Long-term stability from fixation probabilities in finite populations: new perspectives for ESS theory

For mixed strategies in finite populations, long-term stability is defined with respect to the probability of fixation of a mutant. Under weak selection, necessary and sufficient conditions are obtained using a diffusion approximation of the Wright-Fisher model or exact solutions for the Moran model. These differ from the usual ESS conditions if the strategies affect fertility instead of viability, leading to a game matrix depending on the population size, or if the mutant mixed strategy uses a new pure strategy. In this case, the mutant deviation must not exceed some threshold value depending on the population size. In a diploid population, long-term stability may not occur unless there is partial dominance. In the case of sex allocation, continuous stability of an even sex ratio is ascertained. If sex allocation is random, an evolutionary decrease of the variance is predicted.     

Fixation probabilities of random mutants under frequency dependent selection

Evolutionary game dynamics describes frequency dependent selection in asexual, haploid populations. It typically considers predefined strategies and fixed payoff matrices. Mutations occur between these known types only. Here, we consider a situation in which a mutation has produced an entirely new type which is characterized by a random payoff matrix that does not change during the fixation or extinction of the mutant. Based on the probability distribution underlying the payoff values, we address the fixation probability of the new mutant. It turns out that for weak selection, only the first moments of the distribution matter. For strong selection, the probability that a new payoff entry is larger than the wild type's payoff against itself is the crucial quantity.     

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.     

Evolution in heterogeneous populations: From migration models to fixation probabilities

Although dispersal is recognized as a key issue in several fields of population biology (such as behavioral ecology, population genetics, metapopulation dynamics or evolutionary modeling), these disciplines focus on different aspects of the concept and often make different implicit assumptions regarding migration models. Using simulations, we investigate how such assumptions translate into effective gene flow and fixation probability of selected alleles. Assumptions regarding migration type (e.g. source-sink, resident pre-emption, or balanced dispersal) and patterns (e.g. stepping-stone versus island dispersal) have large impacts when demes differ in sizes or selective pressures. The effects of fragmentation, as well as the spatial localization of newly arising mutations, also strongly depend on migration type and patterns. Migration rate also matters: depending on the migration type, fixation probabilities at an intermediate migration rate may lie outside the range defined by the low- and high-migration limits when demes differ in sizes. Given the extreme sensitivity of fixation probability to characteristics of dispersal, we underline the importance of making explicit (and documenting empirically) the crucial ecological/ behavioral assumptions underlying migration models.     

Prediction of additive and dominance effects in selected or unselected populations with inbreeding

Summary A genetic model with either 64 or 1,600 unlinked biallelic loci and complete dominance was used to study prediction of additive and dominance effects in selected or unselected populations with inbreeding. For each locus the initial frequency of the favourable allele was 0.2, 0.5, or 0.8 in different alternatives, while the initial narrow-sense heritability was fixed at 0.30. A population of size 40 (20 males and 20 females) was simulated 1,000 times for five generations. In each generation 5 males and 10 or 20 females were mated, with each mating producing four or two offspring, respectively. Breeding individuals were selected randomly, on own phenotypic performance or such yielding increased inbreeding levels in subsequent generations. A statistical model containing individual additive and dominance effects but ignoring changes in mean and genetic covariances associated with dominance due to inbreeding resulted in significantly biased predictions of both effects in generations with inbreeding. Bias, assessed as the average difference between predicted and simulated genetic effects in each generation, increased almost linearly with the inbreeding coefficient. In a second statistical model the average effect of inbreeding on the mean was accounted for by a regression of phenotypic value on the inbreeding coefficient. The total dominance effect of an individual in that case was the sum of the average effect of inbreeding and an individual effect of dominance. Despite a high mean inbreeding coefficient (up to 0.35), predictions of additive and dominance effects obtained with this model were empirically unbiased for each initial frequency in the absence of selection and 64 unlinked loci. With phenotypic selection of 5 males and only 10 females in each generation and 64 loci, however, predictions of additive and dominance effects were significantly biased. Observed biases disappeared with 1,600 loci for allelic frequencies at 0.2 and 0.5. Bias was due to a considerable change in allelic frequency with phenotypic selection. Ignoring both the covariance between additive and dominance effects with inbreeding and the change in dominance variance due to inbreeding did not significantly bias prediction of additive and dominance effects in selected or unselected populations with inbreeding.     

A compact result for the time-dependent probability of fixation at a neutral locus

A result is derived, in the form of a sum, for the time-dependent probability of fixation of an unlinked neutral locus. The result captures many of the key features of the probability of fixation in a highly compact form. For ‘small’ times (t?4N_e) a single term of the sum accurately determines the time-dependent probability of fixation. This is in contrast to the well-known result of Kimura, which requires the contribution of many terms in a different sum, for ‘small’ times. Going beyond small times, an approximation is derived for the time-dependent probability of fixation which applies for all times when the initial relative allele frequency is small.     

