is approximated by a summation of Gaussian functions. We demonstrate that the theoretical solution well fits the numerical calculation. From the theoretical solution, we obtain a new interpretation of the complex reputation structure. This study provides a novel mathematical basis for cutting-edge studies on indirect reciprocity.

Non-zero-sum neutrality test for the tropical rain forest community using long-term between-census data

For community ecologists, "neutral or not?" is a fundamental question, and thus, rejecting neutrality is an important first step before investigating the deterministic processes underlying community dynamics. Hubbell's neutral model is an important contribution to the exploration of community dynamics, both technically and philosophically. However, the neutrality tests for this model are limited by a lack of statistical power, partly because the zero-sum assumption of the model is unrealistic. In this study, we developed a neutrality test for local communities that implements non-zero-sum community dynamics and determines the number of new species (N sp) between observations. For the non-zero-sum neutrality test, the model distributed the expected N sp, as calculated by extensive simulations, which allowed us to investigate the neutrality of the observed community by comparing the observed N sp with distributions of the expected N sp derived from the simulations. For this comparison, we developed a new "non-zero-sum N sp test," which we validated by running multiple neutral simulations using different parameter settings. We found that the non-zero-sum N sp test rejected neutrality at a near-significance level, which justified the validity of our approach. For an empirical test, the non-zero-sum N sp test was applied to real tropical tree communities in Panama and Malaysia. The non-zero-sum N sp test rejected neutrality in both communities when the observation interval was long and N sp was large. Hence, the non-zero-sum N sp test is an effective way to examine neutrality and has reasonable statistical power to reject the neutral model, especially when the observed N sp is large. This unique and simple approach is statistically powerful, even though it only employs two temporal sequences of community data. Thus, this test can be easily applied to existing datasets. In addition, application of the test will provide significant benefits for detecting changing biodiversity under climate change and anthropogenic disturbance.

Coevolutionary dynamics of genetic traits and their long-term extended effects under non-random interactions

Organisms continuously modify their living conditions via extended genetic effects on their environment, microbiome, and in some species culture. These effects can impact the fitness of current but also future conspecifics due to non-genetic transmission via ecological or cultural inheritance. In this case, selection on a gene with extended effects depends on the degree to which current and future genetic relatives are exposed to modified conditions. Here, we detail the selection gradient on a quantitative trait with extended effects in a patch-structured population, when gene flow between patches is limited and ecological inheritance within patches can be biased towards offspring. Such a situation is relevant to understand evolutionary driven changes in individual condition that can be preferentially transmitted from parent to offspring, such as cellular state, micro-environments (e.g., nests), pathogens, microbiome, or culture. Our analysis quantifies how the interaction between limited gene flow and biased ecological inheritance influences the joint evolutionary dynamics of traits together with the conditions they modify, helping understand adaptation via non-genetic modifications. As an illustration, we apply our analysis to a gene-culture coevolution scenario in which genetically-determined learning strategies coevolve with adaptive knowledge. In particular, we show that when social learning is synergistic, selection can favour strategies that generate remarkable levels of knowledge under intermediate levels of both vertical cultural transmission and limited dispersal. More broadly, our theory yields insights into the interplay between genetic and non-genetic inheritance, with implications for how organisms evolve to transform their environments.

Evolution of self-limited cell division of symbionts

In mutualism between unicellular hosts and their endosymbionts, symbiont's cell division is often synchronized with its host's, ensuring the permanent relationship between endosymbionts and their hosts. The evolution of synchronized cell division thus has been considered to be an essential step in the evolutionary transition from symbionts to organelles. However, if symbionts would accelerate their cell division without regard for the synchronization with the host, they would proliferate more efficiently. Thus, it is paradoxical that symbionts evolve to limit their own division for synchronized cell division. Here, we theoretically explore the condition for the evolution of self-limited cell division of symbionts, by assuming that symbionts control their division rate and that hosts control symbionts' death rate by intracellular digestion and nutrient supply. Our analysis shows that symbionts can evolve to limit their own cell division. Such evolution occurs if not only symbiont's but also host's benefit through symbiosis is large. Moreover, the coevolution of hosts and symbionts leads to either permanent symbiosis where symbionts proliferate to keep pace with their host, or the arms race between symbionts that behave as lytic parasites and hosts that resist them by rapid digestion.

