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.     

Social and spatial effects on genetic variation between foraging flocks in a wild bird population

Social interactions are rarely random. In some instances, animals exhibit homophily or heterophily, the tendency to interact with similar or dissimilar conspecifics, respectively. Genetic homophily and heterophily influence the evolutionary dynamics of populations, because they potentially affect sexual and social selection. Here, we investigate the link between social interactions and allele frequencies in foraging flocks of great tits (Parus major) over three consecutive years. We constructed co‐occurrence networks which explicitly described the splitting and merging of 85,602 flocks through time (fission–fusion dynamics), at 60 feeding sites. Of the 1,711 birds in those flocks, we genotyped 962 individuals at 4,701 autosomal single nucleotide polymorphisms (SNPs). By combining genomewide genotyping with repeated field observations of the same individuals, we were able to investigate links between social structure and allele frequencies at a much finer scale than was previously possible. We explicitly accounted for potential spatial effects underlying genetic structure at the population level. We modelled social structure and spatial configuration of great tit fission–fusion dynamics with eigenvector maps. Variance partitioning revealed that allele frequencies were strongly affected by group fidelity (explaining 27%–45% of variance) as individuals tended to maintain associations with the same conspecifics. These conspecifics were genetically more dissimilar than expected, shown by genomewide heterophily for pure social (i.e., space‐independent) grouping preferences. Genomewide homophily was linked to spatial configuration, indicating spatial segregation of genotypes. We did not find evidence for homophily or heterophily for putative socially relevant candidate genes or any other SNP markers. Together, these results demonstrate the importance of distinguishing social and spatial processes in determining population structure.     

Evolutionary dynamics of continuous strategy games on graphs and social networks under weak selection

Understanding the emergence of cooperation among selfish individuals has been a long-standing puzzle, which has been studied by a variety of game models. Most previous studies presumed that interactions between individuals are discrete, but it seems unrealistic in real systems. Recently, there are increasing interests in studying game models with a continuous strategy space. Existing research work on continuous strategy games mainly focuses on well-mixed populations. Especially, little theoretical work has been conducted on their evolutionary dynamics in a structured population. In the previous work (Zhong et al., BioSystems, 2012), we showed that under strong selection, continuous and discrete strategies have significantly different equilibrium and game dynamics in spatially structured populations. In this paper, we further study evolutionary dynamics of continuous strategy games under weak selection in structured populations. By using the fixation probability based stochastic dynamics, we derive exact conditions of natural selection favoring cooperation for the death–birth updating scheme. We also present a network gain decomposition of the game equilibrium, which might provide a new view of the network reciprocity in a quantitative way. Finally, we make a detailed comparison between games using discrete and continuous strategies. As compared to the former, we find that for the latter (i) the same selection conditions are derived for the general 2 × 2 game; especially, the rule b/c > k in a simplified Prisoner's Dilemma is valid as well; however, (ii) for a coordination game, interestingly, the risk-dominant strategy is disfavored. Numerical simulations have also been conducted to validate our results.     

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 . We study a mutation and selection process. For weak selection strategy A is favored over B if and only if σa+b>c+σd. This means the effect of population structure on strategy selection can be described by a single parameter, σ. We present the values of σ 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 σ, 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 σ and the critical benefit to cost ratio for the evolution of cooperation. The single parameter, σ, allows us to quantify the ability of a population structure to promote the evolution of cooperation or to choose efficient equilibria in coordination games.     

Indirect Genetic Effects and the Dynamics of Social Interactions

BackgroundIndirect genetic effects (IGEs) occur when genes expressed in one individual alter the expression of traits in social partners. Previous studies focused on the evolutionary consequences and evolutionary dynamics of IGEs, using equilibrium solutions to predict phenotypes in subsequent generations. However, whether or not such steady states may be reached may depend on the dynamics of interactions themselves.ResultsIn our study, we focus on the dynamics of social interactions and indirect genetic effects and investigate how they modify phenotypes over time. Unlike previous IGE studies, we do not analyse evolutionary dynamics; rather we consider within-individual phenotypic changes, also referred to as phenotypic plasticity. We analyse iterative interactions, when individuals interact in a series of discontinuous events, and investigate the stability of steady state solutions and the dependence on model parameters, such as population size, strength, and the nature of interactions. We show that for interactions where a feedback loop occurs, the possible parameter space of interaction strength is fairly limited, affecting the evolutionary consequences of IGEs. We discuss the implications of our results for current IGE model predictions and their limitations.     

