Coevolutionary dynamics in large, but finite populations

How social network structure impacts the ability of zealots to promote weak opinions

Social networks are often permeated by agents who promote their opinions without allowing for their own mind to be changed. Understanding how these so-called "zealots" act to increase the prevalence of their promoted opinion over the network is important for understanding opinion dynamics. In this work, we consider these promoted opinions to be "weak" and therefore less likely to be accepted relative to the default opinion in the network. We show how the proportion of zealots in the network, the relative strength of the weak opinion, and the structure of the network impact the long-term proportion of those in the network who subscribe to the weak opinion.

Coevolutionary game dynamics with localized environmental resource feedback

Dynamic environments shape diverse dynamics in evolutionary game systems. We introduce spatial heterogeneity of resources into the prisoner's dilemma game model to explore coevolutionary game dynamics with environmental feedback. The availability of resources significantly affects the survival competitiveness of surrounding individuals. Feedback between individuals' strategies and the resources they can use leads to the oscillating dynamic known as the "oscillatory tragedy of the commons." Our findings indicate that when the influence of individuals' strategies on the update rate of resources is significantly high in systems characterized by environmental heterogeneity, they can attain an equilibrium state that avoids the oscillatory tragedy. In contrast to the numerical results obtained in well-mixed structures, self-organized clustered patterns emerge in simulations utilizing square lattices, further enhancing the stability of the system. We discuss critical phenomena in detail, demonstrating that the aforementioned transition is robust across various system parameters, including the strength of cooperators in restoring the environment, initial distributions of cooperators, system size and structures, and noise.

Frequency-dependent returns in nonlinear public goods games

When individuals interact in groups, the evolution of cooperation is traditionally modelled using the framework of public goods games. These models often assume that the return of the public goods depends linearly on the fraction of contributors. In contrast, in real-life public goods interactions, the return can depend on the size of the investor pool as well. Here, we consider a model in which the multiplication factor (marginal per capita return) for the public goods depends linearly on how many contribute, which results in a nonlinear model of public goods. This simple model breaks the curse of dominant defection found in linear public goods interactions and gives rise to richer dynamical outcomes in evolutionary settings. We provide an in-depth analysis of the more varied decisions by the classical rational player in nonlinear public goods interactions as well as a mechanistic, microscopic derivation of the evolutionary outcomes for the stochastic dynamics in finite populations and in the deterministic limit of infinite populations. This kind of nonlinearity provides a natural way to model public goods with diminishing returns as well as economies of scale.

Explosive Cooperation in Social Dilemmas on Higher-Order Networks

Understanding how cooperative behaviors can emerge from competitive interactions is an open problem in biology and social sciences. While interactions are usually modeled as pairwise networks, the units of many real-world systems can also interact in groups of three or more. Here, we introduce a general framework to extend pairwise games to higher-order networks. By studying social dilemmas on hypergraphs with a tunable structure, we find an explosive transition to cooperation triggered by a critical number of higher-order games. The associated bistable regime implies that an initial critical mass of cooperators is also required for the emergence of prosocial behavior. Our results show that higher-order interactions provide a novel explanation for the survival of cooperation.

Frequency-dependent ecological interactions increase the prevalence, and shape the distribution, of pre-existing drug resistance

The evolution of resistance remains one of the primary challenges for modern medicine from infectious diseases to cancers. Many of these resistance-conferring mutations often carry a substantial fitness cost in the absence of treatment. As a result, we would expect these mutants to undergo purifying selection and be rapidly driven to extinction. Nevertheless, pre-existing resistance is frequently observed from drug-resistant malaria to targeted cancer therapies in non-small cell lung cancer (NSCLC) and melanoma. Solutions to this apparent paradox have taken several forms from spatial rescue to simple mutation supply arguments. Recently, in an evolved resistant NSCLC cell line, we found that frequency-dependent ecological interactions between ancestor and resistant mutant ameliorate the cost of resistance in the absence of treatment. Here, we hypothesize that frequency-dependent ecological interactions in general play a major role in the prevalence of pre-existing resistance. We combine numerical simulations with robust analytical approximations to provide a rigorous mathematical framework for studying the effects of frequency-dependent ecological interactions on the evolutionary dynamics of pre-existing resistance. First, we find that ecological interactions significantly expand the parameter regime under which we expect to observe pre-existing resistance. Next, even when positive ecological interactions between mutants and ancestors are rare, these resistant clones provide the primary mode of evolved resistance because even weak positive interaction leads to significantly longer extinction times. We then find that even in the case where mutation supply alone is sufficient to predict pre-existing resistance, frequency-dependent ecological forces still contribute a strong evolutionary pressure that selects for increasingly positive ecological effects (negative frequency-dependent selection). Finally, we genetically engineer several of the most common clinically observed resistance mechanisms to targeted therapies in NSCLC, a treatment notorious for pre-existing resistance. We find that each engineered mutant displays a positive ecological interaction with their ancestor. As a whole, these results suggest that frequency-dependent ecological effects can play a crucial role in shaping the evolutionary dynamics of pre-existing resistance.

