Effects of neighbourhood size and connectivity on the spatial Continuous Prisoner's Dilemma

Sharp thresholds limit the benefit of defector avoidance in cooperation on networks

Consider a cooperation game on a spatial network of habitat patches, where players can relocate between patches if they judge the local conditions to be unfavorable. In time, the relocation events may lead to a homogeneous state where all patches harbor the same relative densities of cooperators and defectors, or they may lead to self-organized patterns, where some patches become safe havens that maintain an elevated cooperator density. Here we analyze the transition between these states mathematically. We show that safe havens form once a certain threshold in connectivity is crossed. This threshold can be analytically linked to the structure of the patch network and specifically to certain network motifs. Surprisingly, a forgiving defector avoidance strategy may be most favorable for cooperators. Our results demonstrate that the analysis of cooperation games in ecological metacommunity models is mathematically tractable and has the potential to link topics such as macroecological patterns, behavioral evolution, and network topology.

Evolution of Cooperation in Social Dilemmas with Assortative Interactions

Cooperation in social dilemmas plays a pivotal role in the formation of systems at all levels of complexity, from replicating molecules to multi-cellular organisms to human and animal societies. In spite of its ubiquity, the origin and stability of cooperation pose an evolutionary conundrum, since cooperation, though beneficial to others, is costly to the individual cooperator. Thus natural selection would be expected to favor selfish behavior in which individuals reap the benefits of cooperation without bearing the costs of cooperating themselves. Many proximate mechanisms have been proposed to account for the origin and maintenance of cooperation, including kin selection, direct reciprocity, indirect reciprocity, and evolution in structured populations. Despite the apparent diversity of these approaches they all share a unified underlying logic: namely, each mechanism results in assortative interactions in which individuals using the same strategy interact with a higher probability than they would at random. Here we study the evolution of cooperation in both discrete strategy and continuous strategy social dilemmas with assortative interactions. For the sake of tractability, assortativity is modeled by an individual interacting with another of the same type with probability r and interacting with a random individual in the population with probability 1−r, where r is a parameter that characterizes the degree of assortativity in the system. For discrete strategy social dilemmas we use both a generalization of replicator dynamics and individual-based simulations to elucidate the donation, snowdrift, and sculling games with assortative interactions, and determine the analogs of Hamilton’s rule, which govern the evolution of cooperation in these games. For continuous strategy social dilemmas we employ both a generalization of deterministic adaptive dynamics and individual-based simulations to study the donation, snowdrift, and tragedy of the commons games, and determine the effect of assortativity on the emergence and stability of cooperation.

Evolutionary game inspired by Cipolla's basic laws of human stupidity

In this work we present an evolutionary game inspired by the work of Carlo Cipolla entitled The Basic Laws of Human Stupidity. The game expands the classical scheme of two archetypical strategies, collaborators and defectors, by including two additional strategies. One of these strategies is associated with a stupid player that, according to Cipolla, is the most dangerous one as it undermines the global wealth of the population. By considering a spatial evolutionary game and imitation dynamics that go beyond the paradigm of a rational player we explore the impact of Cipolla's ideas and analyze the extent of the damage that stupid players inflict on the population.

Cooperation in a generalized age-structured spatial game

The emergence and prevalence of cooperative behavior within a group of selfish individuals remains a puzzle for evolutionary game theory precisely because it conflicts directly with the central idea of natural selection. Accordingly, in recent years, the search for an understanding of how cooperation can be stimulated, even when it conflicts with individual interest, has intensified. We investigate the emergence of cooperation in an age-structured evolutionary spatial game. In it, players age with time and the payoff that they receive after each round depends on their age. We find that the outcome of the game is strongly influenced by the type of distribution used to modify the payoffs according to the age of each player. The results show that, under certain circumstances, cooperators may not only survive but dominate the population.

Cooperation in Microbial Populations: Theory and Experimental Model Systems

Cooperative behavior, the costly provision of benefits to others, is common across all domains of life. This review article discusses cooperative behavior in the microbial world, mediated by the exchange of extracellular products called public goods. We focus on model species for which the production of a public good and the related growth disadvantage for the producing cells are well described. To unveil the biological and ecological factors promoting the emergence and stability of cooperative traits we take an interdisciplinary perspective and review insights gained from both mathematical models and well-controlled experimental model systems. Ecologically, we include crucial aspects of the microbial life cycle into our analysis and particularly consider population structures where ensembles of local communities (subpopulations) continuously emerge, grow, and disappear again. Biologically, we explicitly consider the synthesis and regulation of public good production. The discussion of the theoretical approaches includes general evolutionary concepts, population dynamics, and evolutionary game theory. As a specific but generic biological example, we consider populations of Pseudomonas putida and its regulation and use of pyoverdines, iron scavenging molecules, as public goods. The review closes with an overview on cooperation in spatially extended systems and also provides a critical assessment of the insights gained from the experimental and theoretical studies discussed. Current challenges and important new research opportunities are discussed, including the biochemical regulation of public goods, more realistic ecological scenarios resembling native environments, cell-to-cell signaling, and multispecies communities.

