Evolutionary stability on graphs

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.

How to promote the adoption of intelligent spray technology in farmers' cooperatives? --Based on the perspective of evolutionary game

In order to accelerate the development of smart agriculture and realize the green transformation of agriculture, the coupling of prospect theory and evolutionary game theory is introduced. Construct a two-party evolutionary game model for the adoption of intelligent spray technology in farmers' cooperatives, analyze the evolution of farmers' cooperatives and government strategy selection and its influencing factors according to the replication dynamic equation, and conduct numerical simulation experiments through Matlab software. The results show that the adoption of intelligent spray technology by farmers' cooperatives and the government's choice of subsidies are the two optimal stable states of the evolutionary system. The government's subsidy policy can effectively stimulate farmers' cooperatives to adopt intelligent spray technology, but when the government subsidy coefficient is greater than or equal to 70%, the decision-making evolution of the government and farmers' cooperatives is unstable. The increase in farmers' awareness of pesticide hazards, the scale of operation of farmers' cooperatives, the price of high-quality agricultural products, and the sensitivity of farmers' cooperatives to profits and losses contribute to the promotion of intelligent spray technology, and the damage probability of intelligent spray technology has a restraining effect.

Periodic orbits in deterministic discrete-time evolutionary game dynamics: An information-theoretic perspective

Even though the existence of nonconvergent evolution of the states of populations in ecological and evolutionary contexts is an undeniable fact, insightful game-theoretic interpretations of such outcomes are scarce in the literature of evolutionary game theory. As a proof-of-concept, we tap into the information-theoretic concept of relative entropy in order to construct a game-theoretic interpretation for periodic orbits in a wide class of deterministic discrete-time evolutionary game dynamics, primarily investigating the two-player two-strategy case. Effectively, we present a consistent generalization of the evolutionarily stable strategy-the cornerstone of the evolutionary game theory-and aptly term the generalized concept "information stable orbit." The information stable orbit captures the essence of the evolutionarily stable strategy in that it compares the total payoff obtained against an evolving mutant with the total payoff that the mutant gets while playing against itself. Furthermore, we discuss the connection of the information stable orbit with the dynamical stability of the corresponding periodic orbit.

Quantitative assessment can stabilize indirect reciprocity under imperfect information

The field of indirect reciprocity investigates how social norms can foster cooperation when individuals continuously monitor and assess each other's social interactions. By adhering to certain social norms, cooperating individuals can improve their reputation and, in turn, receive benefits from others. Eight social norms, known as the "leading eight," have been shown to effectively promote the evolution of cooperation as long as information is public and reliable. These norms categorize group members as either 'good' or 'bad'. In this study, we examine a scenario where individuals instead assign nuanced reputation scores to each other, and only cooperate with those whose reputation exceeds a certain threshold. We find both analytically and through simulations that such quantitative assessments are error-correcting, thus facilitating cooperation in situations where information is private and unreliable. Moreover, our results identify four specific norms that are robust to such conditions, and may be relevant for helping to sustain cooperation in natural populations.

Universal scaling of extinction time in stochastic evolutionary dynamics

Evolutionary dynamics is well captured by the replicator equations when the population is infinite and well-mixed. However, the extinction dynamics is modified with finite and structured populations. Experiments on the non-transitive ecosystem containing three populations of bacteria found that the ecological stability sensitively depends on the spatial structure of the populations. Based on the Reference-Gamble-Birth algorithm, we use agent-based Monte Carlo simulations to investigate the extinction dynamics in the rock-paper-scissors ecosystem with finite and structured populations. On the fully-connected network, the extinction time in stable and unstable regimes falls into two universal functions when plotted with the rescaled variables. On the two dimensional grid, the spatial structure changes the transition boundary between stable and unstable regimes but doesn't change its extinction trend. The finding of universal scaling in extinction dynamics is unexpected, and may provide a powerful method to classify different evolutionary dynamics into universal classes.

The role of evolutionary game theory in spatial and non-spatial models of the survival of cooperation in cancer: a review

Evolutionary game theory (EGT) is a branch of mathematics which considers populations of individuals interacting with each other to receive pay-offs. An individual’s pay-off is dependent on the strategy of its opponent(s) as well as on its own, and the higher its pay-off, the higher its reproductive fitness. Its offspring generally inherit its interaction strategy, subject to random mutation. Over time, the composition of the population shifts as different strategies spread or are driven extinct. In the last 25 years there has been a flood of interest in applying EGT to cancer modelling, with the aim of explaining how cancerous mutations spread through healthy tissue and how intercellular cooperation persists in tumour-cell populations. This review traces this body of work from theoretical analyses of well-mixed infinite populations through to more realistic spatial models of the development of cooperation between epithelial cells. We also consider work in which EGT has been used to make experimental predictions about the evolution of cancer, and discuss work that remains to be done before EGT can make large-scale contributions to clinical treatment and patient outcomes.

