Maximization principles for frequency-dependent selection II: the one-locus multiallele case

In this article we study a single-locus multiallele version of the pairwise-interaction model (PIM) in discrete and continuous time and a density-dependent version of this model (D-PIM) in continuous time. The PIM assumes that the fitnesses of genotypes are proportional to the average amount of competition resulting from pairwise interactions. Hence, fitness is frequency dependent. Our main aim is to provide necessary and sufficient conditions for the validity of maximization principles analogous to Fisher’s Fundamental Theorem for constant selection. We provide a systematic analysis and illustrate our results by concrete examples. We show that in discrete time the mean fitness is nondecreasing along every trajectory provided the interaction coefficients are nonnegative and symmetric. For asymmetric interactions this is in general not true. However, for what we call pseudo-symmetric interactions a function similar to, but in general not identical to, the mean fitness: the adjusted-mean fitness, is nondecreasing along trajectories. For asymmetric interactions, we also provide sufficient conditions for the mean fitness, and more generally for the adjusted-mean fitness, to be nondecreasing and sufficient conditions when it is not. In continuous time, we provide similar but stronger results. If the interaction coefficients are pseudo-symmetric, the adjusted-mean fitness is nondecreasing in the D-PIM.     

Stochastic fluctuations in frequency-dependent selection: a one-locus, two-allele and two-phenotype model

Stochastic fluctuations in a simple frequency-dependent selection model with one-locus, two-alleles and two-phenotypes are investigated. The steady-state statistics of allele frequencies for an interior stable phenotypic equilibrium are shown to be similar to the stochastic fluctuations in standard evolutionary game dynamics [Tao, Y., Cressman, R., 2007. Stochastic fluctuations through intrinsic noise in evolutionary game dynamics. Bull. Math. Biol. 69, 1377-1399]. On the other hand, for an interior stable phenotypic or genotypic equilibrium, our main results show that the deterministic model cannot be used to predict the expectation of phenotypic frequency. The variance of phenotypic frequency for an interior stable genotypic equilibrium is more sensitive to the expected population size than for an interior stable phenotypic equilibrium. Furthermore, the stochastic fluctuations of allele frequency and phenotypic frequency can be considered approximately independent of each other for these genotypic equilibria, but not for phenotypic.     

Complex dynamics occur in a single-locus, multiallelic model of general frequency-dependent selection

We examine the characteristics of non-equilibrium dynamics produced by a simple well-known model of frequency-dependent selection at a single diploid locus. An examination of the parameter space of this “pairwise-interaction model” (PIM) revealed non-equilibrium dynamics for polymorphisms of 3, 4 and 5 alleles; both allele-frequency cycling and aperiodic trajectories were detected. We measured the number, cycle length and domains of attraction of the various attractors produced by the model. The domains of attraction tended to be smaller, and the cycles longer, for systems with larger number of alleles. Fitnesses that parametrized negative frequency-dependent selection were more likely to allow cycling, and these cycles also had larger domains of attraction. Aperiodic trajectories were detected only in cases with 4 or 5 alleles. The genetic cycles produced by the model do not have periods as short as those predicted in ecological models with cycling (such as predator–prey population cycles, etc.). Consequently, in a real-world system, PIM allele-frequency cycling is likely to be indistinguishable from stable equilibria when observed over short time scales.     

Discrete polymorphisms due to disruptive selection on a continuous trait--I: the one-locus case

We have investigated, numerically and analytically, long-term evolution under frequency-dependent disruptive selection of a continuous trait varying in a finite range and controlled by one diploid mendelian locus. We found that evolution converges towards a unique long-term equilibrium where only two extreme phenotypes are present with frequencies identical to those of the mixed strategy that would be the unique ESS of the game defined by the basic fitness function of the model. As long as this precise phenotypic composition is preserved, any genetic configuration of the polymorphism is equally acceptable (selectively neutral) at the equilibrium. Thus the number of alleles and their dominance pattern may vary considerably among different equilibrium populations. If genetic expression of the trait is variable but the amount of variability is genetically modifiable, disruptive selection, acting on such modifiers, produces a steady increase of expression variability before the equilibrium is attained. In this case a population at the long-term equilibrium might even be genetically monomorphic, with the phenotypic dimorphism resulting from purely random individual variation.     