The probability of fixation of a single mutant in an exchangeable selection model

The Cannings exchangeable model for a finite population in discrete time is extended to incorporate selection. The probability of fixation of a mutant type is studied under the assumption of weak selection. An exact formula for the derivative of this probability with respect to the intensity of selection is deduced, and developed in the case of a single mutant. This formula is expressed in terms of mean coalescence times under neutrality assuming that the coefficient of selection for the mutant type has a derivative with respect to the intensity of selection that takes a polynomial form with respect to the frequency of the mutant type. An approximation is obtained in the case where this derivative is a continuous function of the mutant frequency and the population size is large. This approximation is consistent with a diffusion approximation under moment conditions on the number of descendants of a single individual in one time step. Applications to evolutionary game theory in finite populations are presented.      

A numerical method for calculating moments of coalescent times in finite populations with selection

A numerical method is developed for solving a nonstandard singular system of second-order differential equations arising from a problem in population genetics concerning the coalescent process for a sample from a population undergoing selection. The nonstandard feature of the system is that there are terms in the equations that approach infinity as one approaches the boundary. The numerical recipe is patterned after the LU decomposition for tridiagonal matrices. Although there is no analytic proof that this method leads to the correct solution, various examples are presented that suggest that the method works. This method allows one to calculate the expected number of segregating sites in a random sample of n genes from a population whose evolution is described by a model which is not selectively neutral.     

Perturbation expansions of multilocus fixation probabilities for frequency-dependent selection with applications to the Hill-Robertson effect and to the joint evolution of helping and punishment

Natural populations are of finite size and organisms carry multilocus genotypes. There are, nevertheless, few results on multilocus models when both random genetic drift and natural selection affect the evolutionary dynamics. In this paper we describe a formalism to calculate systematic perturbation expansions of moments of allelic states around neutrality in populations of constant size. This allows us to evaluate multilocus fixation probabilities (long-term limits of the moments) under arbitrary strength of selection and gene action. We show that such fixation probabilities can be expressed in terms of selection coefficients weighted by mean first passages times of ancestral gene lineages within a single ancestor. These passage times extend the coalescence times that weight selection coefficients in one-locus perturbation formulas for fixation probabilities. We then apply these results to investigate the Hill-Robertson effect and the coevolution of helping and punishment. Finally, we discuss limitations and strengths of the perturbation approach. In particular, it provides accurate approximations for fixation probabilities for weak selection regimes only (Ns?1), but it provides generally good prediction for the direction of selection under frequency-dependent selection.     

Environmental variation, fluctuating selection and genetic drift in subdivided populations

Although there have many studies of the population genetical consequences of environmental variation, little is known about the combined effects of genetic drift and fluctuating selection in structured populations. Here we use diffusion theory to investigate the effects of temporally and spatially varying selection on a population of haploid individuals subdivided into a large number of demes. Using a perturbation method for processes with multiple time scales, we show that as the number of demes tends to infinity, the overall frequency converges to a diffusion process that is also the diffusion approximation for a finite, panmictic population subject to temporally fluctuating selection. We find that the coefficients of this process have a complicated dependence on deme size and migration rate, and that changes in these demographic parameters can determine both the balance between the dispersive and stabilizing effects of environmental variation and whether selection favors alleles with lower or higher fitness variance.     

一类互惠生态模型解的周期性和爆破

研究了变系数两种群互惠模型解的周期性和爆破性.首先利用上下解和单调迭代方法研究了互惠模型周期解的存在性、稳定性和吸引性,结果表明种群之间的竞争强于互惠时周期解存在.接着利用比较原理得到了解在有限时刻爆破的充分条件,结果表明种群之间的互惠强于之间的竞争时爆破发生.最后通过数值模拟验证了定理的正确性.     

Conditions for the invasion of male‐haploidy in diploid populations

Male‐haploidy has independently evolved several times in different phylogenetic groups and has led to various extant lineages in the insects, Arachnida and Rotifera. Although the stability of male‐haploidy as an evolutionary strategy is not well understood, various theories address the invasion of male‐haploidy in diploid populations. Here two of these theories: (i) the maternal transmission hypothesis (MTH) and (ii) the deleterious mutation hypothesis (DMH), are re‐investigated with an agent‐based model to understand the role of genetic drift as a mechanism facilitating the spread of male‐haploidy. These two hypotheses are analysed separately and comparatively, and the results suggest dominance of the MTH. In addition, comparison of the stochastic results to deterministic results using the same model structure shows how genetic drift can enhance the parameter space where male‐haploidy can be expected to invade.     

Effect of mass selection on the within-family genetic variance in finite populations

Summary The adequacy of an expression for the withinfamily genetic variance under pure random drift in an additive infinitesimal model was tested via simulation in populations undergoing mass selection. Two hundred or one thousand unlinked loci with two alleles at initial frequencies of 1/2 were considered. The size of the population was 100 (50 males and 50 females). Full-sib matings were carried out for 15 generations with only one male and one female chosen as parents each generation, either randomly or on an individual phenotypic value. In the unselected population, results obtained from 200 replicates were in agreement with predictions. With mass selection, within-family genetic variance was overpredicted by theory from the 12th and 4th generations for the 1,000 and 200 loci cases, respectively. Taking into account the observed change in gene frequencies in the algorithm led to a much better agreement with observed values. Results for the distribution of gene frequencies and the withinlocus genetic covariance are presented. It is concluded that the expression for the within-family genetic variance derived for pure random drift holds well for mass selection within the limits of an additive infinitesimal model.     