Replicator-mutator dynamics of the rock-paper-scissors game: Learning through mistakes

We generalize the Bush-Mosteller learning, the Roth-Erev learning, and the social learning to include mistakes, such that the nonlinear replicator-mutator equation with either additive or multiplicative mutation is generated in an asymptotic limit. Subsequently, we exhaustively investigate the ubiquitous rock-paper-scissors game for some analytically tractable motifs of mutation pattern for which the replicator-mutator flow is seen to exhibit rich dynamics that include limit cycles and chaotic orbits. The main result of this paper is that in both symmetric and asymmetric game interactions, mistakes can sometimes help the players learn; in fact, mistakes can even control chaos to lead to rational Nash-equilibrium outcomes. Furthermore, we report a hitherto-unknown Hamiltonian structure of the replicator-mutator equation.

A geometric process of evolutionary game dynamics

Many evolutionary processes occur in phenotype spaces which are continuous. It is therefore of interest to explore how selection operates in continuous spaces. One approach is adaptive dynamics, which assumes that mutants are local. Here we study a different process which also allows non-local mutants. We assume that a resident population is challenged by an invader who uses a strategy chosen from a random distribution on the space of all strategies. We study the repeated donation game of direct reciprocity. We consider reactive strategies given by two probabilities, denoting respectively the probability to cooperate after the co-player has cooperated or defected. The strategy space is the unit square. We derive analytic formulae for the stationary distribution of evolutionary dynamics and for the average cooperation rate as function of the cost-to-benefit ratio. For positive reactive strategies, we prove that cooperation is more abundant than defection if the area of the cooperative region is greater than 1/2 which is equivalent to benefit, b, divided by cost, c, exceeding [Formula: see text]. We introduce the concept of strategies that are stable with probability one. We also study an extended process and discuss other games.

Stochastic competitive release and adaptive chemotherapy

We develop a finite-cell model of tumor natural selection dynamics to investigate the stochastic fluctuations associated with multiple rounds of adaptive chemotherapy. The adaptive cycles are designed to avoid chemoresistance in the tumor by managing the ecological mechanism of competitive release of a resistant subpopulation. Our model is based on a three-component evolutionary game played among healthy (H), sensitive (S), and resistant (R) populations of N cells, with a chemotherapy control parameter, C(t), which we use to dynamically impose selection pressure on the sensitive subpopulation to slow tumor growth and manage competitive release of the resistant population. The adaptive chemoschedule is designed based on the deterministic (N→∞) adjusted replicator dynamical system, then implemented using the finite-cell stochastic frequency dependent Moran process model (N=10K-50K) to ascertain the cumulative effect of the stochastic fluctuations on the efficacy of the adaptive schedules over multiple rounds. We quantify the stochastic fixation probability regions of the R and S populations in the HSR trilinear phase plane as a function of the control parameter C∈[0,1], showing that the size of the R region increases with increasing C. We then implement an adaptive time-dependent schedule C(t) for the stochastic model and quantify the variances (using principal component coordinates) associated with the evolutionary cycles over multiple rounds of adaptive therapy. The variances increase subquadratically through several rounds before the evolutionary cycle begins to break down. Despite this, we show the stochastic adaptive schedules are more effective at delaying resistance than standard maximum tolerated dose and low-dose metronomic schedules. The simplified low-dimensional model provides some insights on how well multiple rounds of adaptive therapies are likely to perform over a range of tumor sizes (i.e., different values of N) if the goal is to maintain a sustained balance among competing subpopulations of cells to avoid chemoresistance via competitive release in a stochastic environment.

A Variational Synthesis of Evolutionary and Developmental Dynamics

This paper introduces a variational formulation of natural selection, paying special attention to the nature of 'things' and the way that different 'kinds' of 'things' are individuated from-and influence-each other. We use the Bayesian mechanics of particular partitions to understand how slow phylogenetic processes constrain-and are constrained by-fast, phenotypic processes. The main result is a formulation of adaptive fitness as a path integral of phenotypic fitness. Paths of least action, at the phenotypic and phylogenetic scales, can then be read as inference and learning processes, respectively. In this view, a phenotype actively infers the state of its econiche under a generative model, whose parameters are learned via natural (Bayesian model) selection. The ensuing variational synthesis features some unexpected aspects. Perhaps the most notable is that it is not possible to describe or model a population of conspecifics per se. Rather, it is necessary to consider populations of distinct natural kinds that influence each other. This paper is limited to a description of the mathematical apparatus and accompanying ideas. Subsequent work will use these methods for simulations and numerical analyses-and identify points of contact with related mathematical formulations of evolution.

Weak selection and stochastic evolutionary stability in a stochastic replicator dynamics

It is a very challenging problem whether natural selection is able to effectively resist the continuous disturbance of environmental noise such that the direction or outcome of evolution determined by the deterministic selection pressure will not be changed. By analyzing the impact of weak selection on the evolutionary stability of a stochastic replicator dynamics with n possible pure strategies, we found that the weak selection is able to enhance the evolutionary stability, that is, under weak selection, the stochastic evolutionary stability of the system is determined by the mean payoff matrix. This finding strongly implies that the weak selection should be regarded as an important mechanism to ensure evolutionary stability in stochastic environments.