Emergence of opinion leaders in reference networks

Individuals often refer to opinions of others when they make decisions in the real world. Our question is how the people’s reference structure self-organizes when people try to provide correct answers by referring to more accurate agents. We constructed an adaptive network model, in which each node represents an agent and each directed link represents a reference. In every iteration round within our model, each agent makes a decision sequentially by following the majority of the reference partners’ opinions and rewires a reference link to a partner if the partner’s performance falls below a given threshold. The value of this threshold is common for all agents and represents the performance assessment severity of the population. We found that the reference network self-organizes into a heterogeneous one with a nearly exponential in-degree (the number of followers) distribution, where reference links concentrate around agents with high intrinsic ability. In this heterogeneous network, the decision-making accuracy of agents improved on average. However, the proportion of agents who provided correct answers showed strong temporal fluctuation compared to that observed in the case in which each agent refers to randomly selected agents. We also found a counterintuitive phenomenon in which reference links concentrate more around high-ability agents and the population became smarter on average when the rewiring threshold was set lower than when it was set higher.

Asymmetric public goods game cooperation through pest control

Cooperation in a public goods game has been studied extensively to find the conditions for sustaining the commons, yet the effect of asymmetry between agents has been explored very little. Here we study a game theoretic model of cooperation for pest control among farmers. In our simple model, each farmer has a paddy of the same size arranged adjacently on a line. A pest outbreak occurs at an abandoned paddy at one end of the line, directly threatening the frontier farmer adjacent to it. Each farmer pays a cost of his or her choice to an agricultural collective, and the total sum held by the collective is used for pest control, with success probability increasing with the sum. Because the farmers' incentives depend on their distance from the pest outbreak, our model is an asymmetric public goods game. We derive each farmer's cost strategy at the Nash equilibrium. We find that asymmetry among farmers leads to a few unexpected outcomes. The individual costs at the equilibrium do not necessarily increase with how much the future is valued but rather show threshold behavior. Moreover, an increase in the number of farmers can sometimes paradoxically undermine pest prevention. A comparison with a symmetric public goods game model reveals that the farmer at the greatest risk pays a disproportionate amount of cost in the asymmetric game, making the use of agricultural lands less sustainable.

Evolution of emotional contagion in group-living animals

Emotional contagion refers to an instantaneous matching of an emotional state between a subject and an object. It is believed to form one of the bases of empathy and it causes consistent group behavior in many animals. However, how this emotional process relates to group size remains unclear. Individuals with the ability of emotional contagion can instantaneously copy the emotion of another group member and can take relevant behavior driven by this emotion, but this would entail both cost and benefit to them because the behavior can be either appropriate or inappropriate depending on the situation. For example, emotional contagion may help them escape from a predator but sometimes induce mass panic. We theoretically study how these two aspects of emotional contagion affect its evolution in group-living animals. We consider a situation where an environmental cue sometimes indicates a serious event and individuals have to make a decision whether to react to them. We show that, as the group size increases, individuals with the ability of emotional contagion would evolutionarily weaken their sensitivity to environmental cues. We also show that a larger group yields a larger benefit to them through such evolutionary change. However, larger group size prevents the invasion of mutants with the ability of emotional contagion into the population of residents who react to environmental cues independently of other group members. These results provide important suggestions on the evolutionary relationship between emotional contagion and group living.

