Selection at a diallelic autosomal locus in a dioecious population

The model used is that of an infinite dioecious population with nonoverlapping discrete generations and random mating. If the fitnesses are constant and heterozygotes are viable, it is proved that the allelic frequencies converge to equilibria as the number of generations tend to infinity. The results complement those of Karlin and Lessard [1] and Selgrade and Ziehe [5] in that hyperbolicity of equilibria is not assumed, use of index theory is avoided and it is determined how the number of equilibria and phase portraits depend on the fitnesses in the most general case. Lessard [2] gives, in the same situation, a condensed proof of convergence of allelic frequencies off the separatrix under the hypothesis that 1 is not an eigenvalue at any equilibrium. Our method of study is elementary.     

The Two-Locus Model with Sex Differences in Recombination

The criteria for stability of the equilibrium with D=0 are obtained for the two locus model with multiplicative or symmetric fitnesses when the recombination values in males and females are different. It is shown that if r is defined to be equal to the average of the recombination values in males and females, then the criteria are exactly the same as in the standard two locus model.-The equilibrium values with D not equal0 are obtained for the symmetric fitness model. At this equilibrium, the absolute value of D is always greater (for the same average recombination value) if the recombination values in males and females differ than if they are equal.     

The evolution of alternative mating strategies in variable environments

Summary We assessed the influence of phenotypic plasticity in age at maturity on the maintenance of alternative mating strategies in male Atlantic salmon,Salmo salar. We calculated the fitness,r, associated with the parr and the anadromous strategies, using age-specific survival data from the field and strategy-specific fertilization data from the laboratory. The fitness of each strategy depended largely on mate competition (numbers of parr per female, i.e. parr frequency) and on age at maturity. Fitness declined with increasing numbers of parr per female with equilibrium frequencies (at which the fitnesses of each strategy are equal) being within the range observed in the wild. Equilibrium parr frequencies declined with decreasing growth rate and increasing age at maturity. Within populations, the existence of multiple age-specific sets of fitness functions suggests that the fitnesses of alternative strategies are best represented as multidimensional surfaces. The points of intersection of these surfaces, whose boundaries encompass natural variation in age at maturity and mate competition, define an evolutionarily stable continuum (ESC) of strategy frequencies along which the fitnesses associated with each strategy are equal. We propose a simple model that incorporates polygenic thresholds of a largely environmentally-controlled trait (age at maturity) to provide a mechanism by which an ESC can be maintained within a population. An indirect test provides support for the prediction that growth-rate thresholds for parr maturation exist and are maintained by stabilizing selection. Evolutionarily stable continua, maintained by negative frequency-dependent selection on threshold traits, provide a theoretical basis for understanding how alternative life histories can evolve in variable environments.     

Fertility interactions in Drosophila: Theoretical model and experimental tests

The results of 11 experiments with Drosophila species show that fertility is not a reducible property: the fertility of a mating pair cannot be predicted from the average fertility of the two genotypes involved. We propose a model of fertility selection that does not assume additivity (or multiplicativity) but assumes random mating and that the genotypic frequencies are in Hardy-Weinberg equilibrium. Numerical simulations show that removal of the assumption of Hardy-Weinberg frequencies does not significantly change the equilibrium frequencies predicted by the model.     

Colony-level selection in the social insects: Single locus additive and nonadditive models

Genetic models of colony-level selection applicable to diploids (termites) and haplodiploids (social Hymenoptera) are analysed. In the Additive model colony fitnesses are just the arithmetic average of the contribution of the worker genotypes. In the Nonadditive model the fitness of the heterogenotypic colonies (those comprised of more than one worker genotype) may be altered due to interaction between the different worker genotypes. This is modelled by multiplying the additive fitness by the variable, e_i. With additive selection the same equilibrium gene frequency occurs in diploids and in haplodiploids with both once and twice mated queens. In haplodiploids if selection is nonadditive and strong, up to three polymorphic equilibria can exist; however, only a maximum of two are possible with weak selection. Multiple mating by queens increases the number of equilibria possible. Worker-produced males alter the conditions for the existence of a polymorphic equilibrium, and shift the male and female equilibrium gene frequencies.     

