A review of evolutionary graph theory with applications to game theory

Evolutionary graph theory (EGT), studies the ability of a mutant gene to overtake a finite structured population. In this review, we describe the original framework for EGT and the major work that has followed it. This review looks at the calculation of the “fixation probability” - the probability of a mutant taking over a population and focuses on game-theoretic applications. We look at varying topics such as alternate evolutionary dynamics, time to fixation, special topological cases, and game theoretic results. Throughout the review, we examine several interesting open problems that warrant further research.     

The one-third law of evolutionary dynamics

Evolutionary game dynamics in finite populations provide a new framework for studying selection of traits with frequency-dependent fitness. Recently, a "one-third law" of evolutionary dynamics has been described, which states that strategy A fixates in a B-population with selective advantage if the fitness of A is greater than that of B when A has a frequency 13. This relationship holds for all evolutionary processes examined so far, from the Moran process to games on graphs. However, the origin of the "number"13 is not understood. In this paper we provide an intuitive explanation by studying the underlying stochastic processes. We find that in one invasion attempt, an individual interacts on average with B-players twice as often as with A-players, which yields the one-third law. We also show that the one-third law implies that the average Malthusian fitness of A is positive.     

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.     

The analysis of bi-level evolutionary graphs

Evolutionary graphs (EGs), in which evolutionary dynamic is arranged on a graph, were initially proposed by Lieberman et al. [Lieberman, E., Hauert, C., Nowak, M.A., 2005. Evolutionary dynamics on graphs. Nature 433, 312–316] in the biological field and many biological phenomena are successfully explained. EGs on two levels (or bi-level EGs), based on some biological phenomena, are considered in this paper. The bi-level EGs are compared with the one-rooted EGs in two cases. One has the identical numbers of the followers, the other with the same numbers of total individuals. Then, some properties of the bi-level EGs are obtained. It is showed that bi-level EGs are more stable, and the bi-level EGs with just two leaders are the most stable, if they have identical followers respectively. The bi-level EGs theory can successfully explain the phenomena of symbiosis in biology.     

The different limits of weak selection and the evolutionary dynamics of finite populations

Evolutionary theory often resorts to weak selection, where different individuals have very similar fitness. Here, we relate two ways to introduce weak selection. The first considers evolutionary games described by payoff matrices with similar entries. This approach has recently attracted a lot of interest in the context of evolutionary game dynamics in finite populations. The second way to introduce weak selection is based on small distances in phenotype space and is a standard approach in kin-selection theory. Whereas both frameworks are interchangeable for constant fitness, frequency-dependent selection shows significant differences between them. We point out the difference between both limits of weak selection and discuss the condition under which the differences vanish. It turns out that this condition is fulfilled by the popular parametrization of the prisoner's dilemma in benefits and costs. However, for general payoff matrices differences between the two frameworks prevail.     

Directionality of contact networks suppresses selection pressure in evolutionary dynamics

Individuals of different types, may it be genetic, cultural, or else, with different levels of fitness often compete for reproduction and survival. A fitter type generally has higher chances of disseminating their copies to other individuals. The fixation probability of a single mutant type introduced in a population of wild-type individuals quantifies how likely the mutant type spreads. How much the excess fitness of the mutant type increases its fixation probability, namely, the selection pressure, is important in assessing the impact of the introduced mutant. Previous studies mostly based on undirected and unweighted contact networks of individuals showed that the selection pressure depends on the structure of networks and the rule of reproduction. Real networks underlying ecological and social interactions are usually directed or weighted. Here we examine how the selection pressure is modulated by directionality of interactions under several update rules. Our conclusions are twofold. First, directionality discounts the selection pressure for different networks and update rules. Second, given a network, the update rules in which death events precede reproduction events significantly decrease the selection pressure than the other rules.     