Effects of genetic architecture on the evolution of assortative mating under frequency-dependent disruptive selection

We consider a model of sympatric speciation due to frequency-dependent competition, in which it was previously assumed that the evolving traits have a very simple genetic architecture. In the present study, we numerically analyze the consequences of relaxing this assumption. First, previous models assumed that assortative mating evolves in infinitesimal steps. Here, we show that the range of parameters for which speciation is possible increases when mutational steps are large. Second, it was assumed that the trait under frequency-dependent selection is determined by a single locus with two alleles and additive effects. As a consequence, the resultant intermediate phenotype is always heterozygous and can never breed true. To relax this assumption, here we add a second locus influencing the trait. We find three new possible evolutionary outcomes: evolution of three reproductively isolated species, a monomorphic equilibrium with only the intermediate phenotype, and a randomly mating population with a steep unimodal distribution of phenotypes. Both extensions of the original model thus increase the likelihood of competitive speciation.     

The influence of density on frequency-dependent food selection: a comparison of four experiments with wild birds

We compare the results of four experiments, conducted at different times and with different protocols, that explored the relationship between frequency-dependent selection and prey density in wild birds feeding on artificial populations of coloured baits. One (experiment 4) used pastry baits that differed only in the presence or absence of a red stripe, and this experiment provided no evidence for any kind of selective behaviour. The other three experiments used green and brown baits, and they all provided evidence for a trend towards increasing anti-apostatic selection with high densities (>100 baits m^–2). However, one of these (experiment 3) provided no evidence for frequency-dependent selection at low densities (0.5–20 baits m^–2), while the other two experiments concurred in suggesting a trend towards increasing apostatic selection with low densities (down to 2 baits m^–2). Together, these experiments both support and qualify the published findings of experiment 1 that frequency- dependent selection by wild birds on bait populations is modified by density. Experiment 4 indicates that frequency-dependent selection may break down entirely if bait types are too similar, while experiment 3 indicates that some details of this trend with density will depend either on the protocol used or on exogenous changes in the birds’ feeding behaviour. Received: 1 September 1999 / Accepted: 22 March 2000     

Impact of density and disease on frequency-dependent selection and genetic polymorphism: experiments with stripe rust and wheat

Frequency-dependent disease impacts may contribute to the maintenance of genetic diversity and sexual reproduction in plant populations. In earlier work with experimental wheat (Triticum aestivum) populations at a single density, we found that stripe rust (caused by Puccinia striiformis) created frequency-dependent selection on its host but competitive interactions between host genotypes reduced the potential for disease to maintain genetic polymorphisms in this highly self-pollinated species; the weaker competitor actually exhibited positive disease-mediated frequency-dependent selection. Based on these results we predicted that at low density, where the overall level of competition is lower, disease would have a stronger impact relative to competition and thus be more likely to maintain genetic polymorphisms; at low densities the greatest effect of disease for negative frequency-dependent selection should be seen in the weak competitor. Here we report on results with wheat stripe rust in which we altered both the frequency and density of host genotypes in factorial combinations of two-way mixtures where each host genotype was attacked by its own specialized race of rust. Within each density disease levels increased with genotype frequencies, creating frequency-dependent disease attack at all densities. Similarly, disease created negative frequency-dependent selection on its host at all densities, as a genotype’s fitness was often greater at low than high frequency when disease was present. Disease levels increased with plant density in 1997 but decreased in 1998. While increasing plant density reduced absolute fitness, presumably as a result of increased competition, a genetic polymorphism was not more likely to be maintained at low than high density as we had predicted. Within each density, the impact of disease was insufficient to reverse the slope of the relationship between absolute fitness and planted frequency from positive to negative for the less competitive host genotype, thus preventing the maintenance of a genetic polymorphism.     

