We study large fluctuations in evolutionary games belonging to the coordination and anti-coordination classes. The dynamics of these games, modeling cooperation dilemmas, is characterized by a coexistence fixed point separating two absorbing states. We are particularly interested in the problem of fixation that refers to the possibility that a few mutants take over the entire population. Here, the fixation phenomenon is induced by large fluctuations and is investigated by a semi-classical WKB (Wentzel-Kramers-Brillouin) theory generalized to treat stochastic systems possessing multiple absorbing states. Importantly, this method allows us to analyze the combined influence of selection and random fluctuations on the evolutionary dynamics \textit{beyond} the weak selection limit often considered in previous works. We accurately compute, including pre-exponential factors, the probability distribution function in the long-lived coexistence state and the mean fixation time necessary for a few mutants to take over the entire population in anti-coordination games, and also the fixation probability in the coordination class. Our analytical results compare excellently with extensive numerical simulations. Furthermore, we demonstrate that our treatment is superior to the Fokker-Planck approximation when the selection intensity is finite.
One of the most striking effect of fluctuations in evolutionary game theory is the possibility for mutants to fixate (take over) an entire population. Here, we generalize a recent WKB-based theory to study fixation in evolutionary games under non-vanishing selection, and investigate the relation between selection intensity w and demographic (random) fluctuations. This allows the accurate treatment of large fluctuations and yields the probability and mean times of fixation beyond the weak selection limit. The power of the theory is demonstrated on prototypical models of cooperation dilemmas wit…
We study the influence of complex graphs on the metastability and fixation properties of a set of evolutionary processes. In the framework of evolutionary game theory, where the fitness and selection are frequency-dependent and vary with the population composition, we analyze the dynamics of snowdrift games (characterized by a metastable coexistence state) on scale-free networks. Using an effective diffusion theory in the weak selection limit, we demonstrate how the scale-free structure affects the system's metastable state and leads to anomalous fixation. In particular, we analytically and nu…
We study evolutionary multi-player games in finite populations, subject to fluctuating environments. The population undergoes a birth-death process with absorbing states, and the environment follows a Markovian process, resulting in a fluctuating payoff matrix for the evolutionary game. Our focus is on the fixation or extinction of a single mutant in a population of wildtypes. We show that the nonlinear nature of fitnesses in multi-player games gives rise to an intricate interplay of selection, genetic drift and environmental fluctuations. This generates effects not seen in simpler two-player …
This paper is based on the complete classification of evolutionary scenarios for the Moran process with two strategies given by Taylor et al. (B. Math. Biol. 66(6): 1621--1644, 2004). Their classification is based on whether each strategy is a Nash equilibrium and whether the fixation probability for a single individual of each strategy is larger or smaller than its value for neutral evolution. We improve on this analysis by showing that each evolutionary scenario is characterized by a definite graph shape for the fixation probability function. A second class of results deals with the behavior…
One of the most fundamental concepts of evolutionary dynamics is the "fixation" probability, i.e. the probability that a mutant spreads through the whole population. Most natural communities are geographically structured into habitats exchanging individuals among each other and can be modeled by an evolutionary graph (EG), where directed links weight the probability for the offspring of one individual to replace another individual in the community. Very few exact analytical results are known for EGs. We show here how by using the techniques of the fixed point of Probability Generating Function…
Evolutionary dynamics in finite populations is known to fixate eventually in the absence of mutation. We here show that a similar phenomenon can be found in stochastic game dynamical batch learning, and investigate fixation in learning processes in a simple 2x2 game, for two-player games with cyclic interaction, and in the context of the best-shot network game. The analogues of finite populations in evolution are here finite batches of observations between strategy updates. We study when and how such fixation can occur, and present results on the average time-to-fixation from numerical simulat…
Different evolutionary models are known to make disparate predictions for the success of an invading mutant in some situations. For example, some evolutionary mechanics lead to amplification of selection in structured populations, while others suppress it. Here, we use computer simulations to study evolutionary populations moved by flows, and show how the speed of this motion impacts the fixation probability of an invading mutant. Flows of different speeds interpolate between evolutionary dynamics on fixed heterogeneous graphs and in well-stirred populations. We find that the motion has an act…