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

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

Evolutionary game dynamics with non-uniform interaction rates in finite population

In this study, we extend evolutionary game dynamics with non-uniform interaction rates to the situation with finite population. Our main goal is to show how the fixation probability is influenced by the non-uniform interaction rates under weak selection. Based on the diffusion approximation of the Moran process and assumption of weak selection, the stochastic dynamic properties of a two-phenotype game with non-uniform interaction rates in a finite population are investigated. By the analysis of some cases, we show that the non-uniform interaction rates may result in the potential evolutionary complexity of game dynamics in finite population.

Evolutionary Game Model of Group Choice Dilemmas on Hypergraphs

We introduce an evolutionary game on hypergraphs in which decisions between a risky alternative and a safe one are taken in social groups of different sizes. The model naturally reproduces choice shifts, namely the differences between the preference of individual decision makers and the consensual choice of a group, that have been empirically observed in choice dilemmas. In particular, a deviation from the Nash equilibrium toward the risky strategy occurs when the dynamics takes place on heterogeneous hypergraphs. These results can explain the emergence of irrational herding and radical behaviors in social groups.

Interindividual cooperation mediated by partisanship complicates Madison's cure for "mischiefs of faction"

Political theorists have long argued that enlarging the political sphere to include a greater diversity of interests would cure the ills of factions in a pluralistic society. While the scope of politics has expanded dramatically over the past 75 y, polarization is markedly worse. Motivated by this paradox, we take a bottom-up approach to explore how partisan individual-level dynamics in a diverse (multidimensional) issue space can shape collective-level factionalization via an emergent dimensionality reduction. We extend a model of cultural evolution grounded in evolutionary game theory, in which individuals accumulate benefits through pairwise interactions and imitate (or learn) the strategies of successful others. The degree of partisanship determines the likelihood of learning from individuals of the opposite party. This approach captures the coupling between individual behavior, partisan-mediated opinion dynamics, and an interaction network that changes endogenously according to the evolving interests of individuals. We find that while expanding the diversity of interests can indeed improve both individual and collective outcomes, increasingly high partisan bias promotes a reduction in issue dimensionality via party-based assortment that leads to increasing polarization. When party bias becomes extreme, it also boosts interindividual cooperation, thereby further entrenching extreme polarization and creating a tug-of-war between individual cooperation and societal cohesion. These dangers of extreme partisanship are highest when individuals' interests and opinions are heavily shaped by peers and there is little independent exploration. Overall, our findings highlight the urgency to study polarization in a coupled, multilevel context.

Impact of the baseline payoff on evolutionary outcomes

Do individuals enjoying a higher baseline payoff behave similarly in competitive scenarios compared to their counterparts? The classical replicator equation does not answer such a question since it is invariant to the background or baseline payoff of individuals. In reality, however, if one's baseline payoff is higher than the possible payoffs of an interaction (or game), the individual may respond generously or indifferently if s(he) is satisfied with the prevailing benchmark payoff. This work intends to explore such a phenomenon within the realm of pairwise interactions-taking the prisoner's dilemma as a metaphor-in well-mixed finite and infinite populations. In this framework, a player uses the payoff (comprising baseline and game payoffs) -expectation difference to estimate a degree of eagerness and, with that degree of eagerness, revises his or her strategy with a certain probability. We adopt two approaches to explore such a context, naming them as the Fermi and imitation processes, in which the former uses a pairwise Femi function and the latter considers the relative fitness to estimate probabilities for strategy revision. In a finite population, we examine the effect of intensities to payoff-expectation and strategic payoff differences (denoted by k_{1} and k_{2}, respectively) as well as the level of contentment (ω) on the fixation probability and fixation time (for a single defector). We observe that the fixation probability surges with the increase of intensity parameters. Nevertheless, the maximum fixation probability may require a substantially larger time to fixate, especially when the expectation is lower than the baseline payoff. This means that cooperators can persist for a longer period of time. A higher expectation or greed, however, considerably reduces the fixation time. Interestingly, our numerical simulation reveals that both approaches are equivalent under weak k_{2}(≪1) in the Fermi process. We further derive mean-field equations for both approaches in the context of an infinite population, where we observe two possible evolutionary consequences: either full-scale defection or the persistence of the initial frequency of cooperators. The latter scenario indicates players' uninterested or neutral behavior in relation to the interaction due to their satisfaction on the baseline payoff.

Hamilton's rule and kin competition in a finite kin population

Kin selection means that individuals can increase their own inclusive fitness through displaying more altruistically toward their relatives. So, Hamilton's rule says kin selection will work if the coefficient of relatedness exceeds the cost-to-benefit ratio of the altruistic act. However, some studies have shown that the kin competition due to the altruism among relatives can reduce, and even totally negate, the kin-selected benefits of altruism toward relatives. In order to understand how the evolution of cooperation is influenced by both kin selection and kin competition under a general theoretical framework, we here consider the evolutionary dynamics of cooperation in a finite kin population, where kin competition is incorporated into a simple Prisoner's Dilemma game between relatives. Differently from the previous studies, we emphasize that the difference between the effects of mutually and unilaterally altruistic acts on kin competition may play an important role for the evolution of cooperation. The main results not only show the conditions that Hamilton's rule still works under the kin competition but also reveal the evolutionary biological mechanism driving the evolution of cooperation in a finite kin population.