Differentiating the effects of origin and frequency in reciprocal transplant experiments used to test negative frequency-dependent selection hypotheses

Reciprocal transplant experiments have been used to estimate the probability that negative frequency-dependent selection by natural enemies has occurred in host populations by determining whether pest populations are less adapted to “foreign” (rare) hosts, which originate from a population with which the pests have not coevolved. However, these experiments usually confound the effects of frequency and origin: the rare genotypes are also genotypes that did not originate at a site. When clonal organisms are used, and the clones occur in more than one population, it is possible to separate the effects of origin and frequency. Here I present the results of an experiment in which Arabis clones of known frequency were reciprocally transplanted among sites. Contrary to expectations, clones at their site of origin had less disease, less herbivory, and higher fitness than foreign clones. However, variation within and among sites in herbivory and infection was large, suggesting that the number of sites and clones needed to thoroughly test the hypothesis of negative frequency-dependent selection in this system is very large: thus, these results are suggestive but not conclusive. Received: 20 October 1997 / Accepted: 8 February 1998     

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.     

A rare model limits the distribution of its more common mimic: a twist on frequency-dependent Batesian mimicry

Batesian mimics are predicted to lose their fitness advantage not only in the absence of an unpalatable model, but also when the mimic becomes relatively abundant. The phenotypic hybrid zone between mimetic and nonmimetic admiral butterflies, comprising the polytypic Limenitis arthemis species complex, offers an ideal opportunity to test these predictions because the position of the hybrid zone is hypothesized to be controlled by the geographic range of Battus philenor , the chemically defended model. We used 29 years of observational field data from a continental-scale butterfly monitoring program, the 4th of July Butterfly Counts, to show that (1) the advantage of mimicry does not extend beyond the range of the model, (2) in contrast to expectations, the mimicry complex is maintained even where the model is rare and (3) the sharp phenotypic transition between mimetic and nonmimetic admiral populations occurs over a very narrow spatial scale corresponding to the limit of the model's range. These results suggest that, even at very low densities, there is selection for Batesian mimicry and it maintains the geographic position of this hybrid zone. Our findings highlight the value of large-scale, long-term citizen science monitoring programs for answering basic ecological and evolutionary questions.     

Evidence for overdominant selection maintaining X-linked fitness variation in Drosophila melanogaster

The role of balancing selection in maintaining genetic variation for fitness is largely unresolved. This reflects the inherent difficulty in distinguishing between models of recurrent mutation versus selection, which produce similar patterns of inbreeding depression, as well as the limitations of testing such hypotheses when fitness variation is averaged across the genome. Signatures of X-linked overdominant selection are less likely to be obscured by mutational variation because X-linked mutations are rapidly eliminated by purifying selection in males. Although models maintaining genetic variation for fitness are not necessarily mutually exclusive, a series of predictions for identifying X-linked overdominant selection can be used to separate its contribution from other underlying processes. We consider the role of overdominant selection in maintaining fitness variation in a sample of 12 X chromosomes from a population of Drosophila melanogaster. Substantial variation was observed for male reproductive success and female fecundity, with heterozygous-X genotypes exhibiting the greatest degree of variance, a finding that agrees well with predictions of the overdominance model. The importance of X-linked overdominant selection is discussed along with models of recurrent mutation and sexually antagonistic selection.     

Spatially extended host-parasite interactions: the role of recovery and immunity

Techniques for determining the long-term dynamics of host-parasite systems are well established for mixed populations. The field of spatial modelling in ecology is more recent but a number of key advances have been made. In this paper, we use state-of-the-art approximation techniques, supported by simulations, in order to investigate the role of recovery and immunity in spatially structured populations. Our approach is to use correlation models, namely pair-wise models, to capture the spatial relationships of contacts and interactions between individuals. We use the pair-wise framework to address a number of key ecological questions; including, the persistence of endemic limit cycles and regions of parasite-driven extinction--features which differentiate spatial from non-spatial models--and the effects on invasion fitness. We demonstrate a loss of limit cycle behaviour, in addition to an increase in the critical transmissibility and extinction thresholds, when recovery is included. This approach allows for a better analytical understanding of the dynamics of host-parasite interactions and demonstrates the importance of recovery and immunity in local interactions.     