From local interactions to population dynamics in site-based models of ecology

A central problem in ecology is relating the interactions of individuals-described in terms of competition, predation, interference, etc.-to the dynamics of the populations of these individuals-in terms of change in numbers of individuals over time. Here, we address this problem for a class of site-based ecological models, where local interactions between individuals take place at a finite number of discrete resource sites over non-overlapping generations and, between generations, individuals move randomly between sites over the entire system. Such site-based models have previously been applied to a wide range of ecological systems: from those involving contest or scramble competition for resources to host-parasite interactions and meta-populations. We show how the population dynamics of site-based models can be accurately approximated by and understood through deterministic and stochastic difference equations. Conversely, we use the inverse of this approximation to show what implicit assumptions are made about individual interactions by modelling of population dynamics in terms of difference equations. To this end, we prove a useful and general theorem: that any model in our class of site-based models has a corresponding stochastic difference equation population model, by which it can be approximated. This theorem allows us to calculate long-term population dynamics, evolutionary stable strategies and, by extending our theory to account for large deviations, extinction probabilities for a wide range of site-based systems. Our methodology is then illustrated to various examples of between species competition, predator-prey interactions and co-operation.     

Phylogentic constraints,adaptive syndromes,and emergent properties: From individuals to population dynamics

The hypothesis is developed that there are causal linkages in evolved insect herbivore life histories and behaviors from phylogenetic constraints to adaptive syndromes to the emergent properties involving ecological interactions and population dynamics. Thus the argument is developed that the evolutionary biology of a species predetermines its current ecology.Phylogenetic Constraints refer to old characters in the phylogeny of a species and a group of species which set limits on the range of life history patterns and behaviors that can evolve. For example, a sawfly is commonly limited to oviposition in soft plant tissue, while plants are growing rapidly.Adaptive Syndromes are evolutionary responses to the phylogenetic constraints that minimize the limitations and maximize larval performance. Such syndromes commonly involve details of female ovipositional behavior and how individuals make choices for oviposition sites relative to plant quality variation which maximize larval survival. Syndromes also involve larval adaptations to the kinds of choices females make in oviposition. The evolutionary biology involved with phylogenetic constraints and adaptive syndromes commonly predetermines the ecological interactions of a species and its population dynamics. Therefore, these ecological interactions are calledEmergent Properties because they are natural consequences of evolved morphology, behavior, and physiology. They commonly strongly influence the three-trophic-level interactions among host plants, insect herbivores, and carnivores, and the relative forces of bottom-up and top-down influences in food webs. The arguments are supported using such examples as galling sawflies and other gallers, shoot-boring moths and beetles, budworms, and forest Macrolepidoptera. The contrasts between outbreak or eruptive species and uncommon and rare species with latent population dynamics are emphasized.     

The danger within: the role of genetic,behavioural and ecological factors in population persistence of colour polymorphic species

Polymorphic species have been the focus of important work in evolutionary biology. It has been suggested that colour polymorphic species have specific evolutionary and population dynamics that enable them to persist through environmental changes better than less variable species. We suggest that recent empirical and theoretical work indicates that polymorphic species may be more vulnerable to extinction than previously thought. This vulnerability arises because these species often have a number of correlated sexual, behavioural, life history and ecological traits, which can have a simple genetic underpinning. When exacerbated by environmental change, these alternate strategies can lead to conflict between morphs at the genomic and population levels, which can directly or indirectly affect population and evolutionary dynamics. In this perspective, we identify a number of ways in which the nature of the correlated traits, their underpinning genetic architecture, and the inevitable interactions between colour morphs can result in a reduction in population fitness. The principles illustrated here apply to all kinds of discrete polymorphism (e.g. behavioural syndromes), but we focus primarily on colour polymorphism because they are well studied. We urge further empirical investigation of the genetic architecture and interactions in polymorphic species to elucidate the impact on population fitness.     