Rock-paper-scissors games metaphorically model cyclic dominance in ecology and microbiology. In a static environment, these models are characterized by fixation probabilities obeying two different "laws" in large and small well-mixed populations. Here, we investigate the evolution of these three-species models subject to a randomly switching carrying capacity modeling the endless change between states of resources scarcity and abundance. Focusing mainly on the zero-sum rock-paper-scissors game, equivalent to the cyclic Lotka-Volterra model, we study how the ${\it coupling}$ of demographic and …
We investigate evolutionary dynamics of two-strategy matrix games with zealots in finite populations. Zealots are assumed to take either strategy regardless of the fitness. When the strategy selected by the zealots is the same, the fixation of the strategy selected by the zealots is a trivial outcome. We study fixation time in this scenario. We show that the fixation time is divided into three main regimes, in one of which the fixation time is short, and in the other two the fixation time is exponentially long in terms of the population size. Different from the case without zealots, there is a…
Evolutionary graph theory models the effects of natural selection and random drift on structured populations of competing mutant and non-mutant individuals. Recent studies have found that fixation times in such systems often have right-skewed distributions. Little is known, however, about how these distributions and their skew depend on mutant fitness. Here we calculate the fitness dependence of the fixation-time distribution for the Moran Birth-death process in populations modeled by two extreme networks: the complete graph and the one-dimensional ring lattice, obtaining exact solutions in th…
In evolutionary dynamics, a key measure of a mutant trait's success is the probability that it takes over the population given some initial mutant-appearance distribution. This "fixation probability" is difficult to compute in general, as it depends on the mutation's effect on the organism as well as the population's spatial structure, mating patterns, and other factors. In this study, we consider weak selection, which means that the mutation's effect on the organism is small. We obtain a weak-selection perturbation expansion of a mutant's fixation probability, from an arbitrary initial config…
Agent-based stochastic models for finite populations have recently received much attention in the game theory of evolutionary dynamics. Both the ultimate fixation and the pre-fixation transient behavior are important to a full understanding of the dynamics. In this paper, we study the transient dynamics of the well-mixed Moran process through constructing a landscape function. It is shown that the landscape playing a central theoretical "device" that integrates several lines of inquiries: the stable behavior of the replicator dynamics, the long-time fixation, and continuous diffusion approxima…
Evolutionary game theory has proved to be a powerful tool to probe the self-organisation of collective behaviour by considering frequency-dependent fitness in evolutionary processes. It has shown that the stability of a strategy depends not only on the payoffs received after each encounter but also on the population's size. Here, we study $2\times2$ games in well-mixed finite populations by analysing the fixation probabilities of single mutants as functions of population size. We proved that 9 out of the 24 possible games always lead to monotonically decreasing functions, similarly to fixed fi…
Without mutation and migration, evolutionary dynamics ultimately leads to the extinction of all but one species. Such fixation processes are well understood and can be characterized analytically with methods from statistical physics. However, many biological arguments focus on stationary distributions in a mutation-selection equilibrium. Here, we address the equilibration time required to reach stationarity in the presence of mutation, this is known as the mixing time in the theory of Markov processes. We show that mixing times in evolutionary games have the opposite behaviour from fixation ti…
Fixation times in evolutionary games under weak selection
在进化博弈动力学中，繁殖成功率随着进化博弈中的表现而增加。如果策略$A$的表现优于策略$B$，则策略$A$将在人群中传播。在随机动力学下，单个突变体迟早会接管整个种群或灭绝。我们分析了与此过程相关的平均退出时间（或平均固定时间）。我们分析表明，这些时间依赖于一个令人惊讶的简单的方式下弱选择，即强随机性的游戏的回报矩阵：回报差$Δπ$是一个线性函数的$A$个人$i$，$Δπ= u i + v$的数量。无条件平均退出时间仅取决于常数项$v$。假设单个$A$突变体接管群体，则相应的条件平均退出时间仅取决于密度依赖项$u$。我们证明了这一发现的两个常用的微观进化过程。
The probability that the frequency of a particular trait will eventually become unity, the so-called fixation probability, is a central issue in the study of population evolution. Its computation, once we are given a stochastic finite population model without mutations and a (possibly frequency dependent) fitness function, is straightforward and it can be done in several ways. Nevertheless, despite the fact that the fixation probability is an important macroscopic property of the population, its precise knowledge does not give any clear information about the interaction patterns among individu…
Whether or not cooperation is favored over defection in evolutionary games can be assigned by structure coefficients for any arrangement of cooperators and defectors on any network modeled as a regular graph. We study how these structure coefficients relate to a scaling of dilemma strength in social dilemma games. It is shown that some graphs permit certain arrangements of cooperators and defectors to possess particularly large structure coefficients. Moreover, these large coefficients imply particularly large sections of a bounded parameter plane spanned by scaling gamble-intending and risk-a…
Evolutionary games on graphs describe how strategic interactions and population structure determine evolutionary success, quantified by the probability that a single mutant takes over a population. Graph structures, compared to the well-mixed case, can act as amplifiers or suppressors of selection by increasing or decreasing the fixation probability of a beneficial mutant. Properties of the associated mean fixation times can be more intricate, especially when selection is strong. The intuition is that fixation of a beneficial mutant happens fast (in a dominance game), that fixation takes very …