Game-theoretical approach for opinion dynamics on social networks

Opinion dynamics on social networks have received considerable attentions in recent years. Nevertheless, just a few works have theoretically analyzed the condition in which a certain opinion can spread in the whole structured population. In this article, we propose an evolutionary game approach for a binary opinion model to explore the conditions for an opinion's spreading. Inspired by real-life observations, we assume that an agent's choice to select an opinion is not random but is based on a score rooted from both public knowledge and the interactions with neighbors. By means of coalescing random walks, we obtain a condition in which opinion A can be favored to spread on social networks in the weak selection limit. We find that the successfully spreading condition of opinion A is closely related to the basic scores of binary opinions, the feedback scores on opinion interactions, and the structural parameters including the edge weights, the weighted degrees of vertices, and the average degree of the network. In particular, when individuals adjust their opinions based solely on the public information, the vitality of opinion A depends exclusively on the difference of basic scores of A and B. When there are no negative (positive) feedback interactions between connected individuals, we find that the success of opinion A depends on the ratio of the obtained positive (negative) feedback scores of competing opinions. To complete our study, we perform computer simulations on fully connected, small-world, and scale-free networks, respectively, which support and confirm our theoretical findings.

The evolution of indirect reciprocity under action and assessment generosity

Indirect reciprocity is a mechanism for the evolution of cooperation based on social norms. This mechanism requires that individuals in a population observe and judge each other's behaviors. Individuals with a good reputation are more likely to receive help from others. Previous work suggests that indirect reciprocity is only effective when all relevant information is reliable and publicly available. Otherwise, individuals may disagree on how to assess others, even if they all apply the same social norm. Such disagreements can lead to a breakdown of cooperation. Here we explore whether the predominantly studied 'leading eight' social norms of indirect reciprocity can be made more robust by equipping them with an element of generosity. To this end, we distinguish between two kinds of generosity. According to assessment generosity, individuals occasionally assign a good reputation to group members who would usually be regarded as bad. According to action generosity, individuals occasionally cooperate with group members with whom they would usually defect. Using individual-based simulations, we show that the two kinds of generosity have a very different effect on the resulting reputation dynamics. Assessment generosity tends to add to the overall noise and allows defectors to invade. In contrast, a limited amount of action generosity can be beneficial in a few cases. However, even when action generosity is beneficial, the respective simulations do not result in full cooperation. Our results suggest that while generosity can favor cooperation when individuals use the most simple strategies of reciprocity, it is disadvantageous when individuals use more complex social norms.

Evolution of cooperation in an epithelium

Cooperation is prevalent in nature, not only in the context of social interactions within the animal kingdom but also on the cellular level. In cancer, for example, tumour cells can cooperate by producing growth factors. The evolution of cooperation has traditionally been studied for well-mixed populations under the framework of evolutionary game theory, and more recently for structured populations using evolutionary graph theory (EGT). The population structures arising due to cellular arrangement in tissues, however, are dynamic and thus cannot be accurately represented by either of these frameworks. In this work, we compare the conditions for cooperative success in an epithelium modelled using EGT, to those in a mechanical model of an epithelium—the Voronoi tessellation (VT) model. Crucially, in this latter model, cells are able to move, and birth and death are not spatially coupled. We calculate fixation probabilities in the VT model through simulation and an approximate analytic technique and show that this leads to stronger promotion of cooperation in comparison with the EGT model.

Parent-preferred dispersal promotes cooperation in structured populations

Dispersal is a key process for the emergence of social and biological behaviours. Yet, little attention has been paid to dispersal's effects on the evolution of cooperative behaviour in structured populations. To address this issue, we propose two new dispersal modes, parent-preferred and offspring-preferred dispersal, incorporate them into the birth-death update rule, and consider the resultant strategy evolution in the prisoner's dilemma on random-regular, small-world, and scale-free networks, respectively. We find that parent-preferred dispersal favours the evolution of cooperation in these different types of population structures, while offspring-preferred dispersal inhibits the evolution of cooperation in homogeneous populations. On scale-free networks when the strength of parent-preferred dispersal is weak, cooperation can be enhanced at intermediate strengths of offspring-preferred dispersal, and cooperators can coexist with defectors at high strengths of offspring-preferred dispersal. Moreover, our theoretical analysis based on the pair-approximation method corroborates the evolutionary outcomes on random-regular networks. We also incorporate the two new dispersal modes into three other update rules (death-birth, imitation, and pairwise comparison updating), and find that similar results about the effects of parent-preferred and offspring-preferred dispersal can again be observed in the aforementioned different types of population structures. Our work, thus, unveils robust effects of preferential dispersal modes on the evolution of cooperation in different interactive environments.

