Ecological public goods games: cooperation and bifurcation

The Public Goods Game is one of the most popular models for studying the origin and maintenance of cooperation. In its simplest form, this evolutionary game has two regimes: defection goes to fixation if the multiplication factor r is smaller than the interaction group size N, whereas cooperation goes to fixation if the multiplication factor r is larger than the interaction group size N. Hauert et al. [Hauert, C., Holmes, M., Doebeli, M., 2006a. Evolutionary games and population dynamics: Maintenance of cooperation in public goods games. Proc. R. Soc. Lond. B 273, 2565-2570] have introduced the Ecological Public Goods Game by viewing the payoffs from the evolutionary game as birth rates in a population dynamic model. This results in a feedback between ecological and evolutionary dynamics: if defectors are prevalent, birth rates are low and population densities decline, which leads to smaller interaction groups for the Public Goods game, and hence to dominance of cooperators, with a concomitant increase in birth rates and population densities. This feedback can lead to stable co-existence between cooperators and defectors. Here we provide a detailed analysis of the dynamics of the Ecological Public Goods Game, showing that the model exhibits various types of bifurcations, including supercritical Hopf bifurcations, which result in stable limit cycles, and hence in oscillatory co-existence of cooperators and defectors. These results show that including population dynamics in evolutionary games can have important consequences for the evolutionary dynamics of cooperation.     

Global mutations and local mutations have very different effects on evolution, illustrated by mixed strategies of asymmetric binary games

We study the evolutionary effect of rare mutations causing global changes in traits. We consider asymmetric binary games between two players. The first player takes two alternative options with probability x and 1−x; and the second player takes options with probability y and 1−y. Due to natural selection and recurrent mutation, the population evolves to have broad distributions of x and y. We analyze three cases showing qualitatively different dynamics, exemplified by (1) vigilance-intrusion game, (2) asymmetric hawk-dove game and (3) cleaner-client game. We found that the evolutionary outcome is strongly dependent upon the distribution of mutants’ traits, more than the mutation rates. For example in the vigilance-intrusion game, the evolutionary dynamics show a perpetual stable oscillation if mutants are always close to the parent (local-mutation mode), whilst the population converges to a stable equilibrium distribution if mutants can be quite different from the parent (global-mutation mode), even for extremely low mutation rate. When common local mutations and rare global mutations occur simultaneously, the evolutionary outcome is controlled by the latter.     

Fixation probabilities in evolutionary game dynamics with a two-strategy game in finite diploid populations

Fixation processes in evolutionary game dynamics in finite diploid populations are investigated. Traditionally, frequency dependent evolutionary dynamics is modeled as deterministic replicator dynamics. This implies that the infinite size of the population is assumed implicitly. In nature, however, population sizes are finite. Recently, stochastic processes in finite populations have been introduced in order to study finite size effects in evolutionary game dynamics. One of the most significant studies on evolutionary dynamics in finite populations was carried out by Nowak et al. which describes “one-third law” [Nowak, et al., 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646-650]. It states that under weak selection, if the fitness of strategy α is greater than that of strategy β when α has a frequency , strategy α fixates in a β-population with selective advantage. In their study, it is assumed that the inheritance of strategies is asexual, i.e. the population is haploid. In this study, we apply their framework to a diploid population that plays a two-strategy game with two ESSs (a bistable game). The fixation probability of a mutant allele in this diploid population is derived. A “three-tenth law” for a completely recessive mutant allele and a “two-fifth law” for a completely dominant mutant allele are found; other cases are also discussed.     

Mutation-selection equilibrium in games with mixed strategies

We develop a new method for studying stochastic evolutionary game dynamics of mixed strategies. We consider the general situation: there are n pure strategies whose interactions are described by an n×n payoff matrix. Players can use mixed strategies, which are given by the vector (p₁,…,p_n). Each entry specifies the probability to use the corresponding pure strategy. The sum over all entries is one. Therefore, a mixed strategy is a point in the simplex S_n. We study evolutionary dynamics in a well-mixed population of finite size. Individuals reproduce proportional to payoff. We consider the case of weak selection, which means the payoff from the game is only a small contribution to overall fitness. Reproduction can be subject to mutation; a mutant adopts a randomly chosen mixed strategy. We calculate the average abundance of every mixed strategy in the stationary distribution of the mutation-selection process. We find the crucial conditions that specify if a strategy is favored or opposed by selection. One condition holds for low mutation rate, another for high mutation rate. The result for any mutation rate is a linear combination of those two. As a specific example we study the Hawk-Dove game. We prove general statements about the relationship between games with pure and with mixed strategies.     