The body-size dependence of mutual interference

The parameters that drive population dynamics typically show a relationship with body size. By contrast, there is no theoretical or empirical support for a body-size dependence of mutual interference, which links foraging rates to consumer density. Here, I develop a model to predict that interference may be positively or negatively related to body size depending on how resource body size scales with consumer body size. Over a wide range of body sizes, however, the model predicts that interference will be body-size independent. This prediction was supported by a new dataset on interference and consumer body size. The stabilizing effect of intermediate interference therefore appears to be roughly constant across size, while the effect of body size on population dynamics is mediated through other parameters.     

How is the equilibrium of continuous strategy game different from that of discrete strategy game?

Cooperation in the prisoner's dilemma (PD) played on various networks has been explained by so-called network reciprocity. Most of the previous studies presumed that players can offer either cooperation (C) or defection (D). This discrete strategy seems unrealistic in the real world, since actual provisions might not be discrete, but rather continuous. This paper studies the differences between continuous and discrete strategies in two aspects under the condition that the payoff function of the former is a linear interpolation of the payoff matrix of the latter. The first part of this paper proves theoretically that for two-player games, continuous and discrete strategies have different equilibria and game dynamics in a well-mixed but finite population. The second part, conducting a series of numerical experiments, reveals that such differences become considerably large in the case of PD games on networks. Furthermore, it shows, using the Wilcoxon sign-rank test, that continuous and discrete strategy games are statistically significantly different in terms of equilibria. Intensive discussion by comparing these two kinds of games elucidates that describing a strategy as a real number blunts D strategy invasion to C clusters on a network in the early stage of evolution. Thus, network reciprocity is enhanced by the continuous strategy.     

Investigating saccade programming in the praying mantis Tenodera aridifolia using distracter interference paradigms

To investigate the saccadic system in the mantis, I applied distracter interference paradigms. These involved presenting the mantis with a fixation target and one or several distracters supposed to affect saccades towards the target. When a single target was presented, a medium-sized target located in its lower visual field elicited higher rates of saccade response. This preference for target size and position was also observed when a target and a distracter were presented simultaneously. That is, the mantis chose and fixated the target rather than a distracter that was much smaller or larger than the target, or was located above the target. Furthermore, the mantis' preference was not affected by increasing the number of distracters. However, the presence of the distracter decreased the occurrence rate of saccade and increased the response time to saccade. I conclude that distracter interference paradigms are an effective way of investigating the visual processing underlying saccade generation in the mantis. Possible mechanisms of saccade generation in the mantis are discussed.     

Fixation probability for a beneficial allele and a mutant strategy in a linear game under weak selection in a finite island model

The effect of population structure on the probability of fixation of a newly introduced mutant under weak selection is studied using a coalescent approach. Wright's island model in a framework of a finite number of demes is assumed and two selection regimes are considered: a beneficial allele model and a linear game among offspring. A first-order approximation of the fixation probability for a single mutant with respect to the intensity of selection is deduced. The approximation requires the calculation of expected coalescence times, under neutrality, for lineages starting from two or three sampled individuals. The results are obtained in a general setting without assumptions on the number of demes, the deme size or the migration rate, which allows for simultaneous coalescence or migration events in the genealogy of the sampled individuals. Comparisons are made with limit cases as the deme size or the number of demes goes to infinity or the migration rate goes to zero for which a diffusion approximation approach is possible. Conditions for selection to favor a mutant strategy replacing a resident strategy in the context of a linear game in a finite island population are addressed.     