Evolutionary emergence and maintenance of horizontally transmitted mutualism that do not rely on the supply of standing variation in symbiont quality

Mutualism based on reciprocal exchange of costly services must avoid exploitation by 'free-rides'. Accordingly, hosts discriminate against free-riding symbionts in many mutualistic relationships. However, as the selective advantage of discriminators comes from the presence of variability in symbiont quality that they eliminate, discrimination and thus mutualism have been considered to be maintained with exogenous supply of free-riders. In this study, we tried to resolve the 'paradoxical' co-evolution of discrimination by hosts and cooperation by symbionts, by comparing two different types of discrimination: 'one-shot' discrimination, where a host does not reacquire new symbionts after evicting free-riders, and 'resampling' discrimination, where a host does from the environment. Our study shows that this apparently minor difference in discrimination types leads to qualitatively different evolutionary outcomes. First, although it has been usually considered that the benefit of discriminators is derived from the variability of symbiont quality, the benefit of a certain type of discriminators (e.g. one-shot discrimination) is proportional to the frequency of free-riders, which is in stark contrast to the case of resampling discrimination. As a result, one-shot discriminators can invade the free-rider/nondiscriminator population, even if standing variation for symbiont quality is absent. Second, our one-shot discriminators can also be maintained without exogenous supply of free-riders and hence is free from the paradox of discrimination. Therefore, our result indicates that the paradox is not a common feature of evolution of discrimination but is a problem of specific types of discrimination.

Forward and backward evolutionary processes and allele frequency spectrum in a cancer cell population

A cancer grows from a single cell, thereby constituting a large cell population. In this work, we are interested in how mutations accumulate in a cancer cell population. We provide a theoretical framework of the stochastic process in a cancer cell population and obtain near exact expressions of allele frequency spectrum or AFS (only continuous approximation is involved) from both forward and backward treatments under a simple setting; all cells undergo cell divisions and die at constant rates, b and d, respectively, such that the entire population grows exponentially. This setting means that once a parental cancer cell is established, in the following growth phase, all mutations are assumed to have no effect on b or d (i.e., neutral or passengers). Our theoretical results show that the difference from organismal population genetics is mainly in the coalescent time scale, and the mutation rate is defined per cell division, not per time unit (e.g., generation). Except for these two factors, the basic logic is very similar between organismal and cancer population genetics, indicating that a number of well established theories of organismal population genetics could be translated to cancer population genetics with simple modifications.

The price of being seen to be just: an intention signalling strategy for indirect reciprocity

Cooperation among strangers is a marked characteristic of human sociality. One prominent evolutionary explanation for this form of human cooperation is indirect reciprocity, whereby each individual selectively helps people with a 'good' reputation, but not those with a 'bad' reputation. Some evolutionary analyses have underscored the importance of second-order reputation information (the reputation of a current partner's previous partner) for indirect reciprocity as it allows players to discriminate justified 'good' defectors, who selectively deny giving help to 'bad' partners, from unjustified 'bad' defectors. Nevertheless, it is not clear whether people in fact make use of second-order information in indirect reciprocity settings. As an alternative, we propose the intention signalling strategy, whereby defectors are given the option to abandon a resource as a means of expunging their 'bad' reputation. Our model deviates from traditional modelling approaches in the indirect reciprocity literature in a crucial way-we show that first-order information is sufficient to maintain cooperation if players are given an option to signal their intention. Importantly, our model is robust against invasion by both unconditionally cooperative and uncooperative strategies, a first step towards demonstrating its viability as an evolutionarily stable strategy. Furthermore, in two behavioural experiments, when participants were given the option to abandon a resource so as to mend a tarnished reputation, participants not only spontaneously began to use this option, they also interpreted others' use of this option as a signal of cooperative intent.

Reputation Effects in Public and Private Interactions

We study the evolution of cooperation in a model of indirect reciprocity where people interact in public and private situations. Public interactions have a high chance to be observed by others and always affect reputation. Private interactions have a lower chance to be observed and only occasionally affect reputation. We explore all second order social norms and study conditions for evolutionary stability of action rules. We observe the competition between “honest” and “hypocritical” strategies. The former cooperate both in public and in private. The later cooperate in public, where many others are watching, but try to get away with defection in private situations. The hypocritical idea is that in private situations it does not pay-off to cooperate, because there is a good chance that nobody will notice it. We find simple and intuitive conditions for the evolution of honest strategies.