The strength of genetic interactions scales weakly with mutational effects

Background

Genetic interactions pervade every aspect of biology, from evolutionary theory, where they determine the accessibility of evolutionary paths, to medicine, where they can contribute to complex genetic diseases. Until very recently, studies on epistatic interactions have been based on a handful of mutations, providing at best anecdotal evidence about the frequency and the typical strength of genetic interactions. In this study, we analyze a publicly available dataset that contains the growth rates of over five million double knockout mutants of the yeast Saccharomyces cerevisiae.

Results

We discuss a geometric definition of epistasis that reveals a simple and surprisingly weak scaling law for the characteristic strength of genetic interactions as a function of the effects of the mutations being combined. We then utilized this scaling to quantify the roughness of naturally occurring fitness landscapes. Finally, we show how the observed roughness differs from what is predicted by Fisher''s geometric model of epistasis, and discuss the consequences for evolutionary dynamics.

Conclusions

Although epistatic interactions between specific genes remain largely unpredictable, the statistical properties of an ensemble of interactions can display conspicuous regularities and be described by simple mathematical laws. By exploiting the amount of data produced by modern high-throughput techniques, it is now possible to thoroughly test the predictions of theoretical models of genetic interactions and to build informed computational models of evolution on realistic fitness landscapes.     

EVOLUTIONARY EPIDEMIOLOGY AND THE DYNAMICS OF ADAPTATION

The mean fitness of a population, often equal to its growth rate, measures its level of adaptation to particular environmental conditions. A better understanding of the evolution of mean fitness could thus provide a natural link between evolution and demography. Yet, after the seminal work of Fisher and its renowned "fundamental theorem of natural selection," the dynamics of mean fitness has attracted little attention, and mostly from theoretical population geneticists. Here we analyze the dynamics of mean fitness in the context of host-parasite interactions. We illustrate the potential relevance of this analysis under different scenarios ranging from a simple situation in which a parasite evolves in a homogeneous host population to a more complex one with host-parasite coevolution. In each case, we contrast the effects of natural selection, recurrent mutations, and the change of the biotic environment, on the dynamics of adaptation. Decoupling these three components helps elucidate the interplay between evolutionary and ecological dynamics. In particular, it offers new perspectives on situations leading to evolutionary suicide. As mean fitness is an easily measurable quantity in microbial systems, this analysis provides new ways to track the dynamics of adaptation in experimental evolution and coevolution studies.     

Wild immunology

In wild populations, individuals are regularly exposed to a wide range of pathogens. In this context, organisms must elicit and regulate effective immune responses to protect their health while avoiding immunopathology. However, most of our knowledge about the function and dynamics of immune responses comes from laboratory studies performed on inbred mice in highly controlled environments with limited exposure to infection. Natural populations, on the other hand, exhibit wide genetic and environmental diversity. We argue that now is the time for immunology to be taken into the wild. The goal of 'wild immunology' is to link immune phenotype with host fitness in natural environments. To achieve this requires relevant measures of immune responsiveness that are both applicable to the host-parasite interaction under study and robustly associated with measures of host and parasite fitness. Bringing immunology to nonmodel organisms and linking that knowledge host fitness, and ultimately population dynamics, will face difficult challenges, both technical (lack of reagents and annotated genomes) and statistical (variation among individuals and populations). However, the affordability of new genomic technologies will help immunologists, ecologists and evolutionary biologists work together to translate and test our current knowledge of immune mechanisms in natural systems. From this approach, ecologists will gain new insight into mechanisms relevant to host health and fitness, while immunologists will be given a measure of the real-world health impacts of the immune factors they study. Thus, wild immunology can be the missing link between laboratory-based immunology and human, wildlife and domesticated animal health.     