Evolutionary game dynamics of combining the imitation and aspiration-driven update rules

So far, most studies on evolutionary game dynamics in finite populations have concentrated on a single update rule. However, given the impacts of the environment and individual cognition, individuals may use different update rules to change their current strategies. In light of this, the current paper reports on a study that constructed a mixed stochastic evolutionary game dynamic by combining the imitation and aspiration-driven update processes. The target was to clarify the influences of the aspiration-driven process on the evolution of the level of cooperation by considering the behavior of a population in which individuals have two strategies available: cooperation and defection. Through a numerical analysis of unstructured populations and simulation analyses of structured populations and of the random-matching model, the following results were found. First, the mean fraction of cooperators varied alongside the probability with which the individual adopted the aspiration-driven update rule. In the Prisoner's Dilemma and coexistence games, the aspiration-driven update process promoted cooperation in the well-mixed population but inhibited it in structured ones and the random-matching model; however, in the coordination game, the aspiration-driven update process was seen to exert the opposite effect on cooperation by inhibiting the latter in a homogeneously mixed population but promoting it in structured ones and in the random-matching model. Second, the mean fraction of cooperators changed with the aspiration level in the differently structured populations and random-matching model, and there appeared a phase transition point. Third, the evolutionary characteristics of the mean fraction of cooperators maintained robustness in the differently structured populations and random-matching model. These results extend evolutionary game theory.

Evolutionary game dynamics of the Wright-Fisher process with different selection intensities

Evolutionary game dynamics in finite populations can be described by a frequency-dependent, stochastic Wright-Fisher process. The fitness of individuals in a population is not only linked to environmental conditions but also tightly coupled to the types and frequencies of competitors, leading to different types of individuals with different selection intensities. We studied a 2 × 2 symmetric game in a finite population and established a dynamic model of the Wright-Fisher process by introducing different selection intensities for different strategies. Thus, we provided another effective way to study the evolutionary dynamics of a finite population and obtained the analytical expressions of fixation probabilities under weak selection. The fixation probability of a strategy is not only related to a game matrix but also to different selection intensities. The conditions required for natural selection to favor one strategy and for that strategy to be an evolutionary stable strategy (ESS_N) are specified in our model. We compared our results with those of a Moran dynamic process with different selection intensities to explore these two processes better. In the two processes, the conditions conducive to the strategy's taking fixation are the same. By simulation analysis, the dynamic relationships between the fixation probabilities and selection intensities were intuitively observed in the prisoner's dilemma, coordination, and coexistence games. The fixation probability of the cooperative strategy in the prisoner's dilemma decreases with the increase of its own selection intensity. In the coexistence and coordination games, the fixation probability of the cooperative strategy increases with its own selection intensity. For the three types of games, the fixation probability of the cooperative strategy decreases with the increase of the selection intensity of the defection strategy.

Replicator equation on networks with degree regular communities

The replicator equation is one of the fundamental tools to study evolutionary dynamics in well-mixed populations. This paper contributes to the literature on evolutionary graph theory, providing a version of the replicator equation for a family of connected networks with communities, where nodes in the same community have the same degree. This replicator equation is applied to the study of different classes of games, exploring the impact of the graph structure on the equilibria of the evolutionary dynamics.

Promotion of cooperation in evolutionary game dynamics with local information

In this paper, we propose a strategy-updating rule driven by local information, which is called Local process. Unlike the standard Moran process, the Local process does not require global information about the strategic environment. By analyzing the dynamical behavior of the system, we explore how the local information influences the fixation of cooperation in two-player evolutionary games. Under weak selection, the decreasing local information leads to an increase of the fixation probability when natural selection does not favor cooperation replacing defection. In the limit of sufficiently large selection, the analytical results indicate that the fixation probability increases with the decrease of the local information, irrespective of the evolutionary games. Furthermore, for the dominance of defection games under weak selection and for coexistence games, the decreasing of local information will lead to a speedup of a single cooperator taking over the population. Overall, to some extent, the local information is conducive to promoting the cooperation.

Competing opinion diffusion on social networks

Opinion competition is a common phenomenon in real life, such as with opinions on controversial issues or political candidates; however, modelling this competition remains largely unexplored. To bridge this gap, we propose a model of competing opinion diffusion on social networks taking into account degree-dependent fitness or persuasiveness. We study the combined influence of social networks, individual fitnesses and attributes, as well as mass media on people's opinions, and find that both social networks and mass media act as amplifiers in opinion diffusion, the amplifying effect of which can be quantitatively characterized. We analytically obtain the probability that each opinion will ultimately pervade the whole society when there are no committed people in networks, and the final proportion of each opinion at the steady state when there are committed people in networks. The results of numerical simulations show good agreement with those obtained through an analytical approach. This study provides insight into the collective influence of individual attributes, local social networks and global media on opinion diffusion, and contributes to a comprehensive understanding of competing diffusion behaviours in the real world.