We study the evolution of cooperation based on reputation. This mechanism is called indirect reciprocity. In a world of binary reputations, people help a good individual but do not help a bad one. They also monitor their own reputation to receive reciprocation from others. We propose a novel model of indirect reciprocity where two types of interactions exist. In a public interaction your behavior is always observed by others. In a private interaction, your behavior is less likely to be observed. We study the competition between honest and hypocritical strategies. The former always help good individuals, whereas the latter do so only in private interactions. We describe conditions for the evolution of honest strategies.

A panel data analysis of the probability of childbirth in a Japanese sample: New evidence of the two-child norm

OBJECTIVES

To reveal the conditions that could facilitate childbirth in modern humans, it is necessary to analyze not only cross-sectional surveys but also panel data that track the same person for a long period. In this study, we analyzed factors that would influence the probability of childbirth.

METHODS

We analyzed Japanese panel data with a Cox proportional hazard model. Subjects of our analysis were married women and their childbirth records from 2004 to 2009.

RESULTS

Contrary to the predictions based on the theory of behavioral ecology, we found no positive relationship between good parental conditions for childcare, such as high income, increase in income, or coresidence with parents (i.e., grandparents of children), and the occurrence of childbirth. We found that the number of existing children had a significant impact on the probability of childbirth. The likelihood of further childbirth by couples with one child was nearly equal to that of childless ones. However, the corresponding likelihood of couples with two children was about five times lower than that of childless ones.

CONCLUSIONS

The total fertility rates in modern developed societies are quite low and couples prefer having two children. This trend is known as the two-child norm, but it is a paradoxical phenomenon in terms of fitness maximization. Our result provided new quantitative evidence of this norm. This study revealed that the number of existing children being less than two was one of the factors associated with further childbearing in our Japanese sample.

Relationship between aggregation of rewards and the possibility of polymorphism in continuous snowdrift games

The existence of intra-population variations in cooperation level has often been reported by some empirical studies. Evolutionary conditions of polymorphism in cooperation have been investigated by using a framework of the continuous snowdrift game. However, our insights from this framework have been limited because of an assumption that the cooperative reward is a function of total amount of investments within an interacting group. In many cases, payoffs may actually depend on the interactions between the effects of such investments, such as members share the sum of beneficial effects that are individually produced from their own investments. Alternatively, payoffs may depend multiplicatively on investment, such as when investments are complementary. In the present paper, we investigated the influence of such difference on the evolution of cooperation with respect to three aspects of the aggregating process of individuals' contributions for reward, i.e. (i) additive or multiplicative, (ii) aggregation of either investments or effects, and (iii) promotion of advantage or suppression of disadvantage. We analytically show that the possibilities of the emergence of polymorphism are different depending on the type of aggregation process classified from these three aspects. Polymorphism of cooperation level never emerges unless the aggregation process is the aggregation of investment or the multiplicative aggregation of effect with suppression of disadvantage. Our results show the necessary condition for the emergence of polymorphic cooperation levels that are observed in various taxonomic groups.

Evolutionary dynamics of n-player games played by relatives

One of the core concepts in social evolution theory is kin selection. Kin selection provides a perspective to understand how natural selection operates when genetically similar individuals are likely to interact. A family-structured population is an excellent example of this, where relatives are engaged in social interactions. Consequences of such social interactions are often described in game-theoretical frameworks, but there is a growing consensus that a naive inclusive fitness accounting with dyadic relatedness coefficients are of limited use when non-additive fitness effects are essential in those situations. Here, I provide a general framework to analyse multiplayer interactions among relatives. Two important results follow from my analysis. First, it is generally necessary to know the n-tuple genetic association of family members when n individuals are engaged in social interactions. However, as a second result, I found that, for a special class of games, we need only measures of lower-order genetic association to fully describe its evolutionary dynamics. I introduce the concept of degree of the game and show how this degree is related to the degree of genetic association.