A common gene drive language eases regulatory process and eco-evolutionary extensions

Background

Synthetic gene drive technologies aim to spread transgenic constructs into wild populations even when they impose organismal fitness disadvantages. The extraordinary diversity of plausible drive mechanisms and the range of selective parameters they may encounter makes it very difficult to convey their relative predicted properties, particularly where multiple approaches are combined. The sheer number of published manuscripts in this field, experimental and theoretical, the numerous techniques resulting in an explosion in the gene drive vocabulary hinder the regulators’ point of view. We address this concern by defining a simplified parameter based language of synthetic drives.

Results

Employing the classical population dynamics approach, we show that different drive construct (replacement) mechanisms can be condensed and evaluated on an equal footing even where they incorporate multiple replacement drives approaches. Using a common language, it is then possible to compare various model properties, a task desired by regulators and policymakers. The generalization allows us to extend the study of the invasion dynamics of replacement drives analytically and, in a spatial setting, the resilience of the released drive constructs. The derived framework is available as a standalone tool.

Conclusion

Besides comparing available drive constructs, our tool is also useful for educational purpose. Users can also explore the evolutionary dynamics of future hypothetical combination drive scenarios. Thus, our results appraise the properties and robustness of drives and provide an intuitive and objective way for risk assessment, informing policies, and enhancing public engagement with proposed and future gene drive approaches.

Are Adaptive Chemotherapy Schedules Robust? A Three-Strategy Stochastic Evolutionary Game Theory Model

We investigate the robustness of adaptive chemotherapy schedules over repeated cycles and a wide range of tumor sizes. Using a non-stationary stochastic three-component fitness-dependent Moran process model (to track frequencies), we quantify the variance of the response to treatment associated with multidrug adaptive schedules that are designed to mitigate chemotherapeutic resistance in an idealized (well-mixed) setting. The finite cell (N tumor cells) stochastic process consists of populations of chemosensitive cells, chemoresistant cells to drug 1, and chemoresistant cells to drug 2, and the drug interactions can be synergistic, additive, or antagonistic. Tumor growth rates in this model are proportional to the average fitness of the tumor as measured by the three populations of cancer cells compared to a background microenvironment average value. An adaptive chemoschedule is determined by using the N→∞ limit of the finite-cell process (i.e., the adjusted replicator equations) which is constructed by finding closed treatment response loops (which we call evolutionary cycles) in the three component phase-space. The schedules that give rise to these cycles are designed to manage chemoresistance by avoiding competitive release of the resistant cell populations. To address the question of how these cycles perform in practice over large patient populations with tumors across a range of sizes, we consider the variances associated with the approximate stochastic cycles for finite N, repeating the idealized adaptive schedule over multiple periods. For finite cell populations, the distributions remain approximately multi-Gaussian in the principal component coordinates through the first three cycles, with variances increasing exponentially with each cycle. As the number of cycles increases, the multi-Gaussian nature of the distribution breaks down due to the fact that one of the three sub-populations typically saturates the tumor (competitive release) resulting in treatment failure. This suggests that to design an effective and repeatable adaptive chemoschedule in practice will require a highly accurate tumor model and accurate measurements of the sub-population frequencies or the errors will quickly (exponentially) degrade its effectiveness, particularly when the drug interactions are synergistic. Possible ways to extend the efficacy of the stochastic cycles in light of the computational simulations are discussed.

Maximizing cooperation in the prisoner's dilemma evolutionary game via optimal control

The prisoner's dilemma (PD) game offers a simple paradigm of competition between two players who can either cooperate or defect. Since defection is a strict Nash equilibrium, it is an asymptotically stable state of the replicator dynamical system that uses the PD payoff matrix to define the fitness landscape of two interacting evolving populations. The dilemma arises from the fact that the average payoff of this asymptotically stable state is suboptimal. Coaxing the players to cooperate would result in a higher payoff for both. Here we develop an optimal control theory for the prisoner's dilemma evolutionary game in order to maximize cooperation (minimize the defector population) over a given cycle time T, subject to constraints. Our two time-dependent controllers are applied to the off-diagonal elements of the payoff matrix in a bang-bang sequence that dynamically changes the game being played by dynamically adjusting the payoffs, with optimal timing that depends on the initial population distributions. Over multiple cycles nT (n>1), the method is adaptive as it uses the defector population at the end of the nth cycle to calculate the optimal schedule over the n+1st cycle. The control method, based on Pontryagin's maximum principle, can be viewed as determining the optimal way to dynamically alter incentives and penalties in order to maximize the probability of cooperation in settings that track dynamic changes in the frequency of strategists, with potential applications in evolutionary biology, economics, theoretical ecology, social sciences, reinforcement learning, and other fields where the replicator system is used.