Nuclear androdioecy and gynodioecy

We formulate two single-locus Mendelian models, one for androdioecy and the other one for gynodioecy, each with 3 parameters: t the male (female) fertility rate of males (females) to hermaphrodites, s the fraction of the progeny derived from selfing; and g the fitness of inbreeders. Each model is expressed as a transformation of a 3 dimensional zygotic algebra, which we interpret as a rational map of the projective plane. We then study the dynamics for the evolution of each reproductive system; and compare our results with similar published models. In this process, we introduce a general concept of fitness and list some of its properties, obtaining a relative measure of population growth, computable as an eigenvalue of a mixed mating transformation for a population in equilibrium. Our results concur with previous models of the evolution of androdioecy and gynodioecy regarding the threshold values above which the sexual polymophism is stable, although the previous models assume constant the fraction of ovules from hermaphrodites that are self pollinated, while we assume constant the fraction of the progeny derived from selfing. A stable androdioecy requires more stringent conditions than a stable gynodioecy if the amount of pollen used for selfing is negligible in comparison with the total amount of pollen produced by hermaphrodites. Otherwise, both models are identical. We show explicitly that the genotype fitnesses depend linearly on their frequencies. Simulations show that any population not at equilibrium always converges to the equilibrium point of higher fitness. However, at intermediate steps, the fitness function occasionally decreases.     

Nuclear–mitochondrial epistasis: a gene's eye view of genomic conflict

We use population genetic models to investigate the cooperative and conflicting synergistic fitness effects between genes from the nucleus and the mitochondrion. By varying fitness parameters, we examine the scope for conflict relative to cooperation among genomes and the utility of the “gene's eye view” analytical approach, which is based on the marginal average fitness of specific alleles. Because sexual conflict can maintain polymorphism of mitochondrial haplotypes, we can explore two types of evolutionary conflict (genomic and sexual) with one epistatic model. We find that the nuclear genetic architecture (autosomal, X‐linked, or Z‐linked) and the mating system change the regions of parameter space corresponding to the evolution by sexual and genomic conflict. For all models, regardless of conflict or cooperation, we find that population mean fitness increases monotonically as evolution proceeds. Moreover, we find that the process of gene frequency change with positive, synergistic fitnesses is self‐accelerating, as the success of an allele in one genome or in one sex increases the frequency of the interacting allele upon which its success depends. This results in runaway evolutionary dynamics caused by the positive intergenomic associations generated by selection. An inbreeding mating system tends to further accelerate these runaway dynamics because it maintains favorable host–symbiont or male–female gene combinations. In contrast, where conflict predominates, the success of an allele in one genome or in one sex diminishes the frequency of the corresponding allele in the other, resulting in considerably slower evolutionary dynamics. The rate of change of mean fitness is also much faster with positive, synergistic fitnesses and much slower where conflict is predominant. Consequently, selection rapidly fixes cooperative gene combinations, while leaving behind a slowing evolving residue of conflicting gene combinations at mutation–selection balance. We discuss how an emphasis on marginal fitness averages may obscure the interdependence of allelic fitness across genomes, making the evolutionary trajectories appear independent of one another when they are not.     

UNIFYING GENETIC MODELS FOR THE EVOLUTION OF FEMALE CHOICE

A best-of-N rule of female mating preferences can give rise to lines of unstable equilibria in a two-locus haploid model of sexual selection. Under the best-of-N rule, which corresponds to choice at a lek, male fitnesses can exhibit a form of positive frequency-dependence that is not seen under fixed-relative-preference rules (Kirkpatrick, 1982). This positive frequency-dependence can be strongly destabilizing. Lande's (1981) criterion for the stability of the equilibria in quantitative-genetic models of sexual selection applies exactly and in general to the related family of simple population-genetic models. This offers some insight into the workings of these models and greatly simplifies their analysis.     