Neighborhood size-effects shape growing population dynamics in evolutionary public goods games

An evolutionary game emerges when a subset of individuals incur costs to provide benefits to all individuals. Public goods games (PGG) cover the essence of such dilemmas in which cooperators are prone to exploitation by defectors. We model the population dynamics of a non-linear PGG and consider density-dependence on the global level, while the game occurs within local neighborhoods. At low cooperation, increases in the public good provide increasing returns. At high cooperation, increases provide diminishing returns. This mechanism leads to diverse evolutionarily stable strategies, including monomorphic and polymorphic populations, and neighborhood-size-driven state changes, resulting in hysteresis between equilibria. Stochastic or strategy-dependent variations in neighborhood sizes favor coexistence by destabilizing monomorphic states. We integrate our model with experiments of cancer cell growth and confirm that our framework describes PGG dynamics observed in cellular populations. Our findings advance the understanding of how neighborhood-size effects in PGG shape the dynamics of growing populations.

Fixation Probabilities for Any Configuration of Two Strategies on Regular Graphs

Population structure and spatial heterogeneity are integral components of evolutionary dynamics, in general, and of evolution of cooperation, in particular. Structure can promote the emergence of cooperation in some populations and suppress it in others. Here, we provide results for weak selection to favor cooperation on regular graphs for any configuration, meaning any arrangement of cooperators and defectors. Our results extend previous work on fixation probabilities of rare mutants. We find that for any configuration cooperation is never favored for birth-death (BD) updating. In contrast, for death-birth (DB) updating, we derive a simple, computationally tractable formula for weak selection to favor cooperation when starting from any configuration containing any number of cooperators. This formula elucidates two important features: (i) the takeover of cooperation can be enhanced by the strategic placement of cooperators and (ii) adding more cooperators to a configuration can sometimes suppress the evolution of cooperation. These findings give a formal account for how selection acts on all transient states that appear in evolutionary trajectories. They also inform the strategic design of initial states in social networks to maximally promote cooperation. We also derive general results that characterize the interaction of any two strategies, not only cooperation and defection.

Evolutionary dynamics of a smoothed war of attrition game

In evolutionary game theory the War of Attrition game is intended to model animal contests which are decided by non-aggressive behavior, such as the length of time that a participant will persist in the contest. The classical War of Attrition game assumes that no errors are made in the implementation of an animal׳s strategy. However, it is inevitable in reality that such errors must sometimes occur. Here we introduce an extension of the classical War of Attrition game which includes the effect of errors in the implementation of an individual׳s strategy. This extension of the classical game has the important feature that the payoff is continuous, and as a consequence admits evolutionary behavior that is fundamentally different from that possible in the original game. We study the evolutionary dynamics of this new game in well-mixed populations both analytically using adaptive dynamics and through individual-based simulations, and show that there are a variety of possible outcomes, including simple monomorphic or dimorphic configurations which are evolutionarily stable and cannot occur in the classical War of Attrition game. In addition, we study the evolutionary dynamics of this extended game in a variety of spatially and socially structured populations, as represented by different complex network topologies, and show that similar outcomes can also occur in these situations.

Evolution of Cooperation in Social Dilemmas on Complex Networks

Cooperation in social dilemmas is essential for the functioning of systems at multiple levels of complexity, from the simplest biological organisms to the most sophisticated human societies. Cooperation, although widespread, is fundamentally challenging to explain evolutionarily, since natural selection typically favors selfish behavior which is not socially optimal. Here we study the evolution of cooperation in three exemplars of key social dilemmas, representing the prisoner's dilemma, hawk-dove and coordination classes of games, in structured populations defined by complex networks. Using individual-based simulations of the games on model and empirical networks, we give a detailed comparative study of the effects of the structural properties of a network, such as its average degree, variance in degree distribution, clustering coefficient, and assortativity coefficient, on the promotion of cooperative behavior in all three classes of games.