Ecology drives intragenomic conflict over menopause

Menopause is the transition from reproductive to non-reproductive life well before natural death. Rather than involving a smooth, rapid change, it is normally preceded by a long period of erratic hormonal fluctuation that is accompanied by a plethora of unpleasant symptoms. Here, we (1) suggest that this turbulent period owes to conflict, between a woman's maternally inherited (MI) and paternally inherited (PI) genes, over the trade-off between reproduction and communal care; (2) perform a theoretical analysis to show that this conflict is resolved either through silencing or fluctuating expression of one of the genes; (3) highlight which of the symptoms preceding menopause may result from antagonistic co-evolution of MI and PI genes; (4) argue that ecological differences between ancestral human populations may explain the variability in menopause among different ethnic groups; (5) discuss how these insights may be used to inform family planning and cancer risk assessment based on a woman's ancestral background.

Evolution of social versus individual learning in a subdivided population revisited: comparative analysis of three coexistence mechanisms using the inclusive-fitness method

Learning abilities are categorized into social (learning from others) and individual learning (learning on one's own). Despite the typically higher cost of individual learning, there are mechanisms that allow stable coexistence of both learning modes in a single population. In this paper, we investigate by means of mathematical modeling how the effect of spatial structure on evolutionary outcomes of pure social and individual learning strategies depends on the mechanisms for coexistence. We model a spatially structured population based on the infinite-island framework and consider three scenarios that differ in coexistence mechanisms. Using the inclusive-fitness method, we derive the equilibrium frequency of social learners and the genetic load of social learning (defined as average fecundity reduction caused by the presence of social learning) in terms of some summary statistics, such as relatedness, for each of the three scenarios and compare the results. This comparative analysis not only reconciles previous models that made contradictory predictions as to the effect of spatial structure on the equilibrium frequency of social learners but also derives a simple mathematical rule that determines the sign of the genetic load (i.e. whether or not social learning contributes to the mean fecundity of the population).

A mathematical description of the inclusive fitness theory

Recent developments in the inclusive fitness theory have revealed that the direction of evolution can be analytically predicted in a wider class of models than previously thought, such as those models dealing with network structure. This paper aims to provide a mathematical description of the inclusive fitness theory. Specifically, we provide a general framework based on a Markov chain that can implement basic models of inclusive fitness. Our framework is based on the probability distribution of "offspring-to-parent map", from which the key concepts of the theory, such as fitness function, relatedness and inclusive fitness, are derived in a straightforward manner. We prove theorems showing that inclusive fitness always provides a correct prediction on which of two competing genes more frequently appears in the long run in the Markov chain. As an application of the theorems, we prove a general formula of the optimal dispersal rate in the Wright's island model with recurrent mutations. We also show the existence of the critical mutation rate, which does not depend on the number of islands and below which a positive dispersal rate evolves. Our framework can also be applied to lattice or network structured populations.

Does synergy rescue the evolution of cooperation? An analysis for homogeneous populations with non-overlapping generations

Recent developments of social evolution theory have revealed conditions under which cooperation is favored by natural selection. Effects of population structure on the evolution of cooperation have been one of the central questions in this issue, and inclusive fitness analyses have unveiled two different selective forces that favor cooperation; the direct fitness effect to the helper and the indirect fitness benefit to the helper via its kin. Although these theoretical frameworks have made a significant contribution to our understanding of cooperative traits, there is still one factor to be taken into account, synergy. Synergy means a nonlinear effect that arises when two individuals help each other. In other words, it represents deviation from additivity, to which inclusive fitness theory has paid relatively little attention. Here I provide a theoretical result on the possibility that synergy favors the evolution of cooperation. For homogeneously structured populations with non-overlapping generations, I show that incorporating synergistic effects does not rescue the evolution of cooperation. Potential factors that could enable synergy to rescue the evolution of cooperation are also discussed.