Quasi-stationary states of game-driven systems: A dynamical approach

Evolutionary game theory is a framework to formalize the evolution of collectives ("populations") of competing agents that are playing a game and, after every round, update their strategies to maximize individual payoffs. There are two complementary approaches to modeling evolution of player populations. The first addresses essentially finite populations by implementing the apparatus of Markov chains. The second assumes that the populations are infinite and operates with a system of mean-field deterministic differential equations. By using a model of two antagonistic populations, which are playing a game with stationary or periodically varying payoffs, we demonstrate that it exhibits metastable dynamics that is reducible neither to an immediate transition to a fixation (extinction of all but one strategy in a finite-size population) nor to the mean-field picture. In the case of stationary payoffs, this dynamics can be captured with a system of stochastic differential equations and interpreted as a stochastic Hopf bifurcation. In the case of varying payoffs, the metastable dynamics is much more complex than the dynamics of the means.

Adaptive learning in large populations

We consider the adaptive learning rule of Harley (J Theor Biol 89:611-633, 1981) for behavior selection in symmetric conflict games in large populations. This rule uses organisms' past, accumulated rewards as the predictor for future behavior, and can be traced in many life forms from bacteria to humans. We derive a partial differential equation for the distribution of agents in the space of stimuli to select a particular strategy which describes the evolution of learning in heterogeneous populations. We analyze the solutions of the PDE model for symmetric [Formula: see text] games. It is found that in games with small residual stimuli, adaptive learning rules with larger memory factor converge faster to the optimal outcome.

Computation and Simulation of Evolutionary Game Dynamics in Finite Populations

The study of evolutionary dynamics increasingly relies on computational methods, as more and more cases outside the range of analytical tractability are explored. The computational methods for simulation and numerical approximation of the relevant quantities are diverging without being compared for accuracy and performance. We thoroughly investigate these algorithms in order to propose a reliable standard. For expositional clarity we focus on symmetric 2 × 2 games leading to one-dimensional processes, noting that extensions can be straightforward and lessons will often carry over to more complex cases. We provide time-complexity analysis and systematically compare three families of methods to compute fixation probabilities, fixation times and long-term stationary distributions for the popular Moran process. We provide efficient implementations that substantially improve wall times over naive or immediate implementations. Implications are also discussed for the Wright-Fisher process, as well as structured populations and multiple types.

Delays in Fitness Adjustment Can Lead to Coexistence of Hierarchically Interacting Species

Organisms that exploit different environments may experience a stochastic delay in adjusting their fitness when they switch habitats. We study two such organisms whose fitness is determined by the species composition of the local environment, as they interact through a public good. We show that a delay in the fitness adjustment can lead to the coexistence of the two species in a metapopulation, although the faster-growing species always wins in well-mixed competition experiments. Coexistence is favored over wide parameter ranges and is independent of spatial clustering. It arises when species are heterogeneous in their fitness and can keep each other balanced.

Demographic noise and cost of greenbeard can facilitate greenbeard cooperation

Cooperation among organisms, where cooperators suffer a personal cost to benefit others, is ubiquitous in nature. Greenbeard is a key mechanism for the evolution of cooperation, where a single gene or a set of linked genes codes for both cooperation and a phenotypic tag (metaphorically called "green beard"). Greenbeard cooperation is typically thought to decline over time since defectors can also evolve the tag. However, models of tag-based cooperation typically ignore two key realistic features: populations are finite, and that phenotypic tags can be costly. We develop an analytical model for coevolutionary dynamics of two evolvable traits in finite populations with mutations: costly cooperation and a costly tag. We show that an interplay of demographic noise and cost of the tag can induce coevolutionary cycling, where the evolving population does not reach a steady state but spontaneously switches between cooperative tag-carrying and noncooperative tagless states. Such dynamics allows the tag to repeatedly reappear even after it is invaded by defectors. Thus, we highlight the surprising possibility that the cost of the tag, together with demographic noise, can facilitate the evolution of greenbeard cooperation. We discuss implications of these findings in the context of the evolution of quorum sensing and multicellularity.

Multiple scales in metapopulations of public goods producers

Multiple scales in metapopulations can give rise to paradoxical behavior: in a conceptual model for a public goods game, the species associated with a fitness cost due to the public good production can be stabilized in the well-mixed limit due to the mere existence of these scales. The scales in this model involve a length scale corresponding to separate patches, coupled by mobility, and separate time scales for reproduction and interaction with a local environment. Contrary to the well-mixed high mobility limit, we find that for low mobilities, the interaction rate progressively stabilizes this species due to stochastic effects, and that the formation of spatial patterns is not crucial for this stabilization.

Unfair and Anomalous Evolutionary Dynamics from Fluctuating Payoffs

Evolution occurs in populations of reproducing individuals. Reproduction depends on the payoff a strategy receives. The payoff depends on the environment that may change over time, on intrinsic uncertainties, and on other sources of randomness. These temporal variations in the payoffs can affect which traits evolve. Understanding evolutionary game dynamics that are affected by varying payoffs remains difficult. Here we study the impact of arbitrary amplitudes and covariances of temporally varying payoffs on the dynamics. The evolutionary dynamics may be "unfair," meaning that, on average, two coexisting strategies may persistently receive different payoffs. This mechanism can induce an anomalous coexistence of cooperators and defectors in the prisoner's dilemma, and an unexpected selection reversal in the hawk-dove game.