Evolutionary games on cycles with strong selection

Evolutionary games on graphs describe how strategic interactions and population structure determine evolutionary success, quantified by the probability that a single mutant takes over a population. Graph structures, compared to the well-mixed case, can act as amplifiers or suppressors of selection by increasing or decreasing the fixation probability of a beneficial mutant. Properties of the associated mean fixation times can be more intricate, especially when selection is strong. The intuition is that fixation of a beneficial mutant happens fast in a dominance game, that fixation takes very long in a coexistence game, and that strong selection eliminates demographic noise. Here we show that these intuitions can be misleading in structured populations. We analyze mean fixation times on the cycle graph under strong frequency-dependent selection for two different microscopic evolutionary update rules (death-birth and birth-death). We establish exact analytical results for fixation times under strong selection and show that there are coexistence games in which fixation occurs in time polynomial in population size. Depending on the underlying game, we observe inherence of demographic noise even under strong selection if the process is driven by random death before selection for birth of an offspring (death-birth update). In contrast, if selection for an offspring occurs before random removal (birth-death update), then strong selection can remove demographic noise almost entirely.

Truncation selection and payoff distributions applied to the replicator equation

The replicator equation has been frequently used in the theoretical literature to explain a diverse array of biological phenomena. However, it makes several simplifying assumptions, namely complete mixing, an infinite population, asexual reproduction, proportional selection, and mean payoffs. Here, we relax the conditions of mean payoffs and proportional selection by incorporating payoff distributions and truncation selection into extensions of the replicator equation and agent-based models. In truncation selection, replicators with fitnesses above a threshold survive. The reproduction rate is equal for all survivors and is sufficient to replace the replicators that did not survive. We distinguish between two types of truncation: independent and dependent with respect to the fitness threshold. If the payoff variances from all strategy pairing are the same, then we recover the replicator equation from the independent truncation equation. However, if all payoff variances are not equal, then any boundary fixed point can be made stable (or unstable) if only the fitness threshold is chosen appropriately. We observed transient and complex dynamics in our models, which are not observed in replicator equations incorporating the same games. We conclude that the assumptions of mean payoffs and proportional selection in the replicator equation significantly impact replicator dynamics.

Speed of evolution on graphs

The likelihood that a mutant fixates in the wild population, i.e., fixation probability, has been intensively studied in evolutionary game theory, where individuals' fitness is frequency dependent. However, it is of limited interest when it takes long to take over. Thus the speed of evolution becomes an important issue. In general, it is still unclear how fixation times are affected by the population structure, although the fixation times have already been addressed in the well-mixed populations. Here we theoretically address this issue by pair approximation and diffusion approximation on regular graphs. It is shown (i) that under neutral selection, both unconditional and conditional fixation time are shortened by increasing the number of neighbors; (ii) that under weak selection, for the simplified prisoner's dilemma game, if benefit-to-cost ratio exceeds the degree of the graph, then the unconditional fixation time of a single cooperator is slower than that in the neutral case; and (iii) that under weak selection, for the conditional fixation time, limited neighbor size dilutes the counterintuitive stochastic slowdown which was found in well-mixed populations. Interestingly, we find that all of our results can be interpreted as that in the well-mixed population with a transformed payoff matrix. This interpretation is also valid for both death-birth and birth-death processes on graphs. This interpretation bridges the fixation time in the structured population and that in the well-mixed population. Thus it opens the avenue to investigate the challenging fixation time in structured populations by the known results in well-mixed populations.

Socioecological regime shifts in the setting of complex social interactions

The coupling between social and ecological system has become more ubiquitous and predominant in the current era. The strong interaction between these systems can bring about regime shifts which in the extreme can lead to the collapse of social cooperation and the extinction of ecological resources. In this paper, we study the occurrence of such regime shifts in the context of a coupled social-ecological system where social cooperation is established by means of sanction that punishes local selfish act and promotes norms that prescribe nonexcessive resource extraction. In particular, we investigate the role of social networks on social-ecological regimes shift and the corresponding hysteresis effects caused by the local ostracism mechanism under different social and ecological parameters. Our results show that a lowering of network degree reduces the hysteresis effect and also alters the tipping point, which is duly verified by our numerical results and analytical estimation. Interestingly, the hysteresis effect is found to be stronger in scale-free network in comparison with random network even when both networks have the same average degree. These results provide deeper insights into the resilience of these systems, and can have important implications on the management of coupled social-ecological systems with complex social interactions.