Evolutionary graph theory studies the evolutionary dynamics of populations structured on graphs. A central problem is determining the probability that a small number of mutants overtake a population. Currently, Monte Carlo simulations are used for estimating such fixation probabilities on general directed graphs, since no good analytical methods exist. In this paper, we introduce a novel deterministic framework for computing fixation probabilities for strongly connected, directed, weighted evolutionary graphs under neutral drift. We show how this framework can also be used to calculate the exp…
In evolutionary dynamics, the probability that a mutation spreads through the whole population, having arisen in a single individual, is known as the fixation probability. In general, it is not possible to find the fixation probability analytically given the mutant's fitness and the topological constraints that govern the spread of the mutation, so one resorts to simulations instead. Depending on the topology in use, a great number of evolutionary steps may be needed in each of the simulation events, particularly in those that end with the population containing mutants only. We introduce two t…
John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）；Department of Molecular and Cellular Biology, Harvard University（哈佛大学生物分子与细胞生物学系）；Department of Organismic and Evolutionary Biology, Harvard University（哈佛大学有机体与进化生物学系）；Department of Mathematics, Northeastern University（东北大学数学系）；Computer Science Institute, Charles University（查理大学计算机研究所）；Department of Mathematics, Harvard University（哈佛大学数学系）
Whether or not cooperation is favored in evolutionary games on graphs depends on the population structure and spatial properties of the interaction network. Population structures can be expressed as configurations. Such configurations extend scenarios with a single cooperator among defectors to any number of cooperators and any arrangement of cooperators and defectors. Thus, as a single cooperator can be interpreted as a lone mutant, the discussion about fixation properties based on configurations also applies to multiple mutants. For interaction networks modeled as regular graphs and for weak…
Gene duplications are one of major primary driving forces for evolutionary novelty. We took population genetics models of genes duplicate to study how evolutionary forces acting during the fixation of mutant allele at duplicate loci. We study the fixation time of mutant allele at duplicate loci under double null recessive model (DNR) and haploinsufficient model (HI). And we also investigate how selection coefficients with other evolutionary force influence the fixation frequency of mutant allele at duplicate loci. Our results suggest that the selection plays a role in the evolutionary fate of …
Studies of fixation dynamics in Markov processes predominantly focus on the mean time to absorption. This may be inadequate if the distribution is broad and skewed. We compute the distribution of fixation times in one-step birth-death processes with two absorbing states. These are expressed in terms of the spectrum of the process, and we provide different representations as forward-only processes in eigenspace. These allow efficient sampling of fixation time distributions. As an application we study evolutionary game dynamics, where invading mutants can reach fixation or go extinct. We also hi…
Fixation probabilities for multi-allele Moran dynamics with weak selection
Center for Advanced Systems Understanding (CASUS) – Helmholtz-Zentrum Dresden-Rossendorf (HZDR)（高级系统理解中心（CASUS）– 德累斯顿-罗斯多夫亥姆霍兹中心（HZDR））；Universidade Federal de Minas Gerais（米纳斯吉拉斯联邦大学）；ICTP South American Institute for Fundamental Research & Instituto de Física Teórica, Universidade Estadual Paulista - UNESP（国际理论物理中心南美基础研究所与理论物理研究所，圣保罗州立大学（UNESP））；Department of Ecology, Institute of Biosciences, University of São Paulo（圣保罗大学生物科学学院生态学系）
We define the incentive process, a natural generalization of the Moran process incorporating evolutionary updating mechanisms corresponding to well-known evolutionary dynamics, such as the logit, projection, and best-reply dynamics. Fixation probabilities and internal stable states are given for a variety of incentives, including new closed-forms, as well as results relating fixation probabilities for members of two one-parameter families of incentive processes. We show that the behaviors of the incentive process can deviate significantly from the analogous properties of deterministic evolutio…
Players of coevolutionary games may update not only their strategies but also their networks of interaction. Based on interpreting the payoff of players as fitness, dynamic landscape models are proposed. The modeling procedure is carried out for Prisoner's Dilemma (PD) and Snowdrift (SD) games that both use either birth--death (BD) or death--birth (DB) strategy updating. The main focus is on using dynamic fitness landscapes as a mathematical model of coevolutionary game dynamics. Hence, an alternative tool for analyzing coevolutionary games becomes available, and landscape measures such as mod…
Although mutations drive the evolutionary process, the rates at which the mutations occur are themselves subject to evolutionary forces. Our purpose here is to understand the role of selection and random genetic drift in the evolution of mutation rates, and we address this question in asexual populations at mutation-selection equilibrium neglecting selective sweeps. Using a multitype branching process, we calculate the fixation probability of a rare nonmutator in a large asexual population of mutators, and find that a nonmutator is more likely to fix when the deleterious mutation rate of the m…
