Broom and Rychtář [Proc. R. Soc. A (2008) 464, 2609--2627] found an exact solution for the fixation probabilities of the Moran process for a structured population, in which the interaction structure among individuals is given by the so-called star graph, i.e. one central vertex and $n$ leaves, the leaves connecting only to the center. We generalize on their solution by allowing individuals' fitnesses to depend on the population frequency, and also by allowing a possible change in the order of reproduction and death draws. In their cited paper, Broom and Rychtář considered the birth-death (BD) process, in which at each time step an individual is first drawn for reproduction and then an individual is selected for death. In the death-birth (DB) process, the order of the draws is reversed. It may be seen that the order of the draws makes a big difference in the fixation probabilities. Our solution method applies to both the BD and the DB cases. As expected, the exact formulae for the fixation probabilities are complicated. We will also illustrate them with some examples and provide results on the asymptotic behavior of the fixation probabilities when the number $n$ of leaves in the graph tends to infinity.
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…
We reconsider the Moran model in continuous time with population size $N$, two allelic types, and selection. We introduce a new particle representation, which we call the labelled Moran model, and which has the same distribution of type frequencies as the original Moran model, provided the initial values are chosen appropriately. In the new model, individuals are labelled $1,2, \dots, N$; neutral resampling events may take place between arbitrary labels, whereas selective events only occur in the direction of increasing labels. With the help of elementary methods only, we not only recover fixa…
We study stochastic evolutionary game dynamics in a population of finite size. Individuals in the population are divided into two dynamically evolving groups. The structure of the population is formally described by a Wright-Fisher type Markov chain with a frequency dependent fitness. In a strong selection regime that favors one of the two groups, we obtain qualitatively matching lower and upper bounds for the fixation probability of the advantageous population. In the infinite population limit we obtain an exact result showing that a single advantageous mutant can invade an infinite populatio…
We study fixation probabilities for the Moran stochastic process for the evolution of a population with three or more types of individuals and frequency-dependent fitnesses. Contrarily to the case of populations with two types of individuals, in which fixation probabilities may be calculated by an exact formula, here we must solve a large system of linear equations. We first show that this system always has a unique solution. Other results are upper and lower bounds for the fixation probabilities obtained by coupling the Moran process with three strategies with birth-death processes with only …
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…
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）
The spread of an advantageous mutation through a population is of fundamental interest in population genetics. While the classical Moran model is formulated for a well-mixed population, it has long been recognized that in real-world applications, the population usually has an explicit spatial structure which can significantly influence the dynamics. In the context of cancer initiation in epithelial tissue, several recent works have analyzed the dynamics of advantageous mutant spread on integer lattices, using the biased voter model from particle systems theory. In this spatial version of the M…
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…
The Moran process is a classic stochastic process that models invasion dynamics on graphs. A single "mutant" (e.g., a new opinion, strain, social trait etc.) invades a population of residents spread over the nodes of a graph. The mutant fitness advantage $δ\geq 0$ determines how aggressively mutants propagate to their neighbors. The quantity of interest is the fixation probability, i.e., the probability that the initial mutant eventually takes over the whole population. However, in realistic settings, the invading mutant has an advantage only in certain locations. E.g., a bacterial mutation al…
We consider the classic Moran process modeling the spread of genetic mutations, as extended to structured populations by Lieberman et al.\ (Nature, 2005). In this process, individuals are the vertices of a connected graph $G$. Initially, there is a single mutant vertex, chosen uniformly at random. In each step, a random vertex is selected for reproduction with a probability proportional to its fitness: mutants have fitness $r>1$, while non-mutants have fitness 1. The vertex chosen to reproduce places a copy of itself to a uniformly random neighbor in $G$, replacing the individual that was ther…
The Moran process is a random process that models the spread of genetic mutations through graphs. If the graph is connected, the process eventually reaches "fixation", where every vertex is a mutant, or "extinction", where no vertex is a mutant. Our main result is an almost-tight bound on expected absorption time. For all epsilon > 0, we show that the expected absorption time on an n-vertex graph is o(n^(3+epsilon)). In fact, we show that it is at most n^3 * exp(O((log log n)^3)) and that there is a family of graphs where it is Omega(n^3). In the course of proving our main result, we also esta…