四川卧龙自然保护区珙桐种群的空间分布格局

采用扩散系数的t检验、Poisson分布的x^2拟合检验及Morisita指数的F检验来研究珙桐种群的空间分布格局类型,用负二项参数（K）、Green指数（GI）和Lloyd的平均拥挤度（m^＊）与聚块性指标（PAI）来研究珙桐的聚集强度。同时利用上述指数研究取样尺度、海拔及克隆繁殖对珙桐种群分布格局和聚集强度的影响。结果表明,总体上珙桐种群的分布格局类型为集群分布,具一定的聚集强度。取样尺度、海拔对其分布格局和聚集强度均有不同程度的影响。基株种群和分株种群的分布格局和聚集强度在不同取样尺度和海拔下出现不同的变化。最后对珙桐种群格局在各种因素影响下的变化原因进行了讨论,认为珙桐种群格局呈集群分布的最主要原因是由其本身的繁殖特性决定的。     

Sexual reproduction, clonal diversity and genetic differentiation in patchily distributed populations of the temperate forest herb Paris quadrifolia (Trilliaceae)

Clonal plant species have been shown to adopt different strategies to persist in heterogeneous environments by changing relative investments in sexual reproduction and clonal propagation. As a result, clonal diversity and genetic variation may be different along environmental gradients. We examined the regional and local population structure of the clonal rhizomatous forest herb Paris quadrifolia in a complex of forest fragments in Voeren (Belgium). Relationships between population size (the number of shoots), shoot density (the number of shoots per m²) and local growth conditions were investigated for 47 populations. Clonal diversity and genetic variation within and among 19 populations were investigated using amplified fragment length polymorphism markers. To assess the importance of sexual reproduction, seed set, seed weight and germination success were determined in 18 populations. As predicted, local growth conditions largely affected population distribution, size and density of P. quadrifolia. Populations occurring in moist and relatively productive sites contained significantly more shoots. Here, shoots were also much more sparsely distributed compared to populations occurring in dry and relatively unproductive sites, where shoots showed a strongly aggregated distribution pattern. Clonal diversity was relatively high, compared with other clonal species (G/N ratio = 0.43 and Simpson’s D=0.81). Clonal diversity significantly (P<0.01) decreased with increasing shoot density while molecular genetic variation was significantly (P<0.01) affected by population size and local environmental conditions. Lack of recruitment and out-competition of less-adapted genotypes may explain the decreased genetic variation in dry sites. Analysis of molecular variance revealed significant genetic variation among populations (Φ _ST=0.42, P<0.001), whereas pairwise genetic distances were not correlated to geographic distances, suggesting that gene flow among populations is limited. Finally, the number of generative shoots, the number of seeds per fruit and seed weight were significantly and positively related to population size and local growth conditions. We conclude that under stressful conditions populations of clonal forest plant species can slowly evolve into remnant populations characterized by low levels of genetic variation and limited sexual reproduction. Conservation of suitable habitat conditions is therefore a prerequisite for effective long-term conservation of clonal forest plant species.     

Phenotypic plasticity and specialization in clonal versus non-clonal plants: A data synthesis

Reproductive strategies can be associated with ecological specialization and generalization. Clonal plants produce lineages adapted to the maternal habitat that can lead to specialization. However, clonal plants frequently display high phenotypic plasticity (e.g. clonal foraging for resources), factors linked to ecological generalization. Alternately, sexual reproduction can be associated with generalization via increasing genetic variation or specialization through rapid adaptive evolution. Moreover, specializing to high or low quality habitats can determine how phenotypic plasticity is expressed in plants. The specialization hypothesis predicts that specialization to good environments results in high performance trait plasticity and specialization to bad environments results in low performance trait plasticity. The interplay between reproductive strategies, phenotypic plasticity, and ecological specialization is important for understanding how plants adapt to variable environments. However, we currently have a poor understanding of these relationships. In this study, we addressed following questions: 1) Is there a relationship between phenotypic plasticity, specialization, and reproductive strategies in plants? 2) Do good habitat specialists express greater performance trait plasticity than bad habitat specialists? We searched the literature for studies examining plasticity for performance traits and functional traits in clonal and non-clonal plant species from different habitat types. We found that non-clonal (obligate sexual) plants expressed greater performance trait plasticity and functional trait plasticity than clonal plants. That is, non-clonal plants exhibited a specialist strategy where they perform well only in a limited range of habitats. Clonal plants expressed less performance loss across habitats and a more generalist strategy. In addition, specialization to good habitats did not result in greater performance trait plasticity. This result was contrary to the predictions of the specialization hypothesis. Overall, reproductive strategies are associated with ecological specialization or generalization through phenotypic plasticity. While specialization is common in plant populations, the evolution of specialization does not control the nature of phenotypic plasticity as predicted under the specialization hypothesis.     

