Amplifiers of selection for the Moran process with both Birth-death and death-Birth updating

Populations evolve by accumulating advantageous mutations. Every population has some spatial structure that can be modeled by an underlying network. The network then influences the probability that new advantageous mutations fixate. Amplifiers of selection are networks that increase the fixation probability of advantageous mutants, as compared to the unstructured fully-connected network. Whether or not a network is an amplifier depends on the choice of the random process that governs the evolutionary dynamics. Two popular choices are Moran process with Birth-death updating and Moran process with death-Birth updating. %Moran process has two popular versions called Birth-death updating and death-Birth updating. Interestingly, while some networks are amplifiers under Birth-death updating and other networks are amplifiers under death-Birth updating, no network is known to function as an amplifier under both types of updating simultaneously. In this work, we identify networks that act as amplifiers of selection under both versions of the Moran process. The amplifiers are robust, modular, and increase fixation probability for any mutant fitness advantage in a range $r\in(1,1.2)$. To complement this positive result, we also prove that for certain quantities closely related to fixation probability, it is impossible to improve them simultaneously for both versions of the Moran process. Together, our results highlight how the two versions of the Moran process differ and what they have in common

Discrete stochastic processes, replicator and Fokker-Planck equations of coevolutionary dynamics in finite and infinite populations

Finite-size fluctuations in coevolutionary dynamics arise in models of biological as well as of social and economic systems. This brief tutorial review surveys a systematic approach starting from a stochastic process discrete both in time and state. The limit $N\to \infty$ of an infinite population can be considered explicitly, generally leading to a replicator-type equation in zero order, and to a Fokker-Planck-type equation in first order in $1/\sqrt{N}$. Consequences and relations to some previous approaches are outlined.Comment: Banach Center publications, in pres

Strategy intervention for the evolution of fairness

Masses of experiments have shown individual preference for fairness which seems irrational. The reason behind it remains a focus for research. The effect of spite (individuals are only concerned with their own relative standing) on the evolution of fairness has attracted increasing attention from experiments, but only has been implicitly studied in one evolutionary model. The model did not involve high-offer rejections, which have been found in the form of non-monotonic rejections (rejecting offers that are too high or too low) in experiments. Here, we introduce a high offer and a non-monotonic rejection in structured populations of finite size, and use strategy intervention to explicitly study how spite influences the evolution of fairness: five strategies are in sequence added into the competition of a fair strategy and a selfish strategy. We find that spite promotes fairness, altruism inhibits fairness, and the non-monotonic rejection can cause fairness to overcome selfishness, which cannot happen without high-offer rejections. Particularly for the group-structured population with seven discrete strategies, we analytically study the effect of population size, mutation, and migration on fairness, selfishness, altruism, and spite. A larger population size cannot change the dominance of fairness, but it promotes altruism and inhibits selfishness and spite. Intermediate mutation maximizes selfishness and fairness, and minimizes spite; intermediate mutation maximizes altruism for intermediate migration and minimizes altruism otherwise. The existence of migration inhibits selfishness and fairness, and promotes altruism; sufficient migration promotes spite. Our study may provide important insights into the evolutionary origin of fairness.Comment: 15 pages, 7 figures. Comments welcom

Nongaussian fluctuations arising from finite populations: Exact results for the evolutionary Moran process

The appropriate description of fluctuations within the framework of evolutionary game theory is a fundamental unsolved problem in the case of finite populations. The Moran process recently introduced into this context [Nowak et al., Nature (London) 428, 646 (2004)] defines a promising standard model of evolutionary game theory in finite populations for which analytical results are accessible. In this paper, we derive the stationary distribution of the Moran process population dynamics for arbitrary $2\times{}2$ games for the finite size case. We show that a nonvanishing background fitness can be transformed to the vanishing case by rescaling the payoff matrix. In contrast to the common approach to mimic finite-size fluctuations by Gaussian distributed noise, the finite size fluctuations can deviate significantly from a Gaussian distribution.Comment: 4 pages (2 figs). Published in Physical Review E (Rapid Communications

Evolutionary game dynamics in phenotype space

Evolutionary dynamics can be studied in well-mixed or structured populations. Population structure typically arises from the heterogeneous distribution of individuals in physical space or on social networks. Here we introduce a new type of space to evolutionary game dynamics: phenotype space. The population is well-mixed in the sense that everyone is equally likely to interact with everyone else, but the behavioral strategies depend on distance in phenotype space. Individuals might behave differently towards those who look similar or dissimilar. Individuals mutate to nearby phenotypes. We study the `phenotypic space walk' of populations. We present analytic calculations that bring together ideas from coalescence theory and evolutionary game dynamics. As a particular example, we investigate the evolution of cooperation in phenotype space. We obtain a precise condition for natural selection to favor cooperators over defectors: for a one-dimensional phenotype space and large population size the critical benefit-to-cost ratio is given by b/c=1+2/sqrt{3}. We derive the fundamental condition for any evolutionary game and explore higher dimensional phenotype spaces.Comment: version 2: minor changes; equivalent to final published versio

Stochastic evolutionary game dynamics

In this review, we summarize recent developments in stochastic evolutionary game dynamics of finite populations.Comment: To appear in "Reviews of Nonlinear Dynamics and Complexity" Vol. II, Wiley-VCH, 2009, edited by H.-G. Schuste