Impact of heterogeneous activity and community structure on the evolutionary success of cooperators in social networks

Recent empirical studies suggest that heavy-tailed distributions of human activities are universal in real social dynamics [L. Muchnik, S. Pei, L. C. Parra, S. D. S. Reis, J. S. Andrade Jr., S. Havlin, and H. A. Makse, Sci. Rep. 3, 1783 (2013)]. On the other hand, community structure is ubiquitous in biological and social networks [M. E. J. Newman, Nat. Phys. 8, 25 (2012)]. Motivated by these facts, we here consider the evolutionary prisoner's dilemma game taking place on top of a real social network to investigate how the community structure and the heterogeneity in activity of individuals affect the evolution of cooperation. In particular, we account for a variation of the birth-death process (which can also be regarded as a proportional imitation rule from a social point of view) for the strategy updating under both weak and strong selection (meaning the payoffs harvested from games contribute either slightly or heavily to the individuals' performance). By implementing comparative studies, where the players are selected either randomly or in terms of their actual activities to play games with their immediate neighbors, we figure out that heterogeneous activity benefits the emergence of collective cooperation in a harsh environment (the action for cooperation is costly) under strong selection, whereas it impairs the formation of altruism under weak selection. Moreover, we find that the abundance of communities in the social network can evidently foster the formation of cooperation under strong selection, in contrast to the games evolving on randomized counterparts. Our results are therefore helpful for us to better understand the evolution of cooperation in real social systems.

The evolutionary and ecological consequences of animal social networks: emerging issues

The first generation of research on animal social networks was primarily aimed at introducing the concept of social networks to the fields of animal behaviour and behavioural ecology. More recently, a diverse body of evidence has shown that social fine structure matters on a broader scale than initially expected, affecting many key ecological and evolutionary processes. Here, we review this development. We discuss the effects of social network structure on evolutionary dynamics (genetic drift, fixation probabilities, and frequency-dependent selection) and social evolution (cooperation and between-individual behavioural differences). We discuss how social network structure can affect important coevolutionary processes (host-pathogen interactions and mutualisms) and population stability. We also discuss the potentially important, but poorly studied, role of social network structure on dispersal and invasion. Throughout, we highlight important areas for future research.

Chemical evolutionary games

Inspired by the use of hybrid cellular automata in modeling cancer, we introduce a generalization of evolutionary games in which cells produce and absorb chemicals, and the chemical concentrations dictate the death rates of cells and their fitnesses. Our long term aim is to understand how the details of the interactions in a system with n species and m chemicals translate into the qualitative behavior of the system. Here, we study two simple 2×2 games with two chemicals and revisit the two and three species versions of the one chemical colicin system studied earlier by Durrett and Levin (1997). We find that in the 2×2 examples, the behavior of our new spatial model can be predicted from that of the mean field differential equation using ideas of Durrett and Levin (1994). However, in the three species colicin model, the system with diffusion does not have the coexistence which occurs in the lattices model in which sites interact with only their nearest neighbors.

Evolution of cooperation in a heterogeneous graph: fixation probabilities under weak selection

It has been shown that natural selection favors cooperation in a homogenous graph if the benefit-to-cost ratio exceeds the degree of the graph. However, most graphs related to interactions in real populations are heterogeneous, in which some individuals have many more neighbors than others. In this paper, we introduce a new state variable to measure the time evolution of cooperation in a heterogeneous graph. Based on the diffusion approximation, we find that the fixation probability of a single cooperator depends crucially on the number of its neighbors. Under weak selection, a cooperator with more neighbors has a larger probability of fixation in the population. We then investigate the average fixation probability of a randomly chosen cooperator. If a cooperator pays a cost for each of its neighbors (the so called fixed cost per game case), natural selection favors cooperation if the benefit-to-cost ratio is larger than the average degree. In contrast, if a cooperator pays a fixed cost and all its neighbors share the benefit (the fixed cost per individual case), cooperation is favored if the benefit-to-cost ratio is larger than the harmonic mean of the degree distribution. Moreover, increasing the graph heterogeneity will reduce the effect of natural selection.

Adaptive dynamics with interaction structure

Evolutionary dynamics depend critically on a population's interaction structure-the pattern of which individuals interact with which others, depending on the state of the population and the environment. Previous research has shown, for example, that cooperative behaviors disfavored in well-mixed populations can be favored when interactions occur only between spatial neighbors or group members. Combining the adaptive dynamics approach with recent advances in evolutionary game theory, we here introduce a general mathematical framework for analyzing the long-term evolution of continuous game strategies for a broad class of evolutionary models, encompassing many varieties of interaction structure. Our main result, the canonical equation of adaptive dynamics with interaction structure, characterizes expected evolutionary trajectories resulting from any such model, thereby generalizing a central tool of adaptive dynamics theory. Interestingly, the effects of different interaction structures and update rules on evolutionary trajectories are fully captured by just two real numbers associated with each model, which are independent of the considered game. The first, a structure coefficient, quantifies the effects on selection pressures and thus on the shapes of expected evolutionary trajectories. The second, an effective population size, quantifies the effects on selection responses and thus on the expected rates of adaptation. Applying our results to two social dilemmas, we show how the range of evolutionarily stable cooperative behaviors systematically varies with a model's structure coefficient.