Intermediate Levels of Network Heterogeneity Provide the Best Evolutionary Outcomes

Complex networks impact the diffusion of ideas and innovations, the formation of opinions, and the evolution of cooperative behavior. In this context, heterogeneous structures have been shown to generate a coordination-like dynamics that drives a population towards a monomorphic state. In contrast, homogeneous networks tend to result in a stable co-existence of multiple traits in the population. These conclusions have been reached through the analysis of networks with either very high or very low levels of degree heterogeneity. In this paper, we use methods from Evolutionary Game Theory to explore how different levels of degree heterogeneity impact the fate of cooperation in structured populations whose individuals face the Prisoner’s Dilemma. Our results suggest that in large networks a minimum level of heterogeneity is necessary for a society to become evolutionary viable. Moreover, there is an optimal range of heterogeneity levels that maximize the resilience of the society facing an increasing number of social dilemmas. Finally, as the level of degree heterogeneity increases, the evolutionary dominance of either cooperators or defectors in a society increasingly depends on the initial state of a few influential individuals. Our findings imply that neither very unequal nor very equal societies offer the best evolutionary outcome.

Basins of distinct asymptotic states in the cyclically competing mobile five species game

We study the dynamics of cyclic competing mobile five species on spatially extended systems originated from asymmetric initial populations and investigate the basins for the three possible asymptotic states, coexistence of all species, existences of only two independent species, and the extinction. Through extensive numerical simulations, we find a prosperous dependence on initial conditions for species biodiversity. In particular, for fixed given equal densities of two relevant species, we find that only five basins for the existence of two independent species exist and they are spirally entangled for high mobility. A basin of coexistence is outbreaking when the mobility parameter is decreased through a critical value and surrounded by the other five basins. For fixed given equal densities of two independent species, however, we find that basin structures are not spirally entangled. Further, final states of two independent species are totally different. For all possible considerations, the extinction state is not witnessed which is verified by the survival probability. To provide the validity of basin structures from lattice simulations, we analyze the system in mean-field manners. Consequently, results on macroscopic levels are matched to direct lattice simulations for high mobility regimes. These findings provide a good insight into the fundamental issue of the biodiversity among many species than previous cases.

Extinction rates in tumour public goods games

Cancer evolution and progression are shaped by cellular interactions and Darwinian selection. Evolutionary game theory incorporates both of these principles, and has been proposed as a framework to understand tumour cell population dynamics. A cornerstone of evolutionary dynamics is the replicator equation, which describes changes in the relative abundance of different cell types, and is able to predict evolutionary equilibria. Typically, the replicator equation focuses on differences in relative fitness. We here show that this framework might not be sufficient under all circumstances, as it neglects important aspects of population growth. Standard replicator dynamics might miss critical differences in the time it takes to reach an equilibrium, as this time also depends on cellular turnover in growing but bounded populations. As the system reaches a stable manifold, the time to reach equilibrium depends on cellular death and birth rates. These rates shape the time scales, in particular, in coevolutionary dynamics of growth factor producers and free-riders. Replicator dynamics might be an appropriate framework only when birth and death rates are of similar magnitude. Otherwise, population growth effects cannot be neglected when predicting the time to reach an equilibrium, and cell-type-specific rates have to be accounted for explicitly.

A Goal-Directed Bayesian Framework for Categorization

Categorization is a fundamental ability for efficient behavioral control. It allows organisms to remember the correct responses to categorical cues and not for every stimulus encountered (hence eluding computational cost or complexity), and to generalize appropriate responses to novel stimuli dependant on category assignment. Assuming the brain performs Bayesian inference, based on a generative model of the external world and future goals, we propose a computational model of categorization in which important properties emerge. These properties comprise the ability to infer latent causes of sensory experience, a hierarchical organization of latent causes, and an explicit inclusion of context and action representations. Crucially, these aspects derive from considering the environmental statistics that are relevant to achieve goals, and from the fundamental Bayesian principle that any generative model should be preferred over alternative models based on an accuracy-complexity trade-off. Our account is a step toward elucidating computational principles of categorization and its role within the Bayesian brain hypothesis.

A simple rule of direct reciprocity leads to the stable coexistence of cooperation and defection in the Prisoner's Dilemma game

The long-term coexistence of cooperation and defection is a common phenomenon in nature and human society. However, none of the theoretical models based on the Prisoner's Dilemma (PD) game can provide a concise theoretical model to explain what leads to the stable coexistence of cooperation and defection in the long-term even though some rules for promoting cooperation have been summarized (Nowak, 2006, Science 314, 1560-1563). Here, based on the concept of direct reciprocity, we develop an elementary model to show why stable coexistence of cooperation and defection in the PD game is possible. The basic idea behind our theoretical model is that all players in a PD game prefer a cooperator as an opponent, and our results show that considering strategies allowing opting out against defection provide a general and concise way of understanding the fundamental importance of direct reciprocity in driving the evolution of cooperation.