Anti-social punishment can prevent the co-evolution of punishment and cooperation

The evolution of cooperation is one of the great puzzles in evolutionary biology. Punishment has been suggested as one solution to this problem. Here punishment is generally defined as incurring a cost to inflict harm on a wrong-doer. In the presence of punishers, cooperators can gain higher payoffs than non-cooperators. Therefore cooperation may evolve as long as punishment is prevalent in the population. Theoretical models have revealed that spatial structure can favor the co-evolution of punishment and cooperation, by allowing individuals to only play and compete with those in their immediate neighborhood. However, those models have usually assumed that punishment is always targeted at non-cooperators. In light of recent empirical evidence of punishment targeted at cooperators, we relax this assumption and study the effect of so-called 'anti-social punishment'. We find that evolution can favor anti-social punishment, and that when anti-social punishment is possible costly punishment no longer promotes cooperation. As there is no reason to assume that cooperators cannot be the target of punishment during evolution, our results demonstrate serious restrictions on the ability of costly punishment to allow the evolution of cooperation in spatially structured populations. Our results also help to make sense of the empirical observation that defectors will sometimes pay to punish cooperators.

Stochastic evolutionary dynamics of bimatrix games

Evolutionary game dynamics of two-player asymmetric games in finite populations is studied. We consider two roles in the game, roles alpha and beta. alpha-players and beta-players interact and gain payoffs. The game is described by a pair of matrices, which is called bimatrix. One's payoff in the game is interpreted as its fecundity, thus strategies are subject to natural selection. In addition, strategies can randomly mutate to others. We formulate a stochastic evolutionary game dynamics of bimatrix games as a frequency-dependent Moran process with mutation. We analytically derive the stationary distribution of strategies under weak selection. Our result provides a criterion for equilibrium selection in general bimatrix games.

Prelife catalysts and replicators

Life is based on replication and evolution. But replication cannot be taken for granted. We must ask what there was prior to replication and evolution. How does evolution begin? We have proposed prelife as a generative system that produces information and diversity in the absence of replication. We model prelife as a binary soup of active monomers that form random polymers. 'Prevolutionary' dynamics can have mutation and selection prior to replication. Some sequences might have catalytic activity, thereby enhancing the rates of certain prelife reactions. We study the selection criteria for these prelife catalysts. Their catalytic efficiency must be above certain critical values. We find a maintenance threshold and an initiation threshold. The former is a linear function of sequence length, and the latter is an exponential function of sequence length. Therefore, it is extremely hard to select for prelife catalysts that have long sequences. We compare prelife catalysis with a simple model for replication. Assuming fast template-based elongation reactions, we can show that replicators have selection thresholds that are independent of their sequence length. Our calculation demonstrates the efficiency of replication and provides an explanation of why replication was selected over other forms of prelife catalysis.

Strategy selection in structured populations

Evolutionary game theory studies frequency dependent selection. The fitness of a strategy is not constant, but depends on the relative frequencies of strategies in the population. This type of evolutionary dynamics occurs in many settings of ecology, infectious disease dynamics, animal behavior and social interactions of humans. Traditionally evolutionary game dynamics are studied in well-mixed populations, where the interaction between any two individuals is equally likely. There have also been several approaches to study evolutionary games in structured populations. In this paper we present a simple result that holds for a large variety of population structures. We consider the game between two strategies, A and B, described by the payoff matrix(abcd). We study a mutation and selection process. For weak selection strategy A is favored over B if and only if sigma a+b>c+sigma d. This means the effect of population structure on strategy selection can be described by a single parameter, sigma. We present the values of sigma for various examples including the well-mixed population, games on graphs, games in phenotype space and games on sets. We give a proof for the existence of such a sigma, which holds for all population structures and update rules that have certain (natural) properties. We assume weak selection, but allow any mutation rate. We discuss the relationship between sigma and the critical benefit to cost ratio for the evolution of cooperation. The single parameter, sigma, allows us to quantify the ability of a population structure to promote the evolution of cooperation or to choose efficient equilibria in coordination games.