Consolidating birth-death and death-birth processes in structured populations

Network models extend evolutionary game theory to settings with spatial or social structure and have provided key insights on the mechanisms underlying the evolution of cooperation. However, network models have also proven sensitive to seemingly small details of the model architecture. Here we investigate two popular biologically motivated models of evolution in finite populations: Death-Birth (DB) and Birth-Death (BD) processes. In both cases reproduction is proportional to fitness and death is random; the only difference is the order of the two events at each time step. Although superficially similar, under DB cooperation may be favoured in structured populations, while under BD it never is. This is especially troubling as natural populations do not follow a strict one birth then one death regimen (or vice versa); such constraints are introduced to make models more tractable. Whether structure can promote the evolution of cooperation should not hinge on a simplifying assumption. Here, we propose a mixed rule where in each time step DB is used with probability and BD is used with probability . We derive the conditions for selection favouring cooperation under the mixed rule for all social dilemmas. We find that the only qualitatively different outcome occurs when using just BD (). This case admits a natural interpretation in terms of kin competition counterbalancing the effect of kin selection. Finally we show that, for any mixed BD-DB update and under weak selection, cooperation is never inhibited by population structure for any social dilemma, including the Snowdrift Game.

Evolutionary shift dynamics on a cycle

We present a new model of evolutionary dynamics in one-dimensional space. Individuals are arranged on a cycle. When a new offspring is born, another individual dies and the rest shift around the cycle to make room. This rule, which is inspired by spatial evolution in somatic tissue and microbial colonies, has the remarkable property that, in the limit of large population size, evolution acts to maximize the payoff of the whole population. Therefore, social dilemmas, in which some individuals benefit at the expense of others, are resolved. We demonstrate this principle for both discrete and continuous games. We also discuss extensions of our model to other one-dimensional spatial configurations. We conclude that shift dynamics in one dimension is an unusually strong promoter of cooperative behavior.

Stochastic dynamics of the prisoner's dilemma with cooperation facilitators

In the framework of the paradigmatic prisoner's dilemma game, we investigate the evolutionary dynamics of social dilemmas in the presence of "cooperation facilitators." In our model, cooperators and defectors interact as in the classical prisoner's dilemma, where selection favors defection. However, here the presence of a small number of cooperation facilitators enhances the fitness (reproductive potential) of cooperators, while it does not alter that of defectors. In a finite population of size N, the dynamics of the prisoner's dilemma with facilitators is characterized by the probability that cooperation takes over (fixation probability) by the mean times to reach the absorbing states. These quantities are computed exactly using Fokker-Planck equations. Our findings, corroborated by stochastic simulations, demonstrate that the influence of facilitators crucially depends on the difference between their density z and the game's cost-to-benefit ratio r. When z > r, the fixation of cooperators is likely in a large population and, under weak selection pressure, invasion and replacement of defection by cooperation is favored by selection if b(z - r)(1 - z) > N(-1), where 0<b ≤ 1 is the cooperation payoff benefit. When z < r, the fixation probability of cooperators is exponentially enhanced by the presence of facilitators but defection is the dominating strategy.

How mutation affects evolutionary games on graphs

Evolutionary dynamics are affected by population structure, mutation rates and update rules. Spatial or network structure facilitates the clustering of strategies, which represents a mechanism for the evolution of cooperation. Mutation dilutes this effect. Here we analyze how mutation influences evolutionary clustering on graphs. We introduce new mathematical methods to evolutionary game theory, specifically the analysis of coalescing random walks via generating functions. These techniques allow us to derive exact identity-by-descent (IBD) probabilities, which characterize spatial assortment on lattices and Cayley trees. From these IBD probabilities we obtain exact conditions for the evolution of cooperation and other game strategies, showing the dual effects of graph topology and mutation rate. High mutation rates diminish the clustering of cooperators, hindering their evolutionary success. Our model can represent either genetic evolution with mutation, or social imitation processes with random strategy exploration.

Questioning the cultural evolution of altruism

The evolutionary foundations of helping among nonkin in humans have been the object of intense debates in the past decades. One thesis has had a prominent influence in this debate: the suggestion that genuine altruism, strictly defined as a form of help that comes at a net fitness cost for the benefactor, might have evolved owing to cultural transmission. The gene-culture coevolution literature is wont to claim that cultural evolution changes the selective pressures that normally act to limit the emergence of altruistic behaviours. This paper aims to recall, however, that cultural transmission yields altruism only to the extent that it relies on maladaptive mechanisms, such as conformist imitation and (in some cases) payoff-biased transmission. This point is sometimes obscured in the literature by a confusion between genuine altruism, maladaptive by definition, and mutualistic forms of cooperation, that benefit all parties in the long run. Theories of cultural altruism do not lift the selective pressures weighing on strictly altruistic actions; they merely shift the burden of maladaptation from social cognition to cultural transmission.