Population structures can be crucial determinants of evolutionary processes. For the Moran process on graphs certain structures suppress selective pressure, while others amplify it (Lieberman et al. 2005 Nature 433 312-316). Evolutionary amplifiers suppress random drift and enhance selection. Recently, some results for the most powerful known evolutionary amplifier, the superstar, have been invalidated by a counter example (Díaz et al. 2013 Proc. R. Soc. A 469 20130193). Here we correct the original proof and derive improved upper and lower bounds, which indicate that the fixation probability …
Evolutionary dynamics has been classically studied for homogeneous populations, but now there is a growing interest in the non-homogenous case. One of the most important models has been proposed by Lieberman, Hauert and Nowak, adapting to a weighted directed graph the classical process described by Moran. The Markov chain associated with the graph can be modified by erasing all non-trivial loops in its state space, obtaining the so-called Embedded Markov chain (EMC). The fixation probability remains unchanged, but the expected time to absorption (fixation or extinction) is reduced. In this pap…
In evolutionary game theory, an important measure of a mutant trait (strategy) is its ability to invade and take over an otherwise-monomorphic population. Typically, one quantifies the success of a mutant strategy via the probability that a randomly occurring mutant will fixate in the population. However, in a structured population, this fixation probability may depend on where the mutant arises. Moreover, the fixation probability is just one quantity by which one can measure the success of a mutant; fixation time, for instance, is another. We define a notion of homogeneity for evolutionary ga…
We study fixation in large, but finite, populations with two types, and dynamics governed by birth-death processes. By considering a restricted class of such processes, we derive a continuous approximation for the probability of fixation that is valid beyond the weak-selection (WS) limit. From the continuous approximations, we then obtain asymptotic approximations for evolutionary dynamics with at most one equilibrium, in the selection-driven regime, that does not preclude a weak-selection regime. As an application, we study the fixation pattern when the infinite population limit has an interi…
Game theory ideas provide a useful framework for studying evolutionary dynamics in a well-mixed environment. This approach, however, typically enforces a strictly fixed overall population size, deemphasizing natural growth processes. We study a competitive Lotka-Volterra model, with number fluctuations, that accounts for natural population growth and encompasses interaction scenarios typical of evolutionary games. We show that, in an appropriate limit, the model describes standard evolutionary games with both genetic drift and overall population size fluctuations. However, there are also regim…
Temporal environmental variations are ubiquitous in nature, yet most of the theoretical works in population genetics and evolution assume fixed environment. Here we analyze the effect of variations in carrying capacity on the fate of a mutant type. We consider a two-state Moran model, where selection intensity at equilibrium may differ (in amplitude and in sign) from selection during periods of sharp growth and sharp decline. Using Kimura's diffusion approximation we present simple formulae for effective population size and effective selection, and use it to calculate the chance of ultimate fi…
In large clonal populations, several clones generally compete which results in complex evolutionary and ecological dynamics: experiments show successive selective sweeps of favorable mutations as well as long-term coexistence of multiple clonal strains. The mechanisms underlying either coexistence or fixation of several competing strains have rarely been studied altogether. Conditions for coexistence have mostly been studied by population and community ecology, while rates of invasion and fixation have mostly been studied by population genetics. In order to provide a global understanding of th…
The environment in which a population evolves can have a crucial impact on selection. We study evolutionary dynamics in finite populations of fixed size in a changing environment. The population dynamics are driven by birth and death events. The rates of these events may vary in time depending on the state of the environment, which follows an independent Markov process. We develop a general theory for the fixation probability of a mutant in a population of wild-types, and for mean unconditional and conditional fixation times. We apply our theory to evolutionary games for which the payoff struc…
We study the stochastic dynamics of evolutionary games, and focus on the so-called `stochastic slowdown' effect, previously observed in (Altrock et. al, 2010) for simple evolutionary dynamics. Slowdown here refers to the fact that a beneficial mutation may take longer to fixate than a neutral one. More precisely, the fixation time conditioned on the mutant taking over can show a maximum at intermediate selection strength. We show that this phenomenon is present in the prisoner's dilemma, and also discuss counterintuitive slowdown and speedup in coexistence games. In order to establish the micr…
The fixation probability of a mutant in the evolutionary dynamics of Moran process is calculated by the Monte-Carlo method on a few families of clique-based graphs. It is shown that the complete suppression of fixation can be realized with the generalized clique-wheel graph in the limit of small wheel-clique ratio and infinite size. The family of clique-star is an amplifier, and clique-arms graph changes from amplifier to suppressor as the fitness of the mutant increases. We demonstrate that the overall structure of a graph can be more important to determine the fixation probability than the d…