Sex and evolutionary stability

We study evolutionary games in which the rest points of the evolutionary dynamic cluster in connected components, focusing on what we call the Resource Game as a canonical example. The long-term outcome in such games can depend critically on second-order forces that were excluded from the evolutionary dynamics because they are typically insignificant compared with selection pressures. We show that the influence of second-order forces on long-term outcomes can depend on whether the reproduction underlying the evolutionary dynamics is sexual or asexual. An implication is that care is needed in adopting the convenience of an asexual model when examining the behavior of a sexual population in games with nontrivial components of rest points.     

The interaction energy of charged filaments in an electrolyte: Results for all filament spacings

Recent studies have explored interactions between evolutionary game dynamics and population structure. Yet most studies so far mainly paid attention to unweighted and static networks. Here we explore evolutionary games played on dynamically weighted networks. Players update their strategies according to the payoffs they obtain. Players also update weights of their adjacent links depending on payoffs they gain through those links; profitable links are reinforced whereas unprofitable ones are weakened. The system is characterized by two time scales, the one for strategy update, β_S, and the other for weight adjustment, β_W. We find that, under a mean-field approximation, the asymptotic behavior of the system is described by the replicator equation with an effective payoff matrix, which is a combination of the original game matrix A and its transpose, A^T. Both analytical and numerical results show that such an adaptive weight adjustment mechanism dramatically promotes evolution of cooperation.     

Strategy selection in structured populations

Evolutionary game theory studies frequency dependent selection. The fitness of a strategy is not constant, but depends on the relative frequencies of strategies in the population. This type of evolutionary dynamics occurs in many settings of ecology, infectious disease dynamics, animal behavior and social interactions of humans. Traditionally evolutionary game dynamics are studied in well-mixed populations, where the interaction between any two individuals is equally likely. There have also been several approaches to study evolutionary games in structured populations. In this paper we present a simple result that holds for a large variety of population structures. We consider the game between two strategies, A and B, described by the payoff matrix . We study a mutation and selection process. For weak selection strategy A is favored over B if and only if σa+b>c+σd. This means the effect of population structure on strategy selection can be described by a single parameter, σ. We present the values of σ for various examples including the well-mixed population, games on graphs, games in phenotype space and games on sets. We give a proof for the existence of such a σ, which holds for all population structures and update rules that have certain (natural) properties. We assume weak selection, but allow any mutation rate. We discuss the relationship between σ and the critical benefit to cost ratio for the evolution of cooperation. The single parameter, σ, allows us to quantify the ability of a population structure to promote the evolution of cooperation or to choose efficient equilibria in coordination games.     

Genetic diversity of microsatellite loci in hierarchically structured populations

Microsatellite loci are widely used for investigating patterns of genetic variation within and among populations. Those patterns are in turn determined by population sizes, migration rates, and mutation rates. We provide exact expressions for the first two moments of the allele frequency distribution in a stochastic model appropriate for studying microsatellite evolution with migration, mutation, and drift under the assumption that the range of allele sizes is bounded. Using these results, we study the behavior of several measures related to Wright’s F_ST, including Slatkin’s R_ST. Our analytical approximations for F_ST and R_ST show that familiar relationships between N_em and F_ST or R_ST hold when the migration and mutation rates are small. Using the exact expressions for F_ST and R_ST, our numerical results show that, when the migration and mutation rates are large, these relationships no longer hold. Our numerical results also show that the diversity measures most closely related to F_ST depend on mutation rates, mutational models (stepwise versus two-phase), migration rates, and population sizes. Surprisingly, R_ST is relatively insensitive to the mutation rates and mutational models. The differing behaviors of R_ST and F_ST suggest that properties of the among-population distribution of allele frequencies may allow the roles of mutation and migration in producing patterns of diversity to be distinguished, a topic of continuing investigation.     