The spatial structure of an evolving population affects which mutations become fixed. Some structures amplify selection, increasing the likelihood that beneficial mutations become fixed while deleterious mutations do not. Other structures suppress selection, reducing the effect of fitness differences and increasing the role of random chance. This phenomenon can be modeled by representing spatial structure as a graph, with individuals occupying vertices. Births and deaths occur stochastically, according to a specified update rule. We study death-Birth updating: An individual is chosen to die an…
Many mathematical models of evolution assume that all individuals experience the same environment. Here, we study the Moran process in heterogeneous environments. The population is of finite size with two competing types, which are exposed to a fixed number of environmental conditions. Reproductive rate is determined by both the type and the environment. We first calculate the condition for selection to favor the mutant relative to the resident wild type. In large populations, the mutant is favored if and only if the mutant's spatial average reproductive rate exceeds that of the resident. But …
Biological fitness is not an observable quantity and must be inferred from population dynamics. Bayesian inference applied to the Moran process and variants yields a robust inference method that can infer fitness in populations evolving via a Moran dynamic and generalizations. Information about fitness is derived solely from birth-events in birth-death and death-birth processes in which selection acts proportionally to fitness, which allows the method to be applied to populations on a network where the network itself may be changing in time. Populations may also be allowed to change size while…
The fixation probability of a single mutant invading a population of residents is among the most widely-studied quantities in evolutionary dynamics. Amplifiers of natural selection are population structures that increase the fixation probability of advantageous mutants, compared to well-mixed populations. Extensive studies have shown that many amplifiers exist for the Birth-death Moran process, some of them substantially increasing the fixation probability or even guaranteeing fixation in the limit of large population size. On the other hand, no amplifiers are known for the death-Birth Moran p…
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…
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 …
The evolutionary advantage of replacers in the Moran process
Computer Science Institute, Charles University（查尔斯大学计算机科学学院）；Department of Organismic and Evolutionary Biology, Harvard University（哈佛大学生物演化系）；Department of Mathematics, Harvard University（哈佛大学数学系）
University of California, Irvine（加州大学尔湾分校）；University of Southern California（南加州大学）；University of Minnesota（明尼苏达大学）；Sharif University of Technology（谢里夫理工大学）
Evolution in finite populations is often modelled using the classical Moran process. Over the last ten years this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such population, is whether a rare mutant has a higher or lower chance of fixating (the fixation probability) than the Moran probability, i.e. that from the original Moran model, which represents an unstructured population. As evolutionary graph theory has developed, different ways of considering the interactions between individuals through a graph and an associated…
The Moran process, as studied by [Lieberman, E., Hauert, C. and Nowak, M. Evolutionary dynamics on graphs. Nature 433, pp. 312-316 (2005)], is a stochastic process modeling the spread of genetic mutations in populations. In this process, agents of a two-type population (i.e. mutants and residents) are associated with the vertices of a graph. Initially, only one vertex chosen uniformly at random is a mutant, with fitness $r > 0$, while all other individuals are residents, with fitness $1$. In every step, an individual is chosen with probability proportional to its fitness, and its state (mutant…
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…
We investigate the dynamics of the voter model in which the population itself changes endogenously via the birth-death process. There are two species of voters, labeled A and B, and the population of each species can grow or shrink by the birth-death process at equal rates $b$. Individuals of opposite species also undergo voter model dynamics in which an AB pair can equiprobably become AA or BB with rate $v$---neutral evolution. In the limit $b/v\to\infty$, the distribution of consensus times varies as $t^{-3}$ and the probability that the population size equals $n$ at the moment of consensus …
We consider the Moran process in a graph called the "star" and obtain the asymptotic expression for the fixation probability of a single mutant when the size of the graph is large. The expression obtained corrects the previously known expression announced in reference [E Lieberman, C Hauert, and MA Nowak. Evolutionary dynamics on graphs. Nature, 433(7023):312-316, 2005] and further studied in [M. Broom and J. Rychtar. An analysis of the fixation probability of a mutant on special classes of non-directed graphs. Proc. R. Soc. A-Math. Phys. Eng. Sci., 464(2098):2609-2627, 2008]. We also show tha…
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 …