Joint effect of selection, linkage and partial inbreeding on additive genetic variance in an infinite population

Under the inifinitesimal model of gene effects, selection reduces the additive genetic variance by inducing negative linkage disequilibrium among selected genes. If the selected genes are linked, the decay of linkage disequilibrium is delayed, and the reduction of additive genetic variance is enhanced. Inbreeding in an infinite population also alters the additive genetic variance through the generation of positive association among genes within a locus. In the present study, the joint effect of selection, linkage and partial inbreeding (partial selfing or partial full-sib mating) on the additive genetic variance was modeled. The recurrence relations of the additive genetic variance between successive generations and the prediction equation of the asymptotic additive genetic variance were derived. Numerical computation showed that although partially inbred populations initially maintain larger genetic variances, the accumulated effect of selection overrides the effect of inbreeding. Stochastic simulation was carried out to check the precision of prediction, showing that the obtained equations give a satisfactory prediction during initial generations. However, the predicted values always overestimate the simulated values, especially in later generations. Based on these results, possible extensions and perspectives of the assumed model were discussed.     

Evolution of linkage disequilibrium of the founders in exponentially growing populations

Evolution of linkage disequilibrium of the founders in exponentially growing populations was studied using a time-inhomogeneous It? process model. The model is an extension of the diffusion approximation of the Wright-Fisher model. As a measure of linkage disequilibrium, the squared standard linkage deviation, which is defined by a ratio of the moments, was considered. A system of ordinary differential equations that these moments obey was obtained. This system can be solved numerically. By simulations, it was shown that the squared standard linkage deviation gives a good approximation of the expectation of the squared correlation coefficient of gamete frequencies. In addition, a perturbative solution was obtained when the growth rate is not large. By using the perturbation, an asymptotic formula for the squared standard linkage deviation after a large number of generations was obtained. According to the formula, the squared standard linkage deviation tends to be 1/(4Nc), where N is the current size of the population and c is the recombination fraction between two loci. It is dependent on neither the initial effective size, the growth rate, nor the mutation rate. In exponentially growing populations, linkage disequilibrium will be asymptotically the same as that in a constant size population, the effective size of which is the current effective size.     

A general approach to proving the minimality of phylogenetic trees illustrated by an example with a set of 23 vertebrates

Summary We have recently described a method of building phylogenetic trees and have outlined an approach for proving whether a particular tree is optimal for the data used. In this paper we describe in detail the method of establishing lower bounds on the length of a minimal tree by partitioning the data set into subsets. All characters that could be involved in duplications in the data are paired with all other such characters. A matching algorithm is then used to obtain the pairing of characters that reveals the most duplications in the data. This matching may still not account for all nucleotide substitutions on the tree. The structure of the tree is then used to help select subsets of three or more. characters until the lower bound found by partitioning is equal to the length of the tree. The tree must then be a minimal tree since no tree can exist with a length less than that of the lower bound.The method is demonstrated using a set of 23 vertebrate cytochrome c sequences with the criterion of minimizing the total number of nucleotide substitutions. There are 131130 7045768798 9603440625 topologically distinct trees that can be constructed from this data set. The method described in this paper does identify 144 minimal tree variants. The method is general in the sense that it can be used for other data and other criteria of length. It need not however always be possible to prove a tree minimal but the method will give an upper and lower bound on the length of minimal trees.     

Relative effects of segregation and recombination on the evolution of sex in finite diploid populations

The mechanism of reproducing more viable offspring in response to selection is a major factor influencing the advantages of sex. In diploids, sexual reproduction combines genotype by recombination and segregation. Theoretical studies of sexual reproduction have investigated the advantage of recombination in haploids. However, the potential advantage of segregation in diploids is less studied. This study aimed to quantify the relative contribution of recombination and segregation to the evolution of sex in finite diploids by using multilocus simulations. The mean fitness of a sexually or asexually reproduced population was calculated to describe the long-term effects of sex. The evolutionary fate of a sex or recombination modifier was also monitored to investigate the short-term effects of sex. Two different scenarios of mutations were considered: (1) only deleterious mutations were present and (2) a combination of deleterious and beneficial mutations. Results showed that the combined effects of segregation and recombination strongly contributed to the evolution of sex in diploids. If deleterious mutations were only present, segregation efficiently slowed down the speed of Muller''s ratchet. As the recombination level was increased, the accumulation of deleterious mutations was totally inhibited and recombination substantially contributed to the evolution of sex. The presence of beneficial mutations evidently increased the fixation rate of a recombination modifier. We also observed that the twofold cost of sex was easily to overcome in diploids if a sex modifier caused a moderate frequency of sex.     

三种群互惠模型抛物系统解的整体存在与爆破

研究了三种群互惠模型的抛物系统,用上下解方法研究解的整体存在性与爆破问题,并给出了相应的条件.结果表明当种群自身的竞争强时解整体存在,反之则有可能爆破.