Universal scaling for the dilemma strength in evolutionary games

Why would natural selection favor the prevalence of cooperation within the groups of selfish individuals? A fruitful framework to address this question is evolutionary game theory, the essence of which is captured in the so-called social dilemmas. Such dilemmas have sparked the development of a variety of mathematical approaches to assess the conditions under which cooperation evolves. Furthermore, borrowing from statistical physics and network science, the research of the evolutionary game dynamics has been enriched with phenomena such as pattern formation, equilibrium selection, and self-organization. Numerous advances in understanding the evolution of cooperative behavior over the last few decades have recently been distilled into five reciprocity mechanisms: direct reciprocity, indirect reciprocity, kin selection, group selection, and network reciprocity. However, when social viscosity is introduced into a population via any of the reciprocity mechanisms, the existing scaling parameters for the dilemma strength do not yield a unique answer as to how the evolutionary dynamics should unfold. Motivated by this problem, we review the developments that led to the present state of affairs, highlight the accompanying pitfalls, and propose new universal scaling parameters for the dilemma strength. We prove universality by showing that the conditions for an ESS and the expressions for the internal equilibriums in an infinite, well-mixed population subjected to any of the five reciprocity mechanisms depend only on the new scaling parameters. A similar result is shown to hold for the fixation probability of the different strategies in a finite, well-mixed population. Furthermore, by means of numerical simulations, the same scaling parameters are shown to be effective even if the evolution of cooperation is considered on the spatial networks (with the exception of highly heterogeneous setups). We close the discussion by suggesting promising directions for future research including (i) how to handle the dilemma strength in the context of co-evolution and (ii) where to seek opportunities for applying the game theoretical approach with meaningful impact.

The expansion of neighborhood and pattern formation on spatial prisoner's dilemma

The prisoner's dilemma (PD), in which players can either cooperate or defect, is considered a paradigm for studying the evolution of cooperation in spatially structured populations. There the compact cooperator cluster is identified as a characteristic pattern and the probability of forming such pattern in turn depends on the features of the networks. In this paper, we investigate the influence of expansion of neighborhood on pattern formation by taking a weak PD game with one free parameter T, the temptation to defect. Two different expansion methods of neighborhood are considered. One is based on a square lattice and expanses along four directions generating networks with degree increasing with K=4m. The other is based on a lattice with Moore neighborhood and expanses along eight directions, generating networks with degree of K=8m. Individuals are placed on the nodes of the networks, interact with their neighbors and learn from the better one. We find that cooperator can survive for a broad degree 4≤K≤70 by taking a loose type of cooperator clusters. The former simple corresponding relationship between macroscopic patterns and the microscopic PD interactions is broken. Under a condition that is unfavorable for cooperators such as large T and K, systems prefer to evolve to a loose type of cooperator clusters to support cooperation. However, compared to the well-known compact pattern, it is a suboptimal strategy because it cannot help cooperators dominating the population and always corresponding to a low cooperation level.

A structured population model suggests that long life and post-reproductive lifespan promote the evolution of cooperation

Social organization correlates with longevity across animal taxa. This correlation has been explained by selection for longevity by social evolution. The reverse causality is also conceivable but has not been sufficiently considered. We constructed a simple, spatially structured population model of asexually reproducing individuals to study the effect of temporal life history structuring on the evolution of cooperation. Individuals employed fixed strategies of cooperation or defection towards all neighbours in a basic Prisoner's Dilemma paradigm. Individuals aged and transitioned through different life history stages asynchronously without migration. An individual's death triggered a reproductive event by one immediate neighbour. The specific neighbour was chosen probabilistically according to the cumulative payoff from all local interactions. Varying the duration of pre-reproductive, reproductive, and post-reproductive life history stages, long-term simulations allowed a systematic evaluation of the influence of the duration of these specific life history stages. Our results revealed complex interactions among the effects of the three basic life history stages and the benefit to defect. Overall, a long post-reproductive stage promoted the evolution of cooperation, while a prolonged pre-reproductive stage has a negative effect. In general, the total length of life also increased the probability of the evolution of cooperation. Thus, our specific model suggests that the timing of life history transitions and total duration of life history stages may affect the evolution of cooperative behaviour. We conclude that the causation of the empirically observed association of life expectancy and sociality may be more complex than previously realized.