Replicator dynamics with diffusion on multiplex networks

In this study we present an extension of the dynamics of diffusion in multiplex graphs, which makes the equations compatible with the replicator equation with mutations. We derive an exact formula for the diffusion term, which shows that, while diffusion is linear for numbers of agents, it is necessary to account for nonlinear terms when working with fractions of individuals. We also derive the transition probabilities that give rise to such macroscopic behavior, completing the bottom-up description. Finally, it is shown that the usual assumption of constant population sizes induces a hidden selective pressure due to the diffusive dynamics, which favors the increase of fast diffusing strategies.

Bribe and Punishment: An Evolutionary Game-Theoretic Analysis of Bribery

Harassment bribes, paid by citizens to corrupt officers for services the former are legally entitled to, constitute one of the most widespread forms of corruption in many countries. Nation states have adopted different policies to address this form of corruption. While some countries make both the bribe giver and the bribe taker equally liable for the crime, others impose a larger penalty on corrupt officers. We examine the consequences of asymmetric and symmetric penalties by developing deterministic and stochastic evolutionary game-theoretic models of bribery. We find that the asymmetric penalty scheme can lead to a reduction in incidents of bribery. However, the extent of reduction depends on how the players update their strategies over time. If the interacting members change their strategies with a probability proportional to the payoff of the alternative strategy option, the reduction in incidents of bribery is less pronounced. Our results indicate that changing from a symmetric to an asymmetric penalty scheme may not suffice in achieving significant reductions in incidents of harassment bribery.

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.

Inherent randomness of evolving populations

The entropy rates of the Wright-Fisher process, the Moran process, and generalizations are computed and used to compare these processes and their dependence on standard evolutionary parameters. Entropy rates are measures of the variation dependent on both short-run and long-run behaviors and allow the relationships between mutation, selection, and population size to be examined. Bounds for the entropy rate are given for the Moran process (independent of population size) and for the Wright-Fisher process (bounded for fixed population size). A generational Moran process is also presented for comparison to the Wright-Fisher Process. Results include analytic results and computational extensions.

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.

Persistent coexistence of cyclically competing species in spatially extended ecosystems

A fundamental result in the evolutionary-game paradigm of cyclic competition in spatially extended ecological systems, as represented by the classic Reichenbach-Mobilia-Frey (RMF) model, is that high mobility tends to hamper or even exclude species coexistence. This result was obtained under the hypothesis that individuals move randomly without taking into account the suitability of their local environment. We incorporate local habitat suitability into the RMF model and investigate its effect on coexistence. In particular, we hypothesize the use of "basic instinct" of an individual to determine its movement at any time step. That is, an individual is more likely to move when the local habitat becomes hostile and is no longer favorable for survival and growth. We show that, when such local habitat suitability is taken into account, robust coexistence can emerge even in the high-mobility regime where extinction is certain in the RMF model. A surprising finding is that coexistence is accompanied by the occurrence of substantial empty space in the system. Reexamination of the RMF model confirms the necessity and the important role of empty space in coexistence. Our study implies that adaptation/movements according to local habitat suitability are a fundamental factor to promote species coexistence and, consequently, biodiversity.

Frequency-dependent fitness induces multistability in coevolutionary dynamics

Evolution is simultaneously driven by a number of processes such as mutation, competition and random sampling. Understanding which of these processes is dominating the collective evolutionary dynamics in dependence on system properties is a fundamental aim of theoretical research. Recent works quantitatively studied coevolutionary dynamics of competing species with a focus on linearly frequency-dependent interactions, derived from a game-theoretic viewpoint. However, several aspects of evolutionary dynamics, e.g. limited resources, may induce effectively nonlinear frequency dependencies. Here we study the impact of nonlinear frequency dependence on evolutionary dynamics in a model class that covers linear frequency dependence as a special case. We focus on the simplest non-trivial setting of two genotypes and analyse the co-action of nonlinear frequency dependence with asymmetric mutation rates. We find that their co-action may induce novel metastable states as well as stochastic switching dynamics between them. Our results reveal how the different mechanisms of mutation, selection and genetic drift contribute to the dynamics and the emergence of metastable states, suggesting that multistability is a generic feature in systems with frequency-dependent fitness.

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.

Extinction in neutrally stable stochastic Lotka-Volterra models

Populations of competing biological species exhibit a fascinating interplay between the nonlinear dynamics of evolutionary selection forces and random fluctuations arising from the stochastic nature of the interactions. The processes leading to extinction of species, whose understanding is a key component in the study of evolution and biodiversity, are influenced by both of these factors. Here, we investigate a class of stochastic population dynamics models based on generalized Lotka-Volterra systems. In the case of neutral stability of the underlying deterministic model, the impact of intrinsic noise on the survival of species is dramatic: It destroys coexistence of interacting species on a time scale proportional to the population size. We introduce a new method based on stochastic averaging which allows one to understand this extinction process quantitatively by reduction to a lower-dimensional effective dynamics. This is performed analytically for two highly symmetrical models and can be generalized numerically to more complex situations. The extinction probability distributions and other quantities of interest we obtain show excellent agreement with simulations.