Frequency-dependent selection and the evolution of assortative mating

A long-standing goal in evolutionary biology is to identify the conditions that promote the evolution of reproductive isolation and speciation. The factors promoting sympatric speciation have been of particular interest, both because it is notoriously difficult to prove empirically and because theoretical models have generated conflicting results, depending on the assumptions made. Here, we analyze the conditions under which selection favors the evolution of assortative mating, thereby reducing gene flow between sympatric groups, using a general model of selection, which allows fitness to be frequency dependent. Our analytical results are based on a two-locus diploid model, with one locus altering the trait under selection and the other locus controlling the strength of assortment (a "one-allele" model). Examining both equilibrium and nonequilibrium scenarios, we demonstrate that whenever heterozygotes are less fit, on average, than homozygotes at the trait locus, indirect selection for assortative mating is generated. While costs of assortative mating hinder the evolution of reproductive isolation, they do not prevent it unless they are sufficiently great. Assortative mating that arises because individuals mate within groups (formed in time or space) is most conducive to the evolution of complete assortative mating from random mating. Assortative mating based on female preferences is more restrictive, because the resulting sexual selection can lead to loss of the trait polymorphism and cause the relative fitness of heterozygotes to rise above homozygotes, eliminating the force favoring assortment. When assortative mating is already prevalent, however, sexual selection can itself cause low heterozygous fitness, promoting the evolution of complete reproductive isolation (akin to "reinforcement") regardless of the form of natural selection.     

Selection, Generalized Transmission and the Evolution of Modifier Genes. I. the Reduction Principle

Modifier gene models are used to explore the evolution of features of organisms, such as the genetic system, that are not directly involved in the determination of fitness. Recent work has shown that a general "reduction principle" holds in models of selectively neutral modifiers of recombination, mutation, and migration. Here we present a framework for models of modifier genes that shows these reduction results to be part of a more general theory, for which recombination and mutation are special cases. The deterministic forces that affect the genetic composition of a population can be partitioned into two categories: selection and transmission. Selection includes differential viabilities, fertilities, and mating success. Imperfect transmission occurs as a result of such phenomena as recombination, mutation and migration, meiosis, gene conversion, and meiotic drive. Selectively neutral modifier genes affect transmission, and a neutral modifier gene can evolve only by generating association with selected genes whose transmission it affects. We show that, in randomly mating populations at equilibrium, imperfect transmission of selected genes allows a variance in their marginal fitnesses to be maintained. This variance in the marginal fitnesses of selected genes is what drives the evolution of neutral modifier genes. Populations with a variance in marginal fitnesses at equilibrium are always subject to invasion by modifier genes that bring about perfect transmission of the selected genes. It is also found, within certain constraints, that for modifier genes producing what we call "linear variation" in the transmission processes, a new modifier allele can invade a population at equilibrium if it reduces the level of imperfect transmission acting on the selected genes, and will be expelled if it increases the level of imperfect transmission. Moreover, the strength of the induced selection on the modifier gene is shown to range up to the order of the departure of the genetic system from perfect transmission.     

Evolution of dispersal in density regulated populations: A haploid model

The evolution of dispersal is explored in a density-dependent framework. Attention is restricted to haploid populations in which the genotypic fitnesses at a single diallelic locus are decreasing functions of the changing number of individuals in the population. It is shown that migration between two populations in which the genotypic response to density is reversed can maintain both alleles when the intermigration rates are constant or nondecreasing functions of the population densities. There is always a unique symmetric interior equilibrium with equal numbers but opposite gene frequencies in the two populations, provided the system is not degenerate. Numerical examples with exponential and hyperbolic fitnesses suggest that this is the only stable equilibrium state under constant positive migration rates (m) less than . Practically speaking, however, there is only convergence after a reasonable number of generations for relatively small migration rates ( ). A migration-modifying mutant at a second, neutral locus, can successfully enter two populations at a stable migration-selection balance if and only if it reduces the intermigration rates of its carriers at the original equilibrium population size. Moreover, migration modification will always result in a higher equilibrium population size, provided the system approaches another symmetric interior equilibrium. The new equilibrium migration rate will be lower than that at the original equilibrium, even when the modified migration rate is a nondecreasing function of the population sizes. Therefore, as in constant viability models, evolution will lead to reduced dispersal.     