We consider the so called Moran process with frequency dependent fitness given by a certain pay-off matrix. For finite populations, we show that the final state must be homogeneous, and show how to compute the fixation probabilities. Next, we consider the infinite population limit, and discuss the appropriate scalings for the drift-diffusion limit. In this case, a degenerated parabolic PDE is formally obtained that, in the special case of frequency independent fitness, recovers the celebrated Kimura equation in population genetics. We then show that the corresponding initial value problem is w…
Evolutionary dynamics on graphs can lead to many interesting and counterintuitive findings. We study the Moran process, a discrete time birth-death process, that describes the invasion of a mutant type into a population of wild-type individuals. Remarkably, the fixation probability of a single mutant is the same on all regular networks. But non-regular networks can increase or decrease the fixation probability. While the time until fixation formally depends on the same transition probabilities as the fixation probabilities, there is no obvious relation between them. For example, an amplifier o…
Configurational arrangement of network architecture and interaction character of individuals are two most influential factors on the mechanisms underlying the evolutionary outcome of cooperation, which is explained by the well-established framework of evolutionary game theory. In the current study, not only qualitatively but also quantitatively, we measure Moran-evolution of cooperation to support an analytical agreement based on the consequences of the replicator equation in a finite population. The validity of the measurement has been double-checked in the well-mixed network by the Langevin …
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 …
The Moran process on graphs is a popular model to study the dynamics of evolution in a spatially structured population. Exact analytical solutions for the fixation probability and time of a new mutant have been found for only a few classes of graphs so far. Simulations are time-expensive and many realizations are necessary, as the variance of the fixation times is high. We present an algorithm that numerically computes these quantities for arbitrary small graphs by an approach based on the transition matrix. The advantage over simulations is that the calculation has to be executed only once. B…
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…
The ancestral selection graph in population genetics was introduced by KroneNeuhauser (1997) as an analogue of the coalescent genealogy of a sample of genes from a neutrally evolving population. The number of particles in this graph, followed backwards in time, is a birth and death process with quadratic death and linear birth rates. In this paper an explicit form of the probability distribution of the number of particles is obtained by using the density of the allele frequency in the corresponding diffusion model obtained by Kimura (1955). It is shown that the process of fixation of the allel…
We study the evolution of cooperation in a finite population interacting according to a simple model of like-with-like assortment. Evolution proceeds as a Moran process, and payoffs from the underlying cooperator-defector game are translated to positive fitnesses by an exponential transformation. These evolutionary dynamics can arise, for example, in a nest-structured population with rare migration. The use of the exponential transformation, rather than the usual linear one, is appropriate when interactions have multiplicative fitness effects, and allows for a tractable characterization of the…
We analyze evolutionary dynamics on graphs, where the nodes represent individuals of a population. The links of a node describe which other individuals can be displaced by the offspring of the individual on that node. Amplifiers of selection are graphs for which the fixation probability is increased for advantageous mutants and decreased for disadvantageous mutants. A few examples of such amplifiers have been developed, but so far it is unclear how many such structures exist and how to construct them. Here, we show that almost any undirected random graph is an amplifier of selection for Birth-…
Evolutionary models on graphs, as an extension of the Moran process, have two major implementations: birth-death (BD) models (or the invasion process) and death- birth (DB) models (or voter models). The isothermal theorem states that the fixation probability of mutants in a large group of graph structures (known as isothermal graphs, which include regular graphs) coincides with that for the mixed population. This result has been proven by Lieberman et al (Nature 433: 312-316, 2005) in the case of BD processes, where mutants differ from the wild types by their birth rate (and not by their death…
Resources are rarely distributed uniformly within a population. Heterogeneity in the concentration of a drug, the quality of breeding sites, or wealth can all affect evolutionary dynamics. In this study, we represent a collection of properties affecting the fitness at a given location using a color. A green node is rich in resources while a red node is poorer. More colors can represent a broader spectrum of resource qualities. For a population evolving according to the birth-death Moran model, the first question we address is which structures, identified by graph connectivity and graph colorin…
The Moran process, as studied by Lieberman, Hauert and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen u.a.r.) possesses a mutation, with fitness r>1. All other individuals have fitness 1. During each step of the algorithm, an individual is chosen with probability proportional to its fitness, and its state (mutant or non-mutant) is passed on to an out-neighbour which is chosen u.a.r. If the underlying graph is strongly connected then the a…