Evolutionary dynamics of the traveler's dilemma and minimum-effort coordination games on complex networks

The traveler's dilemma game and the minimum-effort coordination game are social dilemmas that have received significant attention resulting from the fact that the predictions of classical game theory are inconsistent with the results found when the games are studied experimentally. Moreover, both the traveler's dilemma and the minimum-effort coordination games have potentially important applications in evolutionary biology. Interestingly, standard deterministic evolutionary game theory, as represented by the replicator dynamics in a well-mixed population, is also inadequate to account for the behavior observed in these games. Here we study the evolutionary dynamics of both these games in populations with interaction patterns described by a variety of complex network topologies. We investigate the evolutionary dynamics of these games through agent-based simulations on both model and empirical networks. In particular, we study the effects of network clustering and assortativity on the evolutionary dynamics of both games. In general, we show that the evolutionary behavior of the traveler's dilemma and minimum-effort coordination games on complex networks is in good agreement with that observed experimentally. Thus, formulating the traveler's dilemma and the minimum-effort coordination games on complex networks neatly resolves the paradoxical aspects of these games.

Stable heterogeneity for the production of diffusible factors in cell populations

The production of diffusible molecules that promote survival and growth is common in bacterial and eukaryotic cell populations, and can be considered a form of cooperation between cells. While evolutionary game theory shows that producers and non-producers can coexist in well-mixed populations, there is no consensus on the possibility of a stable polymorphism in spatially structured populations where the effect of the diffusible molecule extends beyond one-step neighbours. I study the dynamics of biological public goods using an evolutionary game on a lattice, taking into account two assumptions that have not been considered simultaneously in existing models: that the benefit of the diffusible molecule is a non-linear function of its concentration, and that the molecule diffuses according to a decreasing gradient. Stable coexistence of producers and non-producers is observed when the benefit of the molecule is a sigmoid function of its concentration, while strictly diminishing returns lead to coexistence only for very specific parameters and linear benefits never lead to coexistence. The shape of the diffusion gradient is largely irrelevant and can be approximated by a step function. Since the effect of a biological molecule is generally a sigmoid function of its concentration (as described by the Hill equation), linear benefits or strictly diminishing returns are not an appropriate approximations for the study of biological public goods. A stable polymorphism of producers and non-producers is in line with the predictions of evolutionary game theory and likely to be common in cell populations.

Diffusive public goods and coexistence of cooperators and cheaters on a 1D lattice

Many populations of cells cooperate through the production of extracellular materials. These materials (enzymes, siderophores) spread by diffusion and can be applied by both the cooperator and cheater (non-producer) cells. In this paper the problem of coexistence of cooperator and cheater cells is studied on a 1D lattice where cooperator cells produce a diffusive material which is beneficial to the individuals according to the local concentration of this public good. The reproduction success of a cell increases linearly with the benefit in the first model version and increases non-linearly (saturates) in the second version. Two types of update rules are considered; either the cooperative cell stops producing material before death (death-production-birth, DpB) or it produces the common material before it is selected to die (production-death-birth, pDB). The empty space is occupied by its neighbors according to their replication rates. By using analytical and numerical methods I have shown that coexistence of the cooperator and cheater cells is possible although atypical in the linear version of this 1D model if either DpB or pDB update rule is assumed. While coexistence is impossible in the non-linear model with pDB update rule, it is one of the typical behaviors in case of the non-linear model with DpB update rule.

Stochastic evolution of staying together

Staying together means that replicating units do not separate after reproduction, but remain attached to each other or in close proximity. Staying together is a driving force for evolution of complexity, including the evolution of multi-cellularity and eusociality. We analyze the fixation probability of a mutant that has the ability to stay together. We assume that the size of the complex affects the reproductive rate of its units and the probability of staying together. We examine the combined effect of natural selection and random drift on the emergence of staying together in a finite sized population. The number of states in the underlying stochastic process is an exponential function of population size. We develop a framework for any intensity of selection and give closed form solutions for special cases. We derive general results for the limit of weak selection.

Evolution of staying together in the context of diffusible public goods

We study the coevolution of staying together and cooperation. Staying together means that replicating units do not separate after reproduction, but remain in proximity. For example, following cell division the two daughter cells may not fully separate but stay attached to each other. Repeated cell division thereby can lead to a simple multi-cellular complex. We assume that cooperators generate a diffusible public good, which can be absorbed by any cell in the system. The production of the public good entails a cost, while the absorption leads to a benefit. Defectors produce no public good. Defectors have a selective advantage unless a mechanism for evolution of cooperation is at work. Here we explore the idea that the public good produced by a cooperating cell is absorbed by cells of the same complex with a probability that depends on the size of the complex. Larger complexes are better at absorbing the public goods produced by their own individuals. We derive analytical conditions for the evolution of staying together, thereby studying the coevolution of clustering and cooperation. If cooperators and defectors differ in their intrinsic efficiency to absorb the public good, then we find multiple stable equilibria and the possibility for coexistence between cooperators and defectors. Finally we study the implications of disadvantages that might arise if complexes become too large.