A review of evolutionary graph theory with applications to game theory

Evolutionary graph theory (EGT), studies the ability of a mutant gene to overtake a finite structured population. In this review, we describe the original framework for EGT and the major work that has followed it. This review looks at the calculation of the "fixation probability" - the probability of a mutant taking over a population and focuses on game-theoretic applications. We look at varying topics such as alternate evolutionary dynamics, time to fixation, special topological cases, and game theoretic results. Throughout the review, we examine several interesting open problems that warrant further research.

Simulating natural selection as a culling mechanism on finite populations with the hawk-dove game

The behaviors of individuals and species are often explained in terms of evolutionary stable strategies (ESSs). The analysis of ESSs determines which, if any, combinations of behaviors cannot be invaded by alternative strategies. Two assumptions required to generate an ESS (i.e., an infinite population and payoffs described only on the average) do not hold under natural conditions. Previous experiments indicated that under more realistic conditions of finite populations and stochastic payoffs, populations may evolve in trajectories that are unrelated to an ESS, even in very simple games. The simulations offered here extend earlier research by employing truncation selection with random parental selection in a hawk-dove game. Payoffs are determined in pairwise contests using either the expected outcome, or the result of a random variable. In each case, however, the mean fraction of hawks over many generations and across many independent trials does not conform to the expected ESS. Implications of these results and philosophical underpinnings of ESS theory are offered.

Local replicator dynamics: a simple link between deterministic and stochastic models of evolutionary game theory

Classical replicator dynamics assumes that individuals play their games and adopt new strategies on a global level: Each player interacts with a representative sample of the population and if a strategy yields a payoff above the average, then it is expected to spread. In this article, we connect evolutionary models for infinite and finite populations: While the population itself is infinite, interactions and reproduction occurs in random groups of size N. Surprisingly, the resulting dynamics simplifies to the traditional replicator system with a slightly modified payoff matrix. The qualitative results, however, mirror the findings for finite populations, in which strategies are selected according to a probabilistic Moran process. In particular, we derive a one-third law that holds for any population size. In this way, we show that the deterministic replicator equation in an infinite population can be used to study the Moran process in a finite population and vice versa. We apply the results to three examples to shed light on the evolution of cooperation in the iterated prisoner's dilemma, on risk aversion in coordination games and on the maintenance of dominated strategies.

Invasion and expansion of cooperators in lattice populations: prisoner's dilemma vs. snowdrift games

The evolution of cooperation is an enduring conundrum in biology and the social sciences. Two social dilemmas, the prisoner's dilemma and the snowdrift game have emerged as the most promising mathematical metaphors to study cooperation. Spatial structure with limited local interactions has long been identified as a potent promoter of cooperation in the prisoner's dilemma but in the spatial snowdrift game, space may actually enhance or inhibit cooperation. Here we investigate and link the microscopic interaction between individuals to the characteristics of the emerging macroscopic patterns generated by the spatial invasion process of cooperators in a world of defectors. In our simulations, individuals are located on a square lattice with Moore neighborhood and update their strategies by probabilistically imitating the strategies of better performing neighbors. Under sufficiently benign conditions, cooperators can survive in both games. After rapid local equilibration, cooperators expand quadratically until global saturation is reached. Under favorable conditions, cooperators expand as a large contiguous cluster in both games with minor differences concerning the shape of embedded defectors. Under less favorable conditions, however, distinct differences arise. In the prisoner's dilemma, cooperators break up into isolated, compact clusters. The compact clustering reduces exploitation and leads to positive assortment, such that cooperators interact more frequently with other cooperators than with defectors. In contrast, in the snowdrift game, cooperators form small, dendritic clusters, which results in negative assortment and cooperators interact more frequently with defectors than with other cooperators. In order to characterize and quantify the emerging spatial patterns, we introduce a measure for the cluster shape and demonstrate that the macroscopic patterns can be used to determine the characteristics of the underlying microscopic interactions.

A conceptual cellular interaction model of left ventricular remodelling post-MI: dynamic network with exit-entry competition strategy

Background

Progressive remodelling of the left ventricle (LV) following myocardial infarction (MI) is an outcome of spatial-temporal cellular interactions among different cell types that leads to heart failure for a significant number of patients. Cellular populations demonstrate temporal profiles of flux post-MI. However, little is known about the relationship between cell populations and the interaction strength among cells post-MI. The objective of this study was to establish a conceptual cellular interaction model based on a recently established graph network to describe the interaction between two types of cells.

Results

We performed stability analysis to investigate the effects of the interaction strengths, the initial status, and the number of links between cells on the cellular population in the dynamic network. Our analysis generated a set of conditions on interaction strength, structure of the network, and initial status of the network to predict the evolutionary profiles of the network. Computer simulations of our conceptual model verified our analysis.

Conclusions

Our study introduces a dynamic network to model cellular interactions between two different cell types which can be used to model the cellular population changes post-MI. The results on stability analysis can be used as a tool to predict the responses of particular cell populations.

Evolutionary dynamics in structured populations

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

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.