The interaction between mobile DNAs and their hosts in a fluctuating environment

The interaction between mobile DNA sequences and their hosts raises important questions in the context of hosts which reproduce clonally with only rare horizontal transmission between clones. The activity of some mobile DNAs as reversible mutators of genes raises the possibility that, in a fluctuating environment, cells may gain an advantage if they have mobile DNAs which mutate genes whose inactivation is favoured in one of the environments that the population encounters. Here we analyse a model of this process and ask what would be the optimal rate of transposition in a population whose elements are maintained by this mechanism. We also examine the impact of horizontal transfer on such a population. With movement of elements between cells, we can imagine elements with differing rates of transposition and host cells with differing rates of transposition. We find that evolution in the population of elements favours a rapid rate of transposition, and evolution of the host cells favours cells in which this rapid rate of element-dependent transposition results in an optimal rate of transposition per cell. However, when horizontal transfer rates are high, some unexpected features of the model are observed. In particular, a polymorphism between cell types (some with an optimal rate of transposition and some with no transposition at all from endogenous elements) can be stably maintained. We consider the possible biological predictions of this analysis.     

Floral distribution,clonal structure,and their effects on pollination success in a self-incompatible <Emphasis Type="Italic">Convallaria keiskei</Emphasis> population in northern Japan

In plant species, when clonal growth produces a patchy structure and flowering ramets are clustered, the amount of pollen contributing to reproductive success is often regulated by pollinator efficiency and geitonogamy. The spatial population structure may influence reproductive success. We examined the clonal structure, the spatial ramet distribution, and their combined effects on fruit set in a natural population of the insect-pollinated, self-incompatible clonal herb, Convallaria keiskei, in northern Japan. The number of shoots, flowers, and fruits in 1-m² quadrats were counted at every 5 m grid point in an established 100 × 90-m study plot. From all the quadrats where shoots existed, leaf samples were collected for allozyme analysis. Using the two spatial parameters of flowering ramet densities and genotypes, we then constructed individual-based fruit-set models. A total of 236 quadrats contained shoots, and 135 contained flowering ramets, which indicated expanded distribution of this plant throughout the study plot, while shoots, flowers and fruits all showed clustering distributions. Allozyme analysis of 282 samples revealed 94 multilocus genotypes. The largest clone extended to more than 40 m, whereas 56 genotypes were detected in only one sample. Several large clones and many small clones were distributed close to each other. Fine-scale spatial modelling revealed that the neighbouring flower numbers of different genotypes, compared with local genet or flower diversity, more influenced fruit set, in which the range of the neighbour was 14.5 m. These findings indicate that the compatible pollen dispersed by insect pollinators has a significant effect on sexual reproduction, in this C. keiskei population. Consequently, the spatial structure, which includes both genet distribution and clonal expansion by ramets, had a significant effect on pollination success.     