An application of evolutionary game theory to social dilemmas: the traveler's dilemma and the minimum effort coordination game

The Traveler's Dilemma game and the Minimum Effort Coordination game are two social dilemmas that have attracted considerable attention due to the fact that the predictions of classical game theory are at odds with the results found when the games are studied experimentally. Moreover, a direct application of deterministic evolutionary game theory, as embodied in the replicator dynamics, to these games does not explain the observed behavior. In this work, we formulate natural variants of these two games as smoothed continuous-strategy games. We study the evolutionary dynamics of these continuous-strategy games, both analytically and through agent-based simulations, and show that the behavior predicted theoretically is in accord with that observed experimentally. Thus, these variants of the Traveler's Dilemma and the Minimum Effort Coordination games provide a simple resolution of the paradoxical behavior associated with the original games.

Cooperation in networks where the learning environment differs from the interaction environment

We study the evolution of cooperation in a structured population, combining insights from evolutionary game theory and the study of interaction networks. In earlier studies it has been shown that cooperation is difficult to achieve in homogeneous networks, but that cooperation can get established relatively easily when individuals differ largely concerning the number of their interaction partners, such as in scale-free networks. Most of these studies do, however, assume that individuals change their behaviour in response to information they receive on the payoffs of their interaction partners. In real-world situations, subjects do not only learn from their interaction partners, but also from other individuals (e.g. teachers, parents, or friends). Here we investigate the implications of such incongruences between the ‘interaction network’ and the ‘learning network’ for the evolution of cooperation in two paradigm examples, the Prisoner's Dilemma game (PDG) and the Snowdrift game (SDG). Individual-based simulations and an analysis based on pair approximation both reveal that cooperation will be severely inhibited if the learning network is very different from the interaction network. If the two networks overlap, however, cooperation can get established even in case of considerable incongruence between the networks. The simulations confirm that cooperation gets established much more easily if the interaction network is scale-free rather than random-regular. The structure of the learning network has a similar but much weaker effect. Overall we conclude that the distinction between interaction and learning networks deserves more attention since incongruences between these networks can strongly affect both the course and outcome of the evolution of cooperation.

Dynamics of growth factor production in monolayers of cancer cells and evolution of resistance to anticancer therapies

Tumor heterogeneity is well documented for many characters, including the production of growth factors, which improve tumor proliferation and promote resistance against apoptosis and against immune reaction. What maintains heterogeneity remains an open question that has implications for diagnosis and treatment. While it has been suggested that therapies targeting growth factors are robust against evolved resistance, current therapies against growth factors, like antiangiogenic drugs, are not effective in the long term, as resistant mutants can evolve and lead to relapse. We use evolutionary game theory to study the dynamics of the production of growth factors by monolayers of cancer cells and to understand the effect of therapies that target growth factors. The dynamics depend on the production cost of the growth factor, on its diffusion range and on the type of benefit it confers to the cells. Stable heterogeneity is a typical outcome of the dynamics, while a pure equilibrium of nonproducer cells is possible under certain conditions. Such pure equilibrium can be the goal of new anticancer therapies. We show that current therapies, instead, can be effective only if growth factors are almost completely eliminated and if the reduction is almost immediate.

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.

Coexistence of mutualists and non-mutualists in a dual-lattice model

Evolution and maintenance of mutualism have been one of the major questions in evolutionary ecology, because it is often susceptible of invasion of non-mutualistic strategy. Some previous studies using dual-lattice model suggest that spatial structures of habitat can prevent non-mutualism from prevailing over mutualism, while the detail of the dynamics is not fully revealed. Here we explore population dynamics of the two strategies (mutualism and non-mutualism) in two species engaged in Prisoner's Dilemma game on a dual-lattice space, especially focusing on whether mutualists and non-mutualists can coexist in long-term dynamics. The habitat consists of two layers, each of which a population of species inhabits, and interspecific interaction is restricted between two corresponding sites of the layers. Each individual of the both species is either a mutualist or a non-mutualist and only the former pay cost c for benefit of the partner b. The payoff of the game affects the individuals' fecundity, while the mortality is constant. Reproduction is restricted to neighboring vacant sites of the focal individuals. Our computer simulations of the model show that even if b/c ratio remains constant, mutualists become dominant in both species over wider ranges of basic reproduction rate (reproduction rate without interspecific interaction) as b and c increase. If basic reproduction rates are asymmetric between the species or basic reproduction rates were sufficiently large, mutualists and non-mutualists can coexist in one or both species, while their population sizes often fluctuate. Transition of the final state between mutualism and non-mutualism happens rather discontinuously, then total population sizes change drastically at the transition. Moreover, we also find paradoxical cases of unilateral exploitation, i.e. one species consists of mutualists and other species non-mutualists. Additional simulations reveal that accidental extinction of the non-mutualists of one species can result in extinction of mutualist of the other species.