Population structure can be modelled by evolutionary graphs, which can have a substantial, but very subtle influence on the fate of the arising mutants. Individuals are located on the nodes of these graphs, competing with each other to eventually take over the graph via the links. Many applications for this framework can be envisioned, from the ecology of river systems and cancer initiation in colonic crypts to biotechnological search for optimal mutations. In all these applications, it is not only important where and when novel variants arise and how likely it is that they ultimately take ove…
We study the combined influence of selection and random fluctuations on the evolutionary dynamics of two-strategy ("cooperation" and "defection") games in populations comprising cooperation facilitators. The latter are individuals that support cooperation by enhancing the reproductive potential of cooperators relative to the fitness of defectors. By computing the fixation probability of a single cooperator in finite and well-mixed populations that include a fixed number of facilitators, and by using mean field analysis, we determine when selection promotes cooperation in the important classes …
Evolutionary game dynamics describes the spreading of successful strategies in a population of reproducing individuals. Typically, the microscopic definition of strategy spreading is stochastic, such that the dynamics becomes deterministic only in infinitely large populations. Here, we introduce a new microscopic birth--death process that has a fully deterministic strong selection limit in well--mixed populations of any size. Additionally, under weak selection, from this new process the frequency dependent Moran process is recovered. This makes it a natural extension of the usual evolutionary …
Evolutionary game dynamics in finite populations is typically subject to noise, inducing effects which are not present in deterministic systems, including fixation and extinction. In the first part of this paper we investigate the phenomenon of drift reversal in finite populations, taking into account that drift is a local quantity in strategy space. Secondly, we study a simple imitation dynamics, and show that it can lead to fixation at internal mixed-strategy fixed points even in finite populations. Imitation in infinite populations is adequately described by conventional replicator dynamics…
In population genetics, diffusions on the unit interval are often used to model the frequency path of an allele. In this setting we derive approximations for fixation probabilities, expected hitting times and the expected site-frequency-spectrum under small frequency-dependent selection. Specifically, we rederive and extend the one-third rule of evolutionary game theory (Nowak et al., 2004) and effects of stochastic slowdown (Altrock and Traulsen, 2009). Since similar effects are of interest in other application areas, we formulate our results for general one-dimensional diffusions.
In subdivided populations, migration acts together with selection and genetic drift and determines their evolution. Building up on a recently proposed method, which hinges on the emergence of a time scale separation between local and global dynamics, we study the fixation properties of subdivided populations in the presence of balancing selection. The approximation implied by the method is accurate when the effective selection strength is small and the number of subpopulations is large. In particular, it predicts a phase transition between species coexistence and biodiversity loss in the infin…
Horizontal transfer (HT) of heritable information or `traits' (carried by genetic elements, endosymbionts, or culture) is widespread among living organisms. Yet current ecological and evolutionary theory addressing HT is limited. We present a modeling framework for the dynamics of two populations that compete for resources and exchange horizontally (transfer) an otherwise vertically inherited trait. Competition influences individual demographics, affecting population size, which feeds back on the dynamics of transfer. We capture this feedback with a stochastic individual-based model, from whic…
Cooperation is ubiquitous across all levels of biological systems ranging from microbial communities to human societies. It, however, seemingly contradicts the evolutionary theory, since cooperators are exploited by free-riders and thus are disfavored by natural selection. Many studies based on evolutionary game theory have tried to solve the puzzle and figure out the reason why cooperation exists and how it emerges. Network reciprocity is one of the mechanisms to promote cooperation, where nodes refer to individuals and links refer to social relationships. The spatial arrangement of mutant in…
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 single, randomly placed mutants. We find that for any configuration cooperation is never favored for birth-death (BD) updat…
Population structure can have a significant effect on evolution. For some systems with sufficient symmetry, analytic results can be derived within the mathematical framework of evolutionary graph theory which relate to the outcome of the evolutionary process. However, for more complicated heterogeneous structures, computationally intensive methods are required such as individual-based stochastic simulations. By adapting methods from statistical physics, including moment closure techniques, we first show how to derive existing homogenised pair approximation models and the exact neutral drift mo…