Neutral biogeography of phylogenetically structured interaction networks

Little is known about how biogeographic processes affect the dynamics of species interactions in space and time, although it is widely accepted that they drive community assemblage. In functional interactions, such as pollination and seed dispersal, species that share common ancestry tend to retain a common number of interactions and interact with similar sets of species, a pattern more commonly observed for animals than plants. On the one hand, the most coherent explanation for the phylogenetic structure of pollination and seed dispersal networks is that species retain ecological traits over evolution, which would cause the conservation of interaction partners. On the other hand, fundamental processes of biodiversity, such as dispersal and evolutionary rates seem to have important roles shaping the observed phylogenetic structure of mutualistic networks, but no model has been created to study the effect of these processes in the phylogenetic structure of mutualistic interactions. Here, we developed a stochastic simulation model to study the evolution of two interacting groups of species, which evolve independently over the same geographical domain. In our model, individuals of the same interaction group share ecological traits, whereas individuals of different trophic groups are ecologically distinct. We show that even in the absence of ecological differences between individuals, and disregarding any conservation of phenotypical and phenological traits between species, the interplay of dispersal and speciation is still a major driver of complex phylogenetic structure of functional interactions, such as pollination and seed dispersal.     

Evolutionary game theory in an agent-based brain tumor model: exploring the 'Genotype-Phenotype' link

To investigate the genotype-phenotype link in a polyclonal cancer cell population, here we introduce evolutionary game theory into our previously developed agent-based brain tumor model. We model the heterogeneous cell population as a mixture of two distinct genotypes: the more proliferative Type A and the more migratory Type B. Our agent-based simulations reveal a phase transition in the tumor's velocity of spatial expansion linking the tumor fitness to genotypic composition. Specifically, velocity initially falls as rising payoffs reward the interactions among the more stationary Type A cells, but unexpectedly accelerates again when these A-A payoffs increase even further. At this latter accelerating stage, fewer migratory Type B cells appear to confer a competitive advantage in terms of the tumor's spatial aggression over the overall numerically dominating Type A cells, which in turn leads to an acceleration of the overall tumor dynamics while its surface roughness declines. We discuss potential implications of our findings for cancer research.     

Genetic correlations and the coevolutionary dynamics of three-species systems

The majority of species interact with at least several others. We develop simple genetic models of coevolution between three species where interactions are mediated by quantitative traits. We assume that one of the species has two quantitative traits, each of which governs its interaction with one of the other two species. We use this model to explore how genetic correlations between the two traits in the multivariate species shape the evolutionary dynamics and outcomes of three species interactions. Our results suggest that genetic correlations are most important when at least one of the interactions is between a predator and prey or parasite and host. In these cases, genetic correlations between traits lead to a wide variety of novel coevolutionary outcomes and dynamics. In particular, genetic correlations can affect the existence and stability of coevolutionary equilibrium points, and they can lead to recurrent or permanent maladaptation. When the three species interact only as competitors or mutualists, however, genetic correlations have no effect on the outcome of coevolution. In all cases, our results reveal the surprising conclusion that both positive and negative genetic correlations between traits have qualitatively identical effects on coevolutionary dynamics.     

Spectacular phenomena and limits to rationality in genetic and cultural evolution

In studies of both animal and human behaviour, game theory is used as a tool for understanding strategies that appear in interactions between individuals. Game theory focuses on adaptive behaviour, which can be attained only at evolutionary equilibrium. We suggest that behaviour appearing during interactions is often outside the scope of such analysis. In many types of interaction, conflicts of interest exist between players, fuelling the evolution of manipulative strategies. Such strategies evolve out of equilibrium, commonly appearing as spectacular morphology or behaviour with obscure meaning, to which other players may react in non-adaptive, irrational ways. We present a simple model to show some limitations of the game-theory approach, and outline the conditions in which evolutionary equilibria cannot be maintained. Evidence from studies of biological interactions seems to support the view that behaviour is often not at equilibrium. This also appears to be the case for many human cultural traits, which have spread rapidly despite the fact that they have a negative influence on reproduction.     

The influence of habitat boundaries on evolutionary branching along environmental gradients