Mobility, fitness collection, and the breakdown of cooperation

The spatial arrangement of individuals is thought to overcome the dilemma of cooperation: When cooperators engage in clusters, they might share the benefit of cooperation while being more protected against noncooperating individuals, who benefit from cooperation but save the cost of cooperation. This is paradigmatically shown by the spatial prisoner's dilemma model. Here, we study this model in one and two spatial dimensions, but explicitly take into account that in biological setups, fitness collection and selection are separated processes occurring mostly on vastly different time scales. This separation is particularly important to understand the impact of mobility on the evolution of cooperation. We find that even small diffusive mobility strongly restricts cooperation since it enables noncooperative individuals to invade cooperative clusters. Thus, in most biological scenarios, where the mobility of competing individuals is an irrefutable fact, the spatial prisoner's dilemma alone cannot explain stable cooperation, but additional mechanisms are necessary for spatial structure to promote the evolution of cooperation. The breakdown of cooperation is analyzed in detail. We confirm the existence of a phase transition, here controlled by mobility and costs, which distinguishes between purely cooperative and noncooperative absorbing states. While in one dimension the model is in the class of the voter model, it belongs to the directed percolation universality class in two dimensions.

Comparing neutral and trade-off community models in shaping the community biomass-diversity relationship under different disturbance levels

Among numerous mechanisms shaping the unimodal relationship between diversity and community biomass, the trade-off model of "CRS" theory is the most famous one. However, recent researches indicate that this relationship may also emerge under the neutral model where all species are identical with each other. By using an individual-based spatially-explicit model, we evaluated the underlying mechanisms shaping this curve for both models under different disturbance levels. We found unimodal relationships emerged for both models at low and medium disturbance levels; the richness for the trade-off community was lower than the neutral community for most of the environment severity levels, especially at the benign environment due to the strong competitive exclusions among species. Whereas under high disturbance level, the positive relationships emerged for both models; both communities had similar richness with their curves nearly overlapped with each other, that is, because the high disturbance intensity strongly decreased the competitive exclusions within the trade-off community. Our results indicate that although the underlying mechanisms are totally different, both models will produce the similar relationship between diversity and community biomass under different disturbance levels.

Relationship between clustering coefficient and the success of cooperation in networks

In recent years the prisoner's dilemma has become a paradigm for the study of the emergence of cooperation in spatially structured populations. Such a structure is usually assumed to be given by a graph. In general, the success of cooperative strategies is associated with the possibility of forming globular clusters, which in turn depends on a feature of the network that is measured by its clustering coefficient. In this work we study the dependence of the success of cooperation on this coefficient for regular networks. Additionally, for both stochastic and deterministic dynamics we show that there is a strong dependence on the initial composition of the population. This hints at the existence of several different mechanisms that could promote or hinder cluster expansion. We have studied in detail some of these mechanisms by concentrating on completely ordered networks (large clustering coefficient) or completely random networks (vanishing clustering coefficient).

Punishment leads to cooperative behavior in structured societies

Altruistic punishment occurs when an agent incurs a cost to punish another but receives no material benefit for doing so. Despite the seeming irrationality of such behavior, humans in laboratory settings routinely pay to punish others even in anonymous, one-shot settings. Costly punishment is ubiquitous among social organisms in general and is increasingly accepted as a mechanism for the evolution of cooperation. Yet if it is true that punishment explains cooperation, the evolution of altruistic punishment remains a mystery. In a series of computer simulations I give agents the ability to punish one another while playing a continuous prisoner's dilemma. In simulations without social structure, expected behavior evolves-agents do not punish and consequently no cooperation evolves. Likewise, in simulations with social structure but no ability to punish, no cooperation evolves. However, in simulations where agents are both embedded in a social structure and have the option to inflict costly punishment, cooperation evolves quite readily. This suggests a simple and broadly applicable explanation of cooperation for social organisms that have nonrandom social structure and a predisposition to punish one another. Results with scale-free networks further suggest that nodal degree distribution plays an important role in determining whether cooperation will evolve in a structured population.