Evolutionary stability and quasi-stationary strategy in stochastic evolutionary game dynamics

Stochastic evolutionary game dynamics for finite populations has recently been widely explored in the study of evolutionary game theory. It is known from the work of Traulsen et al. [2005. Phys. Rev. Lett. 95, 238701] that the stochastic evolutionary dynamics approaches the deterministic replicator dynamics in the limit of large population size. However, sometimes the limiting behavior predicted by the stochastic evolutionary dynamics is not quite in agreement with the steady-state behavior of the replicator dynamics. This paradox inspired us to give reasonable explanations of the traditional concept of evolutionarily stable strategy (ESS) in the context of finite populations. A quasi-stationary analysis of the stochastic evolutionary game dynamics is put forward in this study and we present a new concept of quasi-stationary strategy (QSS) for large but finite populations. It is shown that the consistency between the QSS and the ESS implies that the long-term behavior of the replicator dynamics can be predicted by the quasi-stationary behavior of the stochastic dynamics. We relate the paradox to the time scales and find that the contradiction occurs only when the fixation time scale is much longer than the quasi-stationary time scale. Our work may shed light on understanding the relationship between the deterministic and stochastic methods of modeling evolutionary game dynamics.     

Evolution and instability in ring species complexes: An in silico approach to the study of speciation

Ring species are a biological complex that theoretically forms when an ancestral population extends its range around a geographic barrier and, despite low-level gene flow, differentiates until reproductive isolation exists when terminal populations come into secondary contact. Due to their rarity in nature, little is known about the biological factors that promote the formation of ring species. We use evolutionary algorithms operating on two simple computational problems (SAW and K-max) to study the process of speciation under the conditions which may yield ring species. We vary evolutionary parameters to measure their influence on ring species’ development and stability over evolutionary time. Using the SAW problem, ring species consistently form, i.e. fertility is negatively correlated with distance (R-values between −0.097 and −0.821, p<0.001), and terminal populations show substantial infertility. However, all SAW simulations demonstrate instability in the complex after sympatric zones are established between terminal populations. Higher mutation rates and larger dispersal/breeding radii promote ring species’ formation and stability. Using a problem with a simple fitness landscape, the K-max problem, ring species do not form. Instead, speciation around the ring occurs before ring closure as good genotypes become locally dominant.     

Can possible evolutionary outcomes be determined directly from the population dynamics?

Traditionally, to determine the possible evolutionary behaviour of an ecological system using adaptive dynamics, it is necessary to calculate the fitness and its derivatives at a singular point. We investigate the claim that the possible evolutionary behaviour can be predicted directly from the population dynamics, without the need for calculation, by applying three criteria — one based on the form of the density dependent rates and two on the role played by the evolving parameters. Taking a general continuous time model, with broad ecological range, we show that the claim is true. Initially, we assume that individuals enter in class 1 and move through population classes sequentially; later we relax these assumptions and find that the criteria still apply. However, when we consider models where the evolving parameters appear non-linearly in the dynamics, we find some aspects of the criteria fail; useful but weaker results on possible evolutionary behaviour now apply.     

Evolutionary dynamics of biological auctions

Many scenarios in the living world, where individual organisms compete for winning positions (or resources), have properties of auctions. Here we study the evolution of bids in biological auctions. For each auction, n individuals are drawn at random from a population of size N. Each individual makes a bid which entails a cost. The winner obtains a benefit of a certain value. Costs and benefits are translated into reproductive success (fitness). Therefore, successful bidding strategies spread in the population. We compare two types of auctions. In “biological all-pay auctions”, the costs are the bid for every participating individual. In “biological second price all-pay auctions”, the cost for everyone other than the winner is the bid, but the cost for the winner is the second highest bid. Second price all-pay auctions are generalizations of the “war of attrition” introduced by Maynard Smith. We study evolutionary dynamics in both types of auctions. We calculate pairwise invasion plots and evolutionarily stable distributions over the continuous strategy space. We find that the average bid in second price all-pay auctions is higher than in all-pay auctions, but the average cost for the winner is similar in both auctions. In both cases, the average bid is a declining function of the number of participants, n. The more individuals participate in an auction the smaller is the chance of winning, and thus expensive bids must be avoided.     

Evolutionary stability on graphs

Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree k>2. Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth-death (BD), death-birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs.     