Diffusion approximations for one-locus multi-allele kin selection,mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics

Diffusion approximations are ascertained from a two-time-scale argument in the case of a group-structured diploid population with scaled viability parameters depending on the individual genotype and the group type at a single multi-allelic locus under recurrent mutation, and applied to the case of random pairwise interactions within groups. The main step consists in proving global and uniform convergence of the distribution of the group types in an infinite population in the absence of selection and mutation, using a coalescent approach. An inclusive fitness formulation with coefficient of relatedness between a focal individual J affecting the reproductive success of an individual I, defined as the expected fraction of genes in I that are identical by descent to one or more genes in J in a neutral infinite population, given that J is allozygous or autozygous, yields the correct selection drift functions. These are analogous to the selection drift functions obtained with pure viability selection in a population with inbreeding. They give the changes of the allele frequencies in an infinite population without mutation that correspond to the replicator equation with fitness matrix expressed as a linear combination of a symmetric matrix for allozygous individuals and a rank-one matrix for autozygous individuals. In the case of no inbreeding, the mean inclusive fitness is a strict Lyapunov function with respect to this deterministic dynamics. Connections are made between dispersal with exact replacement (proportional dispersal), uniform dispersal, and local extinction and recolonization. The timing of dispersal (before or after selection, before or after mating) is shown to have an effect on group competition and the effective population size. In memory of Sam Karlin.     

Genetic variation in a host-parasite association: potential for coevolution and frequency-dependent selection

Abstract.— Models of host‐parasite coevolution assume the presence of genetic variation for host resistance and parasite infectivity, as well as genotype‐specific interactions. We used the freshwater crustacean Daphnia magna and its bacterial microparasite Pasteuria ramosa to study genetic variation for host susceptibility and parasite infectivity within each of two populations. We sought to answer the following questions: Do host clones differ in their susceptibility to parasite isolates? Do parasite isolates differ in their ability to infect different host clones? Are there host clone‐parasite isolate interactions? The analysis revealed considerable variation in both host resistance and parasite infectivity. There were significant host clone‐parasite isolate interactions, such that there was no single host clone that was superior to all other clones in the resistance to every parasite isolate. Likewise, there was no parasite isolate that was superior to all other isolates in infectivity to every host clone. This form of host clone‐parasite isolate interaction indicates the potential for coevolution based on frequency‐dependent selection. Infection success of original host clone‐parasite isolate combinations (i.e., those combinations that were isolated together) was significantly higher than infection success of novel host clone‐parasite isolate combinations (i.e., those combinations that were created in the laboratory). This finding is consistent with the idea that parasites track specific host genotypes under natural conditions. In addition, correspondence analysis revealed that some host clones, although distinguishable with neutral genetic markers, were susceptible to the same set of parasite isolates and thus probably shared resistance genes.     

Riding across the selection landscape: fitness consequences of annual variation in reproductive characteristics

Evolutionary models estimating phenotypic selection in character size usually assume that the character is invariant across reproductive bouts. We show that variation in the size of reproductive traits may be large over multiple events and can influence fitness in organisms where these traits are produced anew each season. With data from populations of two orchid species, Caladenia valida and Tolumnia variegata, we used Bayesian statistics to investigate the effect on the distribution in fitness of individuals when the fitness landscape is not flat and when characters vary across reproductive bouts. Inconsistency in character size across reproductive periods within an individual increases the uncertainty of mean fitness and, consequently, the uncertainty in individual fitness. The trajectory of selection is likely to be muddled as a consequence of variation in morphology of individuals across reproductive bouts. The frequency and amplitude of such changes will certainly affect the dynamics between selection and genetic drift.     

Human life histories and the demographic transition: a case study from Finland, 1870-1949

The life histories of humans who were engaged in reproduction during the demographic transition were investigated. It was discovered that these life histories were subject to great changes during the period involving the birth cohorts from the years 1870-1949. Although the number of all and surviving children decreased during this period, the individual fitness values (lambda) of females remained quite even. The lambda values are sensitive not only to reproductive quantity but also to the timing of reproduction. Therefore, the effective change in female fitness during the demographic transition may not be as dramatic as previously thought. When studying the level of selection (or rather the opportunity for selection), it was found that mortality selection steadily decreased to a very low level. However, fertility selection and total selection, which were relatively low for the cohorts 1870-1889, increased before the steep decrease that was detected for the cohorts 1930-1949. The situation reflects the presence of considerable variance in fertility for the cohorts 1890-1929, when the mean fertility was decreasing. A previously found trade-off between female longevity and reproductive success appeared less significant, apparently due to the presence of more plentiful resources and voluntary limitation of reproduction. The deviation from the potential fitness maximization and the presence of subfertility have become prevalent in human populations.     