Two measures of effective population size for graphs

Effective population size is a key parameter in population ecology because it allows prediction of the dynamics of genetic variation and the rate of genetic drift and inbreeding. It is important for the definition of "nearly neutral" mutations and, hence, has consequences for the fixation or extinction probabilities of advantageous and deleterious mutations. As graph-based population models become increasingly popular for studying evolution in spatially or socially structured populations, a neutral theory for evolution on graphs is called for. Here, we derive formulae for two alternative measures of effective population size, the variance effective and inbreeding effective size of general unweighted and undirected graphs. We show how these two quantities relate to each other and we derive effective sizes for the complete graph the cycle and bipartite graphs. For one-dimensional lattices and small-world graphs, we estimate the inbreeding effective size using simulations. The presented method is suitable for any structured population of haploid individuals with overlapping generations.

How is the equilibrium of continuous strategy game different from that of discrete strategy game?

Cooperation in the prisoner's dilemma (PD) played on various networks has been explained by so-called network reciprocity. Most of the previous studies presumed that players can offer either cooperation (C) or defection (D). This discrete strategy seems unrealistic in the real world, since actual provisions might not be discrete, but rather continuous. This paper studies the differences between continuous and discrete strategies in two aspects under the condition that the payoff function of the former is a linear interpolation of the payoff matrix of the latter. The first part of this paper proves theoretically that for two-player games, continuous and discrete strategies have different equilibria and game dynamics in a well-mixed but finite population. The second part, conducting a series of numerical experiments, reveals that such differences become considerably large in the case of PD games on networks. Furthermore, it shows, using the Wilcoxon sign-rank test, that continuous and discrete strategy games are statistically significantly different in terms of equilibria. Intensive discussion by comparing these two kinds of games elucidates that describing a strategy as a real number blunts D strategy invasion to C clusters on a network in the early stage of evolution. Thus, network reciprocity is enhanced by the continuous strategy.

Evolution of altruistic punishment in heterogeneous populations

Evolutionary models for altruistic behavior typically make the assumption of homogeneity: each individual has the same costs and benefits associated with cooperating with each other and punishing for selfish behavior. In this paper, we relax this assumption by separating the population into heterogeneous classes, such that individuals from different classes differ in their ability to punish for selfishness. We compare the effects of introducing heterogeneity this way across two population models, that each represents a different type of population: the infinite and well-mixed population describes the way workers of social insects such as ants are organized, while a spatially structured population is more related to the way social norms evolve and are maintained in a social network. We find that heterogeneity in the effectiveness of punishment by itself has little to no effect on whether or not altruistic behavior will stabilize in a population. In contrast, heterogeneity in the cost that individuals pay to punish for selfish behavior allows altruistic behavior to be maintained more easily. Fewer punishers are needed to deter selfish behavior, and the individuals that punish will mostly belong to the class that pays a lower cost to do so. This effect is amplified when individuals that pay a lower cost for punishing inflict a higher punishment. The two population models differ when individuals that pay a low cost for punishing also inflict a lower punishment. In this situation, altruistic behavior becomes harder to maintain in an infinite and well-mixed population. However, this effect does not occur when the population is spatially structured.

Co-evolution of behaviour and social network structure promotes human cooperation

The ubiquity of cooperation in nature is puzzling because cooperators can be exploited by defectors. Recent theoretical work shows that if dynamic networks define interactions between individuals, cooperation is favoured by natural selection. To address this, we compare cooperative behaviour in multiple but independent repeated games between participants in static and dynamic networks. In the latter, participants could break their links after each social interaction. As predicted, we find higher levels of cooperation in dynamic networks. Through biased link breaking (i.e. to defectors) participants affected their social environment. We show that this link-breaking behaviour leads to substantial network clustering and we find primarily cooperators within these clusters. This assortment is remarkable because it occurred on top of behavioural assortment through direct reciprocity and beyond the perception of participants, and represents a self-organized pattern. Our results highlight the importance of the interaction between ecological context and selective pressures on cooperation.

Parental investment without kin recognition: Simple conditional rules for parent-offspring behavior

Species differ widely with regard to parental investment strategies and mechanisms underlying those strategies. The passing of benefits to likely offspring can be instantiated with a number of different computational and behavioral systems. We report results from an agent-based model in which offspring maintain proximity with parents and parents transmit benefits to offspring without the capacity of either parent or offspring to 'recognize' one another. Instead, parents follow a simple rule to emit benefits after reproducing and offspring follow a simple rule of moving in the direction of positive benefit gradients. This model differs from previous models of spatial kin-based altruism in that individuals are modeled as having different behavioral rules at different life stages and benefits are transmitted unidirectionally from parents to offspring. High rates of correctly directed parental investment occur when mobility and sociality are low and parental investment occurs over a short period of time. We suggest that strategies based on recognition and bonding/attachment might serve to increase rates of correctly directed parental investment under parameters that are shown here to otherwise lead to high rates of misdirected and wasted parental investment.