The structure of mutations and the evolution of cooperation

Evolutionary game dynamics in finite populations assumes that all mutations are equally likely, i.e., if there are n strategies a single mutation can result in any strategy with probability 1/n. However, in biological systems it seems natural that not all mutations can arise from a given state. Certain mutations may be far away, or even be unreachable given the current composition of an evolving population. These distances between strategies (or genotypes) define a topology of mutations that so far has been neglected in evolutionary game theory. In this paper we re-evaluate classic results in the evolution of cooperation departing from the assumption of uniform mutations. We examine two cases: the evolution of reciprocal strategies in a repeated prisoner's dilemma, and the evolution of altruistic punishment in a public goods game. In both cases, alternative but reasonable mutation kernels shift known results in the direction of less cooperation. We therefore show that assuming uniform mutations has a substantial impact on the fate of an evolving population. Our results call for a reassessment of the "model-less" approach to mutations in evolutionary dynamics.

Stochastic differential equations for evolutionary dynamics with demographic noise and mutations

We present a general framework to describe the evolutionary dynamics of an arbitrary number of types in finite populations based on stochastic differential equations (SDEs). For large, but finite populations this allows us to include demographic noise without requiring explicit simulations. Instead, the population size only rescales the amplitude of the noise. Moreover, this framework admits the inclusion of mutations between different types, provided that mutation rates μ are not too small compared to the inverse population size 1/N. This ensures that all types are almost always represented in the population and that the occasional extinction of one type does not result in an extended absence of that type. For μN≪1 this limits the use of SDEs, but in this case there are well established alternative approximations based on time scale separation. We illustrate our approach by a rock-scissors-paper game with mutations, where we demonstrate excellent agreement with simulation based results for sufficiently large populations. In the absence of mutations the excellent agreement extends to small population sizes.

Free energy, value, and attractors

It has been suggested recently that action and perception can be understood as minimising the free energy of sensory samples. This ensures that agents sample the environment to maximise the evidence for their model of the world, such that exchanges with the environment are predictable and adaptive. However, the free energy account does not invoke reward or cost-functions from reinforcement-learning and optimal control theory. We therefore ask whether reward is necessary to explain adaptive behaviour. The free energy formulation uses ideas from statistical physics to explain action in terms of minimising sensory surprise. Conversely, reinforcement-learning has its roots in behaviourism and engineering and assumes that agents optimise a policy to maximise future reward. This paper tries to connect the two formulations and concludes that optimal policies correspond to empirical priors on the trajectories of hidden environmental states, which compel agents to seek out the (valuable) states they expect to encounter.

Evolutionary and population dynamics: a coupled approach

We study the interplay of population growth and evolutionary dynamics using a stochastic model based on birth and death events. In contrast to the common assumption of an independent population size, evolution can be strongly affected by population dynamics in general. Especially for fast reproducing microbes which are subject to selection, both types of dynamics are often closely intertwined. We illustrate this by considering different growth scenarios. Depending on whether microbes die or stop to reproduce (dormancy), qualitatively different behaviors emerge. For cooperating bacteria, a permanent increase of costly cooperation can occur. Even if not permanent, cooperation can still increase transiently due to demographic fluctuations. We validate our analysis via stochastic simulations and analytic calculations. In particular, we derive a condition for an increase in the level of cooperation.

Evolutionary dynamics on stochastic evolving networks for multiple-strategy games

Evolutionary game theory on dynamical networks has received much attention. Most of the work has been focused on 2×2 games such as prisoner's dilemma and snowdrift, with general n×n games seldom addressed. In particular, analytical methods are still lacking. Here we generalize the stochastic linking dynamics proposed by Wu, Zhou, Fu, Luo, Wang, and Traulsen [PLoS ONE 5, e11187 (2010)] to n×n games. We analytically obtain that the fast linking dynamics results in the replicator dynamics with a rescaled payoff matrix. In the rescaled matrix, intuitively, each entry is the product of the original entry and the average duration time of the corresponding link. This result is shown to be robust to a wide class of imitation processes. As applications, we show both analytically and numerically that the biodiversity, modeled as the stability of a zero-sum rock-paper-scissors game, cannot be altered by the fast linking dynamics. In addition, we show that the fast linking dynamics can stabilize tit-for-tat as an evolutionary stable strategy in the repeated prisoner's dilemma game provided the interaction between the identical strategies happens sufficiently often. Our method paves the way for an analytical study of the multiple-strategy coevolutionary dynamics.

Stability of strategies in payoff-driven evolutionary games on networks

We consider a network of coupled agents playing the Prisoner's Dilemma game, in which players are allowed to pick a strategy in the interval [0, 1], with 0 corresponding to defection, 1 to cooperation, and intermediate values representing mixed strategies in which each player may act as a cooperator or a defector over a large number of interactions with a certain probability. Our model is payoff-driven, i.e., we assume that the level of accumulated payoff at each node is a relevant parameter in the selection of strategies. Also, we consider that each player chooses his∕her strategy in a context of limited information. We present a deterministic nonlinear model for the evolution of strategies. We show that the final strategies depend on the network structure and on the choice of the parameters of the game. We find that polarized strategies (pure cooperator∕defector states) typically emerge when (i) the network connections are sparse, (ii) the network degree distribution is heterogeneous, (iii) the network is assortative, and surprisingly, (iv) the benefit of cooperation is high.