The cellular parameters of leaf development in tobacco: a clonal analysis

The cellular parameters of leaf development in tobacco (Nicotiana tabacum L.) have been characterized using clonal analysis, an approach that provides unequivocal evidence of cell lineage. Our results indicate that the tobacco leaf arises from a group of around 100 cells in the shoot apical meristem. Each of these cells contributes to a unique longitudinal section of the axis and transverse section of the lamina. This pattern of cell lincage indicates that primordial cells contribute more or less equally to the growth of the axis, in contrast to the more traditional view of leaf development in which the leaf is pictured as arising from a group of apical initials. Clones induced prior to the initiation of the lamina demonstrate that the subepidermal layer of the lamina arises from at least six files of cells. Submarginal cells usually divide with their spindles parallel to the margin, and therefore contribute relatively little to the transverse expansion of the lamina. During the expansion of the lamina the orientation and frequency of cell division are highly regulated, as is the duration of meristematic growth. Initially, cell division is polarized so as to produce lineages that are at an oblique angle to the midrib; later cell division is in alternating perpendicular planes. The distribution of clones generated by irradiation at various stages of development indicates that cell division ceases at the tip of the leaf when the leaf is about one tenth its final size, and then ceases in progressively more basal regions of the lamina. Variation in the mutation frequency within the lamina reflects variation in the frequency of mitosis. Prior to the mergence of the leaf the frequency of mutation is maximal near the tip of the leaf and extremely low at its base; after emergence, the frequency of mutation increases at the base of the leaf. In any given region of the lamina the frequency of mutation is highest in interveinal regions, and is relatively low near the margin. Thus, both the orientation and frequency of cell division at the leaf margin indicate that this region plays a minor role in the growth of the lamina.Abbreviation MF mutation frequency     

Environmentally induced clonal reproduction and life history traits of the liana Machaerium cuspidatum in an Amazonian rain forest,Ecuador

The population structure of the canopy liana Machaerium cuspidatum Kuhlm. & Hoehne (Fabaceae) was studied in ten transects covering a total of 2.5 ha of tropical rain forest in Yasuní National Park, eastern Ecuador. The aim was to investigate how trellis availability, topographic position and light availability affected the population density and structure of the most abundant liana species in the area. The environmental variation affected plants in different size classes differently. Large plants (diameter 1 cm) were almost exclusively found in areas with low canopy and dense undergrowth. These areas had the highest density of suitable host plants. Seedling-sized plants (height < 30 cm) and saplings (height 30 cm but diameter < 1 cm) occurred more frequently and withhigher density in steep upland areas than in the floodplains, presumably due to elevated seedling mortality in the periodically flooded areas. Seedling-sized plants and saplings did not grow in areas with elevated light levels, and although plants of all sizes were highly clumped, this clumping rarely coincided with patches with abundant canopy gaps. Machaerium cuspidatum reproduces clonally when branches re-root and break off from the parentplant. The density of clonally produced plants was high in areas where the density of sexually produced plants was low, whereas the total density varied little among habitat types. A higher proportion of the seedling-sized plants were of sexual origin in the upland areas. An environmentally induced increase in the production of independent ramets enables M. cuspidatum to persist in the floodplains. This shows that plants may expand theirrealised niches to habitats with a low potential for seedling survival if high seedling mortality can be compensated for by increased clonal reproduction.     

Strategic analysis in evolutionary genetics and the theory of games

This paper is written in memory of John Maynard Smith. In a brief survey it discusses essential aspects of how game theory in biology relates to its counterpart in economics, the major transition in game theory initiated by Maynard Smith, the discrepancies between genetic and phenotypic models in evolutionary biology, and a balanced way of reconciling these models. In addition, the paper discusses modern problems in understanding games at the genetic level using the examples of conflict between endosymbionts and their hosts, and the molecular interactions between parasites and the mammalian immune system.     

Regeneration capacity and carbohydrate reserves in a clonal plant Rumex alpinus: effect of burial

The morphological and anatomical responses to different depths of burial were examined in Rumex alpinus (Polygonaceae), a perennial plant with monopodial, horizontally growing rhizome. Its segments, which consist of 12–20 internodes, 1 to 2 mm in length each, are products of single growing seasons. The rhizomes regenerated from 5, 10 and 20 cm, but failed to emerge from 30 cm. Number of internodes produced during a growing season was not affected by burial but the length of internodes increased up to about 30-fold. The rhizomes growing up to the surface were subsidized by older rhizome segments. In the case of deeply buried rhizomes the carbohydrate reserves of the last-year-segment were nearly completely depleted. Evolutionary significance of the regeneration capacity is discussed.