Social structure of primate interaction networks facilitates the emergence of cooperation

Animal cooperation has puzzled biologists for a long time as its existence seems to contravene the basic notion of evolutionary biology that natural selection favours 'selfish' genes that promote only their own well-being. Evolutionary game theory has shown that cooperators can prosper in populations of selfish individuals if they occur in clusters, interacting more frequently with each other than with the selfish. Here we show that social networks of primates possess the necessary social structure to promote the emergence of cooperation. By simulating evolutionary dynamics of cooperative behaviour on interaction networks of 70 primate groups, we found that for most groups network reciprocity augmented the fixation probability for cooperation. The variation in the strength of this effect can be partly explained by the groups' community modularity-a network measure for the groups' heterogeneity. Thus, given selective update and partner choice mechanisms, network reciprocity has the potential to explain socially learned forms of cooperation in primate societies.

Social network reciprocity as a phase transition in evolutionary cooperation

In evolutionary dynamics the understanding of cooperative phenomena in natural and social systems has been the subject of intense research during decades. We focus attention here on the so-called "lattice reciprocity" mechanisms that enhance evolutionary survival of the cooperative phenotype in the prisoner's dilemma game when the population of Darwinian replicators interact through a fixed network of social contacts. Exact results on a "dipole model" are presented, along with a mean-field analysis as well as results from extensive numerical Monte Carlo simulations. The theoretical framework used is that of standard statistical mechanics of macroscopic systems, but with no energy considerations. We illustrate the power of this perspective on social modeling, by consistently interpreting the onset of lattice reciprocity as a thermodynamical phase transition that, moreover, cannot be captured by a purely mean-field approach.

Consensus of population systems with community structures

Multicommunity population systems may reach a consensus state where the fractions of each species in different communities agree on a common value. In this paper, by analyzing the evolutionary dynamics based on an extended replicator equation incorporating community effects, the consensus problem of population systems with n communities is studied. In particular, the simple case of two communities is investigated in detail. In general, for n communities, a sufficient and necessary condition for population systems to reach a consensus of coexistent state is provided. Regarding the population dynamics for the four different types of games, whether the population systems can achieve consensus is determined. The dynamics of community-structured populations shows richer features than nonstructured populations, and some nontrivial phenomena arising from different community-structured population systems are illustrated with concrete numerical examples.

Effect of the presence of empty sites on the evolution of cooperation by costly punishment in spatial games

Cooperation and spiteful behavior are still evolutionary puzzles. Costly punishment, for which the game payoff is the same as that of spiteful behavior, is one mechanism for promoting the evolution of cooperation. A spatially structured population facilitates the evolution of either cooperation or spite/punishment if cooperation is linked explicitly or implicitly with spite/punishment; a cooperator cooperates with another cooperator and punishes/spites the other type of player. Different updating rules in the evolutionary game produce different evolutionary outcomes: with one updating rule-the score-dependent viability model, in which a player dies with a probability inversely proportional to the game score and the resulting unoccupied site is colonized by one player chosen randomly-the evolution of spite/punishment is promoted more than with the other updating rule-the score-dependent fertility model, in which, after a player dies randomly, the site is colonized by a player with a higher game score. If the population has empty sites, spiteful players or punishers should have less chance to interact with others and then spite/punish others. Thus the presence of empty sites would affect the evolutionary dynamics of spite/punishment. Here, we investigated whether the presence of empty sites discourages the evolution of spite/punishment in both a lattice-structured population and a completely mixing population where players interact with others randomly, especially when the score-dependent viability model is adopted. In the lattice-structured population adopting this viability model, the presence of empty sites promoted the evolution of cooperation and did not reduce the effect of spite/punishment. In the completely mixing population, the presence of empty sites did not promote evolution of cooperation by punishment. The evolutionary dynamics of the score-dependent viability model with empty sites were close to those of the score-dependent fertility model.

One-third rules with equality: Second-order evolutionary stability conditions in finite populations

The one-third law of evolutionary dynamics [Nowak et al. 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 246-650] describes a robustness criterion for evolution in a finite population: If at an A-frequency of 1/3, the fitness of an A player is greater (smaller) than the fitness of a B player, then a single A mutant that appears in a population of otherwise all B has a fixation probability greater (smaller) than the neutral threshold 1/N, the inverse population size. We examine the case where at an A-frequency of 1/3, the fitness of an A player is exactly equal to the fitness of a B player. We find that in this case the relative magnitude of the cross payoffs matters: If the payoff of A against B is larger (smaller) than the payoff of B against A, then a single A mutant has a fixation probability larger (smaller) than 1/N. If the cross payoffs coincide, we are in the special case of a partnership game, where the deviation cost from an inefficient equilibrium is exactly balanced by the potential gain of switching to the payoff dominant equilibrium. We show that in this case the fixation probability of A is lower than 1/N. Finally, we illustrate our findings by a language game with differentiated costs of signals.