Evolutionary genetics of an S-like polymorphism in Papaveraceae with putative function in self-incompatibility

Background

Papaver rhoeas possesses a gametophytic self-incompatibility (SI) system not homologous to any other SI mechanism characterized at the molecular level. Four previously published full length stigmatic S-alleles from the genus Papaver exhibited remarkable sequence divergence, but these studies failed to amplify additional S-alleles despite crossing evidence for more than 60 S-alleles in Papaver rhoeas alone.

Methodology/Principal Findings

Using RT-PCR we identified 87 unique putative stigmatic S-allele sequences from the Papaveraceae Argemone munita, Papaver mcconnellii, P. nudicuale, Platystemon californicus and Romneya coulteri. Hand pollinations among two full-sib families of both A. munita and P. californicus indicate a strong correlation between the putative S-genotype and observed incompatibility phenotype. However, we also found more than two S-like sequences in some individuals of A. munita and P. californicus, with two products co-segregating in both full-sib families of P. californicus. Pairwise sequence divergence estimates within and among taxa show Papaver stigmatic S-alleles to be the most variable with lower divergence among putative S-alleles from other Papaveraceae. Genealogical analysis indicates little shared ancestral polymorphism among S-like sequences from different genera. Lack of shared ancestral polymorphism could be due to long divergence times among genera studied, reduced levels of balancing selection if some or all S-like sequences do not function in incompatibility, population bottlenecks, or different levels of recombination among taxa. Preliminary estimates of positive selection find many sites under selective constraint with a few undergoing positive selection, suggesting that self-recognition may depend on amino acid substitutions at only a few sites.

Conclusions/Significance

Because of the strong correlation between genotype and SI phenotype, sequences reported here represent either functional stylar S-alleles, tightly linked paralogs of the S-locus or a combination of both. The considerable complexity revealed in this study shows we have much to learn about the evolutionary dynamics of self-incompatibility systems.     

Population dynamics and production of Acanthocyclops robustus (Sars) and Mesocyclops leuckarti (Claus) in Tjeukemeer

The population dynamics and production ecology of the two dominant copepod species, Acanthocyclops robustus and Mesocyclops leuckarti, in Tjeukemeer (The Netherlands) were studied for three successive years. Since copepods in Tjeukemeer show continuous recruitment, a population dynamics model INSTAR was developed and used to integrate field data on population density, population structure and fecundity and laboratory data on development rates and length-weight relationships. The pattern of size specific mortality indicated that both invertebrate and vertebrate predation were important in the regulation of population numbers.     

Genetic analysis of an introduced biological control agent reveals temporal and geographic change,with little evidence of a host mediated shift

The evolution of introduced biological control agents is largely un-explored. Although much is theorized, there is little empirical evidence quantifying the evolutionary dynamics of a biocontrol agent after release into a new environment. In this study we use Diachasmimorpha tryoni, a purposefully introduced biocontrol agent of Ceratitis capitata, to model and quantify spatial, temporal, and host-related evolutionary patterns. This parasitoid has undergone a host shift in its introduced environment, Hawaii, to the gall forming weed biocontrol agent, Eutreta xanthochaeta, an interaction likely mediated by competition for C. capitata with the egg-larval parasitoid Fopius arisanus. To elucidate potential evolutionary patterns we analyzed microsatellites and sequence data extracted from Hawaii and Australian population clusters defined by Structure, in Genepop, Canoco, and IBDWS. Our analysis revealed structuring of Hawaiian D. tryoni populations as defined by significant historic influences related to temporal structure, geographic space, host guild, and augmentative releases. The host-shift parasitoids were not genetically distinct from other Hawaii populations. There were small changes in microsatellite DNA at the population level, but only between Australia and Hawaii populations, not at the host level. These results show that D. tryoni has not undergone host-mediated evolution since introduction to Hawaii, despite the fact that they have expanded their host range in Hawaii to include the gall-forming E. xanthochaeta. To our knowledge this is the first study to quantify genetic differentiation of a biological control agent over geographic space and time using contemporary and museum specimens.     