It is well known that habitat boundaries affect ecological dynamics, but their influence on evolutionary dynamics is less well understood. Here, we study the effects of different kinds of boundaries on evolutionary branching in clonal populations along environmental gradients by systematically analyzing individual-based stochastic models in small- and large-range systems, as well as their large-population-size limits through deterministic approximations. Specifically, we examine four prototypical kinds of boundaries: impermeable boundaries at which individuals stop (“stopping”), or from which they continue back into the interior as if bouncing back mechanically (“reflecting”), or that let them abort the dispersal attempt, return to their original position and try a different direction (“reprising”), and semipermeable boundaries that can be crossed without hindrance, but do not allow the crossing individual to return (“absorbing”).We find that boundary conditions shape branching patterns only in small-range systems, where stopping boundaries generate disruptive selection for a wide range of parameters, whereas absorbing boundaries always generate stabilizing selection. Reflecting and reprising boundaries generate disruptive selection at low individual mobilities, and stabilizing selection at high mobilities. To further analyze these findings, we introduce a simple approximation of the invasion fitness in a mobile population, which predicts the observed outcome. The effect of stochasticity on evolutionary outcomes is small even in small populations: stochasticity causes random branch extinctions at steeper slopes and higher mobilities. In large-range systems, frequency-dependent interactions alone induce evolutionary branching for almost all parameters and independent of boundary conditions.     

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 n×n games in the limit of weak selection.     

Evolution of Cooperation on Stochastic Dynamical Networks

Cooperative behavior that increases the fitness of others at a cost to oneself can be promoted by natural selection only in the presence of an additional mechanism. One such mechanism is based on population structure, which can lead to clustering of cooperating agents. Recently, the focus has turned to complex dynamical population structures such as social networks, where the nodes represent individuals and links represent social relationships. We investigate how the dynamics of a social network can change the level of cooperation in the network. Individuals either update their strategies by imitating their partners or adjust their social ties. For the dynamics of the network structure, a random link is selected and breaks with a probability determined by the adjacent individuals. Once it is broken, a new one is established. This linking dynamics can be conveniently characterized by a Markov chain in the configuration space of an ever-changing network of interacting agents. Our model can be analytically solved provided the dynamics of links proceeds much faster than the dynamics of strategies. This leads to a simple rule for the evolution of cooperation: The more fragile links between cooperating players and non-cooperating players are (or the more robust links between cooperators are), the more likely cooperation prevails. Our approach may pave the way for analytically investigating coevolution of strategy and structure.     

Evolution of nutrient acquisition: when adaptation fills the gap between contrasting ecological theories

Although plant strategies for acquiring nutrients have been widely studied from a functional point of view, their evolution is still not well understood. In this study, we investigate the evolutionary dynamics of these strategies and determine how they influence ecosystem properties. To do so, we use a simple nutrient-limited ecosystem model in which plant ability to take up nutrients is subject to adaptive dynamics. We postulate the existence of a trade-off between this ability and mortality. We show that contrasting strategies are possible as evolutionary outcomes, depending on the shape of the trade-off and, when nitrogen is considered as the limiting nutrient, on the intensity of symbiotic fixation. Our model enables us to bridge these evolutionary outcomes to classical ecological theories such as Hardin''s tragedy of the commons and Tilman''s rule of R*. Evolution does not systematically maximize plant biomass or primary productivity. On the other hand, each evolutionary outcome leads to a decrease in the availability of the limiting mineral nutrient, supporting the work of Tilman on competition between plants for a single resource. Our model shows that evolution can be used to link different classical ecological results and that adaptation may influence ecosystem properties in contrasted ways.     

Evolutionary stability and quasi-stationary strategy in stochastic evolutionary game dynamics

Stochastic evolutionary game dynamics for finite populations has recently been widely explored in the study of evolutionary game theory. It is known from the work of Traulsen et al. [2005. Phys. Rev. Lett. 95, 238701] that the stochastic evolutionary dynamics approaches the deterministic replicator dynamics in the limit of large population size. However, sometimes the limiting behavior predicted by the stochastic evolutionary dynamics is not quite in agreement with the steady-state behavior of the replicator dynamics. This paradox inspired us to give reasonable explanations of the traditional concept of evolutionarily stable strategy (ESS) in the context of finite populations. A quasi-stationary analysis of the stochastic evolutionary game dynamics is put forward in this study and we present a new concept of quasi-stationary strategy (QSS) for large but finite populations. It is shown that the consistency between the QSS and the ESS implies that the long-term behavior of the replicator dynamics can be predicted by the quasi-stationary behavior of the stochastic dynamics. We relate the paradox to the time scales and find that the contradiction occurs only when the fixation time scale is much longer than the quasi-stationary time scale. Our work may shed light on understanding the relationship between the deterministic and stochastic methods of modeling evolutionary game dynamics.