Adaptive reproduction schedule as a cause of worker policing in social hymenoptera: a dynamic game analysis

Evolutionary theories predict conflicts over sex allocation, male parentage, and reproductive allocation in hymenopteran societies. However, no theory to date has considered the evolution when a colony faces these three conflicts simultaneously. We tackled this issue by developing a dynamic game model, focusing especially on worker policing. Whereas a Nash equilibrium predicts male parentage patterns that are basically the same as those of relatedness-based worker-policing theory (queen multiple mating impedes worker reproduction), we also show the potential for worker policing under queen single mating. Worker policing will depend on the stage of colony growth that is caused by interaction with reproductive allocation conflict or a trade-off between current and future reproduction. Male production at an early stage greatly hinders the growth of the work force and undermines future inclusive fitness of colony members, leading to worker policing at the ergonomic stage. This new mechanism can explain much broader ranges of existing worker-policing behavior than that predicted from relatedness. Predictions differ in many respects from those of models assuming operation of only one or two of the three conflicts, suggesting the importance of interactions among conflicts.

Mutation-selection equilibrium in games with multiple strategies

In evolutionary games the fitness of individuals is not constant but depends on the relative abundance of the various strategies in the population. Here we study general games among n strategies in populations of large but finite size. We explore stochastic evolutionary dynamics under weak selection, but for any mutation rate. We analyze the frequency dependent Moran process in well-mixed populations, but almost identical results are found for the Wright-Fisher and Pairwise Comparison processes. Surprisingly simple conditions specify whether a strategy is more abundant on average than 1/n, or than another strategy, in the mutation-selection equilibrium. We find one condition that holds for low mutation rate and another condition that holds for high mutation rate. A linear combination of these two conditions holds for any mutation rate. Our results allow a complete characterization of nxn games in the limit of weak selection.

Evolution of cooperation by phenotypic similarity

The emergence of cooperation in populations of selfish individuals is a fascinating topic that has inspired much work in theoretical biology. Here, we study the evolution of cooperation in a model where individuals are characterized by phenotypic properties that are visible to others. The population is well mixed in the sense that everyone is equally likely to interact with everyone else, but the behavioral strategies can depend on distance in phenotype space. We study the interaction of cooperators and defectors. In our model, cooperators cooperate with those who are similar and defect otherwise. Defectors always defect. Individuals mutate to nearby phenotypes, which generates a random walk of the population in phenotype space. Our analysis brings together ideas from coalescence theory and evolutionary game dynamics. We obtain a precise condition for natural selection to favor cooperators over defectors. Cooperation is favored when the phenotypic mutation rate is large and the strategy mutation rate is small. In the optimal case for cooperators, in a one-dimensional phenotype space and for large population size, the critical benefit-to-cost ratio is given by b/c = 1 + 2/square root(3). We also derive the fundamental condition for any two-strategy symmetric game and consider high-dimensional phenotype spaces.

Originator dynamics

We study the origin of evolution. Evolution is based on replication, mutation, and selection. But how does evolution begin? When do chemical kinetics turn into evolutionary dynamics? We propose "prelife" and "prevolution" as the logical precursors of life and evolution. Prelife generates sequences of variable length. Prelife is a generative chemistry that proliferates information and produces diversity without replication. The resulting "prevolutionary dynamics" have mutation and selection. We propose an equation that allows us to investigate the origin of evolution. In one limit, this "originator equation" gives the classical selection equation. In the other limit, we obtain "prelife." There is competition between life and prelife and there can be selection for or against replication. Simple prelife equations with uniform rate constants have the property that longer sequences are exponentially less frequent than shorter ones. But replication can reverse such an ordering. As the replication rate increases, some longer sequences can become more frequent than shorter ones. Thus, replication can lead to "reversals" in the equilibrium portraits. We study these reversals, which mark the transition from prelife to life in our model. If the replication potential exceeds a critical value, then life replicates into existence.

Direct reciprocity with costly punishment: generous tit-for-tat prevails

The standard model for direct reciprocity is the repeated Prisoner's Dilemma, where in each round players choose between cooperation and defection. Here we extend the standard framework to include costly punishment. Now players have a choice between cooperation, defection and costly punishment. We study the set of all reactive strategies, where the behavior depends on what the other player has done in the previous round. We find all cooperative strategies that are Nash equilibria. If the cost of cooperation is greater than the cost of punishment, then the only cooperative Nash equilibrium is generous-tit-for-tat (GTFT), which does not use costly punishment. If the cost of cooperation is less than the cost of punishment, then there are infinitely many cooperative Nash equilibria and the response to defection can include costly punishment. We also perform computer simulations of evolutionary dynamics in populations of finite size. These simulations show that in the context of direct reciprocity, (i) natural selection prefers generous tit-for-tat over strategies that use costly punishment, and (ii) that costly punishment does not promote the evolution of cooperation. We find quantitative agreement between our simulation results and data from experimental observations.

