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.     

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.     

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.     

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.     

A selection strategy to accommodate genotype-by-environment interaction for grain yield of wheat: managed-environments for selection among genotypes

Selection for grain yield among wheat lines is complicated by large line-by-environment (L × E) interactions in Queensland, Australia. Early generation selection is based on an evaluation of many lines in a few environments. The small sample of environments, together with the large L × E interaction, reduces the realised response to selection. Definition of a series of managed-environments which provides discrimination among lines, which is relevant to the target production-environments, and can be repeated over years, would facilitate early generation selection. Two series of managed-environments were conducted. Eighteen managed-environments were generated in Series-1 by manipulating nitrogen and water availability, together with the sowing date, at three locations. Nine managed-environments based on those from Series-1 were generated in Series-2. Line discrimination for grain yield in the managed-environments was compared to that in a series of 16 random production-environments. The genetic correlation between line discrimination in the managed-environments and that in the production-environments was influenced by the number and combination of managed-environments. Two managed-environment selection regimes, which gave a high genetic correlation in both Series-1 and 2, were identified. The first used three managed-environments, a high input (low water and nitrogen stress) environment with early sowing at three locations. The second used six managed-environments, a combination of a high input (low water and nitrogen stress) and medium input (water and nitrogen stress) with early sowing at three locations. The opportunities for using managed-environments to provide more reliable selection among lines in the Queensland wheat breeding programme and its potential limitations are discussed.     

Density-dependent habitat selection: Testing the theory with fitness data

Summary According to density-dependent habitat selection theory, reproductive success should decline with increased density. Fitness should be similar between habitats if habitat selection follows an ideal free distribution; fitness should be dissimilar between habitats if habitat selection is modified by territorial behavior. I tested these assumptions by examining a variety of fitness estimates obtained from white-footed mice living in nest boxes in forest, forest edge and fencerow habitats in southwestern Ontario. As expected, mean litter size declined with increased population density. Litter sizes, adult longevity and the proportion of adult animals in breeding condition were not significantly different among the three habitats. The success at recruiting at least one offspring to the adult population and the number of recruits per litter were much greater in the forest than in either of the other two habitats. Fitness was thus unequal among habitats and the results confirm both assumptions of density-dependent habitat selection theory for territorial white-footed mice.     

On the evolution of epistasis I: diploids under selection

One interpretation of recent literature on the evolution of phenotypic modularity is that evolution should act to decrease the degree of interaction between genes that contribute to different phenotypes. This issue is addressed directly here using a fitness scheme determined by two genetic loci and a third locus which modifies a measure of statistical interaction between the fitnesses due to the first two. The equilibrium structure of such an epistasis-modifying locus is studied. It is shown that under well-specified conditions a modifying allele that increases epistasis succeeds. In other words, genetic interactions tend to become stronger. It is speculated that this occurs because the mean fitness in such models is locally increasing as a function of the degree of epistasis.     

Inbreeding reveals stronger net selection on Drosophila melanogaster males: implications for mutation load and the fitness of sexual females

Stronger selection on males has the potential to lower the deleterious mutation load of females, reducing the cost of sex. However, few studies have directly quantified the strength of selection for both sexes. As the magnitude of inbreeding depression (ID) is related to the strength of selection, we measured the cost of inbreeding for both males and females in a laboratory population of Drosophila melanogaster. Using a novel technique for inbreeding, we found significant ID for both juvenile viability and adult fitness in both sexes. The genetic variation responsible for this depression in fitness appeared to be recessive for adult fitness (h=0.11) and partially additive for juvenile viability (h=0.29). ID was identical across the sexes in terms of juvenile viability but was significantly more deleterious for males than females as adults, even though female X-chromosome homogamety should predispose them to a higher inbreeding load. We estimated the strength of selection on adult males to be 1.24 greater than on adult females, and this appears to be a consequence of selection arising from competition for mates. Combined with the generally positive intersexual genetic correlation for inbred lines, our results suggest that the mutation load of sexual females could be meaningfully reduced by stronger selection acting on males.     

Sexual selection and non-random mating for shell colour in a natural population of the marine snailLittorina mariae (Gastropoda: Prosobranchia)

In order to estimate the three independent components of mating behaviour, sexual selection in females, sexual selection in males and mating pattern, we studied the distribution of shell colour morphs among mating pairs and between copulating and non-copulating snails in four subsamples of a natural population ofL. mariae. The colour of the shell, the sex and a qualitative estimate of age was recorded for every snail. We found sexual selection acting against one of the two commonest colours (yellow) among the young females. However, in males none of the eight shell colour morphs was favoured during matings. Male sexual choice or differences in female sexual activity may cause the sexual fitness disadvantage of yellow females. Moreover, individuals of different colour morphs did not mate at random, rather dissasortatively. A behavioural choice among shell colour morphs or a non-random microdistribution of the morphs may cause the departure from random mating in this population.     

Long-term genomic selection for heterosis without dominance in multiplicative traits: case study of bunch production in oil palm

Background

To study the potential of genomic selection for heterosis resulting from multiplicative interactions between additive and antagonistic components, we focused on oil palm, where bunch production is the product of bunch weight and bunch number. We simulated two realistic breeding populations and compared current reciprocal recurrent selection (RRS) with reciprocal recurrent genomic selection (RRGS) over four generations. All breeding strategies aimed at selecting the best individuals in parental populations to increase bunch production in hybrids. For RRGS, we obtained the parental genomic estimated breeding values using GBLUP with hybrid phenotypes as data records and population specific allele models. We studied the effects of four RRGS parameters on selection response and genetic parameters: (1) the molecular data used to calibrate the GS model: in RRGS_PAR, we used parental genotypes and in RRGS_HYB we also used hybrid genotypes; (2) frequency of progeny tests (model calibration); (3) number of candidates and (4) number of genotyped hybrids in RRGS_HYB.

Results

We concluded that RRGS could increase the annual selection response compared to RRS by decreasing the generation interval and by increasing the selection intensity. With 1700 genotyped hybrids, calibration every four generations and 300 candidates per generation and population, selection response of RRGS_HYB was 71.8 % higher than RRS. RRGS_PAR with calibration every two generations and 300 candidates was a relevant alternative, as a good compromise between the annual response, risk around the expected response, increased inbreeding and cost. RRGS required inbreeding management because of a higher annual increase in inbreeding than RRS.

Conclusions

RRGS appeared as a valuable method to achieve a long-term increase in the performance for a trait showing heterosis due to the multiplicative interaction between additive and negatively correlated components, such as oil palm bunch production.