Testing for local adaptation in Avena barbata: a classic example of ecotypic divergence

Forty years ago, Robert Allard and colleagues documented that the slender wild oat, Avena barbata , occurred in California as two multi-locus allozyme genotypes, associated with mesic and xeric habitats. This is arguably the first example of ecotypes identified by molecular techniques. Despite widespread citation, however, the inference of local adaptation of these ecotypes rested primarily on the allozyme pattern. This study tests for local adaptation of these ecotypes using reciprocal transplant and quantitative trait locus (QTL) mapping techniques. Both ecotypes and 188 recombinant inbred lines (RILs) derived from a cross between them were grown in common garden plots established at two sites representative of the environments in which the ecotypes were first described. Across four growing seasons at each site, three observations consistently emerged. First, despite significant genotype by environment interaction, the mesic ecotype consistently showed higher lifetime reproductive success across all years and sites. Second, the RILs showed no evidence of a trade-off in performance across sites or years, and fitness was positively correlated across environments. Third, at QTL affecting lifetime reproductive success, selection favoured the same allele in all environments. None of these observations are consistent with local adaptation but suggest that a single genotype is selectively favoured at both moist and dry sites. I propose an alternative hypothesis that A. barbata may be an example of contemporary evolution – whereby the favoured genotype is spreading and increasing in frequency – rather than local adaptation.     

The importance of the timescale of the fitness metric for estimates of selection on phenotypic traits during a period of demographic change

Although fitness is central to the evolutionary process, metrics vary by timescale. Different timescales may give rise to different estimates of selection, especially during demographic transitions caused by rapid environmental and socioeconomic change. In this study, we used a dataset of a human population in Finland from 1775 to 1950 to compare two fitness metrics and their estimates of selection pressures, before and during a demographic transition. Both metrics, lifetime reproductive success and an annual metric of individual performance, declined while selection on the ages at first and last reproduction remained nearly constant, favouring individuals with wider reproductive windows. The ability to partition the annual metric into contributions from reproduction and survival revealed the short‐term effects of a famine and the reversal of selection pressure via the survival component of annual fitness. Although the metrics generally agreed, the annual metric detected the effects of environmental variation and demographic change occurring within a generation.     

Sexual conflict and sexual selection: lost in the chase

The emergent field of evolutionary biology that studies disparities between the evolutionary interests of alleles expressed in the two sexes, or sexual conflict, promises to offer novel insights into male-female coevolution and speciation. Our theoretical understanding of basic concepts is, however, still incomplete. In a recent perspective paper, Pizzari and Snook provided a framework for understanding sexually antagonistic coevolution and for distinguishing this process from other models of male-female coevolution and suggested an experimental protocol to test for sexually antagonistic coevolution. Here, I show that the framework is flawed, primarily because it is built upon the mistaken assumption that male and female fitness can evolve independently. Further, while the empirical strategy advocated has indeed offered important insights in the past, it does not allow unambiguous discrimination between competing hypotheses.     

Accumulation of deleterious mutations: additional Drosophila melanogaster estimates and a simulation of the effects of selection

We report an assay of egg-to-adult viability in full-sibling mutation accumulation (MA) lines derived from a completely homozygous population of Drosophila melanogaster and maintained for 210 generations. A simultaneous evaluation was also made of a large population derived from the same origin and maintained as a control for the same period. We also present computer simulations to explore the possible decline in viability of the control population due to mutation accumulation and the possible effect of selection within and between MA lines. For this purpose, we used two mutational models independent from the data analyzed and based on radically different assumptions. The first model implies a large number of mutations of small effect, whereas the second implies a much smaller number of mutations with much larger effects. The observed rate of decline in mean viability was very small but significant (0.077%). The rate of increase in among line variance (0.189 x 10(-3)) was similar to those obtained previously in the same lines. The simulation results indicated that a model of many mutations of small effect is incompatible with the evolution of the mean viability of the control and MA lines over generations, the distribution of line means after 210 generations of mutation accumulation, and the pattern of line extinction over generations. Basically, this model predicted a large drop in viability, both in the control and particularly the MA lines, that is not observed empirically. It also predicted a rate of line extinction too low in the early generations and too high in the later ones. In contrast, the model based on few mutations of large effect was generally consistent with all the observations.