Evolutionary stability on graphs

Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree k > 2. Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth-death (BD), death-birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs.

Repeated games and direct reciprocity under active linking

Direct reciprocity relies on repeated encounters between the same two individuals. Here we examine the evolution of cooperation under direct reciprocity in dynamically structured populations. Individuals occupy the vertices of a graph, undergoing repeated interactions with their partners via the edges of the graph. Unlike the traditional approach to evolutionary game theory, where individuals meet at random and have no control over the frequency or duration of interactions, we consider a model in which individuals differ in the rate at which they seek new interactions. Moreover, once a link between two individuals has formed, the productivity of this link is evaluated. Links can be broken off at different rates. Whenever the active dynamics of links is sufficiently fast, population structure leads to a simple transformation of the payoff matrix, effectively changing the game under consideration, and hence paving the way for reciprocators to dominate defectors. We derive analytical conditions for evolutionary stability.

The one-third law of evolutionary dynamics

Evolutionary game dynamics in finite populations provide a new framework for studying selection of traits with frequency-dependent fitness. Recently, a "one-third law" of evolutionary dynamics has been described, which states that strategy A fixates in a B-population with selective advantage if the fitness of A is greater than that of B when A has a frequency 13. This relationship holds for all evolutionary processes examined so far, from the Moran process to games on graphs. However, the origin of the "number"13 is not understood. In this paper we provide an intuitive explanation by studying the underlying stochastic processes. We find that in one invasion attempt, an individual interacts on average with B-players twice as often as with A-players, which yields the one-third law. We also show that the one-third law implies that the average Malthusian fitness of A is positive.

Evolutionary graph theory: breaking the symmetry between interaction and replacement

We study evolutionary dynamics in a population whose structure is given by two graphs: the interaction graph determines who plays with whom in an evolutionary game; the replacement graph specifies the geometry of evolutionary competition and updating. First, we calculate the fixation probabilities of frequency dependent selection between two strategies or phenotypes. We consider three different update mechanisms: birth-death, death-birth and imitation. Then, as a particular example, we explore the evolution of cooperation. Suppose the interaction graph is a regular graph of degree h, the replacement graph is a regular graph of degree g and the overlap between the two graphs is a regular graph of degree l. We show that cooperation is favored by natural selection if b/c>hg/l. Here, b and c denote the benefit and cost of the altruistic act. This result holds for death-birth updating, weak-selection and large population size. Note that the optimum population structure for cooperators is given by maximum overlap between the interaction and the replacement graph (g=h=l), which means that the two graphs are identical. We also prove that a modified replicator equation can describe how the expected values of the frequencies of an arbitrary number of strategies change on replacement and interaction graphs: the two graphs induce a transformation of the payoff matrix.

Direct reciprocity on graphs

Direct reciprocity is a mechanism for the evolution of cooperation based on the idea of repeated encounters between the same two individuals. Here we examine direct reciprocity in structured populations, where individuals occupy the vertices of a graph. The edges denote who interacts with whom. The graph represents spatial structure or a social network. For birth-death or pairwise comparison updating, we find that evolutionary stability of direct reciprocity is more restrictive on a graph than in a well-mixed population, but the condition for reciprocators to be advantageous is less restrictive on a graph. For death-birth and imitation updating, in contrast, both conditions are easier to fulfill on a graph. Moreover, for all four update mechanisms, reciprocators can dominate defectors on a graph, which is never possible in a well-mixed population. We also study the effect of an error rate, which increases with the number of links per individual; interacting with more people simultaneously enhances the probability of making mistakes. We provide analytic derivations for all results.