Cooperation enhanced by the 'survival of the fittest' rule in prisoner's dilemma games on complex networks

Prevalence of cooperation within groups of selfish individuals is puzzling in that it contradicts with the basic premise of natural selection, whereby we introduce a model of strategy evolution taking place on evolving networks based on Darwinian 'survival of the fittest' rule. In the present work, players whose payoffs are below a certain threshold will be deleted and the same number of new nodes will be added to the network to maintain the constant system size. Furthermore, the networking effect is also studied via implementing simulations on four typical network structures. Numerical results show that cooperators can obtain the biggest boost if the elimination threshold is fine-tuned. Notably, this coevolutionary rule drives the initial networks to evolve into statistically stationary states with a broad-scale degree distribution. Our results may provide many more insights for understanding the coevolution of strategy and network topology under the mechanism of nature selection whereby superior individuals will prosper and inferior ones be eliminated.

Pavlovian Prisoner's Dilemma-Analytical results, the quasi-regular phase and spatio-temporal patterns

The Prisoner's Dilemma (PD) game is applied in several research fields due to the emergence of cooperation among selfish players. In this work the PD is studied in a one-dimensional lattice, where each cell represents a player, which in turn can interact with the neighbors playing the PD (cooperate or defect). The update of states adopts the Pavlovian Evolutionary Strategy (PES) or Darwinian Evolutionary Strategy (DES). Adopting PES, if a player receives a positive payoff greater than his/her aspiration level, he/she keeps the current state, and switches otherwise. Adopting DES, player compares his/her payoff with payoff of opponents. If it is not the highest, he/she copies the state of fittest player, switching the state if it is different of his/her current state. The critical temptation values obtained analytically are reported, and the cluster patterns that emerge from the interactions among the players are shown. Also we defined analytical functions that calculate the maximum/minimum size of defective/cooperative clusters. Also, the parameter space is explored with exhaustive computational simulations, which confirm the analytical results and reinforce that Pavlovian strategy foments cooperation among players. In steady state, system can reach the cooperative or quasi-regular phases, when adopting the PES, and cooperative, defective or chaotic phases, adopting the DES. The new quasi-regular phase occurs when several players switch their states in each round, but the proportion of cooperators does not show significant variation. Additionally, the present work shows that the lowest temptation level (T=1) may be considered a trivial case only for the particular case where the players interact with only one neighbor, otherwise system presents the same features that for higher temptation values.

Effect of spatial structure on the evolution of cooperation

Spatial structure is known to have an impact on the evolution of cooperation, and so it has been intensively studied during recent years. Previous work has shown the relevance of some features, such as the synchronicity of the updating, the clustering of the network, or the influence of the update rule. This has been done, however, for concrete settings with particular games, networks, and update rules, with the consequence that some contradictions have arisen and a general understanding of these topics is missing in the broader context of the space of 2x2 games. To address this issue, we have performed a systematic and exhaustive simulation in the different degrees of freedom of the problem. In some cases, we generalize previous knowledge to the broader context of our study and explain the apparent contradictions. In other cases, however, our conclusions refute what seems to be established opinions in the field, as for example the robustness of the effect of spatial structure against changes in the update rule, or offer new insights into the subject, e.g., the relation between the intensity of selection and the asymmetry between the effects on games with mixed equilibria.

Beyond dual-lattice models: incorporating plant strategies when modeling the interplay between facilitation and competition along environmental severity gradients

We introduce a spatially explicit model that evaluates how the trade-offs between the life strategies of two interacting plant species affect the outcome of their interaction along environmental severity gradients. In our model, we represent the landscape as a two-dimensional lattice, with environmental severity increasing from left to right. Two species with different strategies, a competitor and a stress-tolerant, interact in the lattice. We find that facilitation expands the realized niche of the competitor into harsh environments by suppressing the stress-tolerant species. Most of their coexisting range is dominated by a positive effect of one species on another, with a reciprocal negative effect from the species receiving the benefits on its benefactor ("+, -"), whereas mutualistic ("+, +") interactions are only found in the harshest part of the environmental gradient. Contrarily as assumed by models commonly used in facilitation research (e.g. dual-lattice models), our results indicate that "+, +" interactions are not dominant, and that their differences with "+, -" interactions along environmental severity gradients depend on the strategies of the interacting species. By integrating the trade-off between competitive ability and stress tolerance, our model provides a new framework to investigate the interplay of facilitative and competitive interactions along environmental gradients and their impacts on processes such as population dynamics and community organization.