Contact Density Affects Protein Evolutionary Rate from Bacteria to Animals

The density of contacts or the fraction of buried sites in a protein structure is thought to be related to a protein’s designability, and genes encoding more designable proteins should evolve faster than other genes. Several recent studies have tested this hypothesis but have found conflicting results. Here, we investigate how a gene’s evolutionary rate is affected by its protein’s contact density, considering the four species Escherichia coli, Saccharomyces cerevisiae, Drosophila melanogaster, and Homo sapiens. We find for all four species that contact density correlates positively with evolutionary rate, and that these correlations do not seem to be confounded by gene expression level. The strength of this signal, however, varies widely among species. We also study the effect of contact density on domain evolution in multidomain proteins and find that a domain’s contact density influences the domain’s evolutionary rate. Within the same protein, a domain with higher contact density tends to evolve faster than a domain with lower contact density. Our study provides evidence that contact density can increase evolutionary rates, and that it acts similarly on the level of entire proteins and of individual protein domains. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Persistent distribution functions for the dispersion of structured populations

This paper primarily expounds upon the problem of persistent age-state distribution functions for the dispersion of structured populations. A general model is introduced, based on the following assumptions: 1) the state of an individual of age a is characterized by a set of random variables X₁, X₂,…, X_Q (weight, size, etc.) obeying a phenomenological master equation; 2) the birth function λ depends on the age a' of parents and on the state variables X₁,…, X_Q of the newborns; 3) the mortality function is composed of two additive terms—the first contribution depends only on age while the second contribution depends on the total population density; 4) the population diffuses to avoid crowding. These hypotheses define a nonlinear population model for which time- and space-persistent age-state distribution functions eventually may occur even if the total population density is time- and space-dependent. A biological interpretation of the main results is given in terms of the distribution function of the state vector at birth. In the last part of the paper a generalized model is presented, assuming that the behavior of an individual is described by a system of age-dependent master equations [29].     

Effects of density dependent sex allocation on the dynamics of a simultaneous hermaphroditic population: Modelling and analysis

In this work we present a mathematical model describing the dynamics of a population where sex allocation remains flexible throughout adult life and so can be adjusted to current environmental conditions. We consider that the fractions of immature individuals acquiring male and female sexual roles are density dependent through nonlinear functions of a weighted total population size. The main goal of this work is to understand the role of life-history parameters on the stabilization or destabilization of the population dynamics.The model turns out to be a nonlinear discrete model which is analysed by studying the existence of fixed points as well as their stability conditions in terms of model parameters. The existence of more complex asymptotic behaviours of system solutions is shown by means of numerical simulations.Females have larger fertility rate than males. On the other hand, increasing population density favours immature individuals adopting the male role. A positive equilibrium of the system exists whenever fertility and survival rates of one of the sexual roles, if shared by all adults, allow population growing while the opposite happens with the other sexual role. In terms of the female inherent net reproductive number, η_F, it is shown that the positive equilibria are stable when η_F is larger and closed to 1 while for larger values of η_F a certain asymptotic assumption on the investment rate in the female function implies that the population density is permanent. Depending on the other parameters values, the asymptotic behaviour of solutions becomes more complex, even chaotic. In this setting the stabilization/destabilization effects of the abruptness rate in density dependence, of the survival rates and of the competition coefficients are analysed.     

O2 consumption and acute salinity exposure in the freshwater shrimp Macrobrachium olfersii (Wiegmann) (Crustacea:Decapoda): whole animal and tissue respiration

The time course of O₂ consumption after acute salinity exposure (1, 3, 6, 12, and 24 h to 0, 7, 14, 21, 28, and 35 S) was examined in isolated supraesophageal ganglia, gills, and intact Macrobrachium olfersii (Wiegmann), a hyperosmoregulating freshwater palaemonid shrimp, to establish patterns of metabolic adjustment during salinity adaptation. In whole shrimps, O₂ uptake rates decline with salinity increase to 21 S, subsequently increasing with further salinity increase. The rates increase to maxima after 6–12-h exposure in low salinities, decreasing steadily with time in high salinities. In gill preparations, O₂ consumption rates increase to a maximum in 14 S, then decline; they are maximal after 3–6-h exposure to low salinities and diminish with time in high salinities. In the supraesophageal ganglion, rates of O₂ uptake, always measured in seawater of 18 S, are also maximal when shrimps are exposed to 14 S, subsequently declining or levelling off. Rates decrease with time in shrimps exposed to very low salinities, and are stable in 21 S, reaching maxima after 3–6-h exposure of shrimps to all other media. Both tissues thus exhibit characteristic response patterns of O₂ consumption rate which appear to depend on their functional significance within the context of the whole organism. Such data are interpreted to indicate an interrelationship between O₂ consumption and osmoregulatory capability.