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…
This work is a systematic study of discrete Markov chains that are used to describe the evolution of a two-types population. Motivated by results valid for the well-known Moran (M) and Wright-Fisher (WF) processes, we define a general class of Markov chains models which we term the Kimura class. It comprises the majority of the models used in population genetics, and we show that many well-known results valid for M and WF processes are still valid in this class. In all Kimura processes, a mutant gene will either fixate or become extinct, and we present a necessary and sufficient condition for …
We consider the Moran model in continuous time with two types, mutation, and selection. We concentrate on the ancestral line and its stationary type distribution. Building on work by Fearnhead (J. Appl. Prob. 39 (2002), 38-54) and Taylor (Electron. J. Probab. 12 (2007), 808-847), we characterise this distribution via the fixation probability of the offspring of all individuals of favourable type (regardless of the offspring's types). We concentrate on a finite population and stay with the resulting discrete setting all the way through. This way, we extend previous results and gain new insight …
The Moran process models the spread of genetic mutations through a population. A mutant with relative fitness $r$ is introduced into a population and the system evolves, either reaching fixation (in which every individual is a mutant) or extinction (in which none is). In a widely cited paper (Nature, 2005), Lieberman, Hauert and Nowak generalize the model to populations on the vertices of graphs. They describe a class of graphs (called "superstars"), with a parameter $k$. Superstars are designed to have an increasing fixation probability as $k$ increases. They state that the probability of fix…
We study the fixation probability of a mutant type when introduced into a resident population. As opposed to the usual assumption of constant pop- ulation size, we allow for stochastically varying population sizes. This is implemented by a stochastic competitive Lotka-Volterra model. The compe- tition coefficients are interpreted in terms of inverse payoffs emerging from an evolutionary game. Since our study focuses on the impact of the competition values, we assume the same birth and death rates for both types. In this gen- eral framework, we derive an approximate formula for the fixation pro…
In evolutionary processes, population structure has a substantial effect on natural selection. Here, we analyze how motion of individuals affects constant selection in structured populations. Motion is relevant because it leads to changes in the distribution of types as mutations march toward fixation or extinction. We describe motion as the swapping of individuals on graphs, and more generally as the shuffling of individuals between reproductive updates. Beginning with a one-dimensional graph, the cycle, we prove that motion suppresses natural selection for death-birth updating or for any pro…
We study the influence of the complex topology of scale-free graphs on the dynamics of anti-coordination games (e.g. snowdrift games). These reference models are characterized by the coexistence (evolutionary stable mixed strategy) of two competing species, say "cooperators" and "defectors", and, in finite systems, by metastability and large-fluctuation-driven fixation. In this work, we use extensive computer simulations and an effective diffusion approximation (in the weak selection limit) to determine under which circumstances, depending on the individual-based update rules, the topology dra…
Evolutionary models for populations of constant size are frequently studied using the Moran model, the Wright-Fisher model, or their diffusion limits. When evolution is neutral, a random genealogy given through Kingman's coalescent is used in order to understand basic properties of such models. Here, we address the use of a genealogical perspective for models with weak frequency-dependent selection, i.e. N s =: α is small, and s is the fitness advantage of a fit individual and N is the population size. When computing fixation probabilities, this leads either to the approach proposed by Rousset…
Power-law-distributed species counts or clone counts arise in many biological settings such as multispecies cell populations, population genetics, and ecology. This empirical observation that the number of species $c_{k}$ represented by $k$ individuals scales as negative powers of $k$ is also supported by a series of theoretical birth-death-immigration (BDI) models that consistently predict many low-population species, a few intermediate-population species, and very high-population species. However, we show how a simple global population-dependent regulation in a neutral BDI model destroys the…
Evolutionary graph theory has grown to be an area of intense study. Despite the amount of interest in the field, it seems to have grown separate from other subfields of population genetics and evolution. In the current work I introduce the concept of Fisher's (1930) reproductive value into the study of evolution on graphs. Reproductive value is a measure of the expected genetic contribution of an individual to a distant future generation. In a heterogeneous graph-structured population, differences in the number of connections among individuals translates into differences in the expected number…