A model of polymorphism with several seasons and several habitats, and its application to the mosquito Aedes aegypti

Polymorphism has been shown to be possible but unlikely with a different selection intensity in each of several niches, or with varying selection intensity during successive generations. We show that polymorphism is likely with the combination of several niches and several seasons. The model contains two seasons, three habitats, many generations per season, habitat selection, positive assortative mating, movement between habitats, and different fitnesses of each genotype in each habitat. It is a stepping stone model with differential migration of genotypes. It is applied to the polymorphism of the indoor and the outdoor genotypes of the yellow fever mosquito Aedes aegypti. Matrix methods and simple models of population genetics comprised the computer model. Polymorphism is likely with most reasonable values of the parameters. Fitnesses and rate of movement are the most important parameters influencing the character and likelihood of polymorphism; habitat selection and positive assortative mating have much less effect. The model indicates that polymorphism of A. aegypti in east Africa results from: (1) the presence of a dry season when breeding occurs only in the human habitat; (2) greater fitness of the indoor ecotype in the human habitat and of the outdoor ecotype in the natural habitat; and (3) less than random movement between human and natural habitats.     

The significance of over- and underdominance for the maintenance of genetic polymorphisms. I. Underdominance and stability

It is pointed out that the standard selection models in population genetics all require some form of heterozygote advantage in fitness in order to guarantee the maintenance or stability of genetic polymorphisms. Even more recent results demonstrating the existence of stable two-locus polymorphisms with marginal underdominance at both loci are based on certain epistatically acting heterosis assumptions. This raises the question as to whether heterozygote advantage in fitness is indeed a generally valid principle of maintaining polymorphisms. To avoid ambiguity in definition of heterozygote advantage (overdominance) as it appears in multiallele or multilocus systems, a one-locus-two-allele model is considered. This model allows for sexually asymmetric selection and random mating. It is shown that the model produces globally stable polymorphisms exhibiting underdominance in fitness for a considerable and biologically reasonable range of selection values. Having thus properly refuted the general validity of the common overdominance principle, a modified version is suggested which covers the classical viability selection model and its extension to arbitrary, sexually asymmetric viability and fertility selection. This modified overdominance principle is based on the notion of fractional fitnesses and relates protectedness of biallelic polymorphisms to the extent to which each genotype reproduces its own type. The fact that the model treated displays frequency dependent fitnesses which may change in ranking while approaching equilibrium is discussed in relation to problems of the evolution of overdominance and underdominance.     

Protected polymorphism in the two-locus haploid model with unpredictable fitnesses

 In an unpredictable environment, the distributions of alleles from which polymorphism can be maintained forever belong to a certain set, the C-viability kernel. Such a set is calculated in the two-locus haploid model, as well as the corresponding fitnesses at any time which make this maintenance possible. The dependence of the C-viability kernel on the set U of admissible fitnesses and on the recombination rate r is studied. Notably, the C-viability kernel varies rapidly in the neighborhood of equal fitness of AB and ab; it becomes empty when ab has a fitness below a certain function, which is delineated, of the recombination rate. The properties of the two-locus model under constraints, out of equilibrium and with unpredictable selection are thus presented. Received: 20 May 1999     

Adaptive Topography in Fertility-Viability Selection Models: The Haplodiploid Case

A linear combination of partial changes of mean fitnesses from one generation to the next one is shown to be approximately equal to the additive genetic variance in fitness after enough generations and away from equilibrium in random mating haplodiploid populations under arbitrary weak frequency-dependent selection on sex-differentiated viability of individuals and sex-differentiated fertility of matings controlled at a single multiallelic locus. The result can be applied to X-linked locus models in diploid populations. The result is used to deduce approximate adaptive topographies far frequency-independent selection models in the cases of nonsex-differentiated fertilities and multiplicative sex-differentiated fertilities and for kin selection models in family-structured populations under the assumptions of single insemination and multiple insemination of females. Multiple insemination creates frequency-dependent selection regimes.     

Male mating success in a North American pitviper: influence of body size,testosterone, and spatial metrics

Males with enhanced traits relative to conspecifics often show increased mating and reproductive success and thus have a fitness advantage. The opportunity or potential for sexual selection is predicted to occur under these conditions. Here, we investigated proximate determinants of mating success in male copperhead snakes (Agkistrodon contortrix), a medium‐sized pitviper of North America. Specifically, we investigated the relationships of body size (snout‐vent length, body mass), body condition index, spatial metrics (total distance moved, home range size), and plasma testosterone concentration on mating success in males. The single mating season lasts from August through September. We compared a set of candidate linear mixed models and selected the best‐fitting one using the adjusted Akaike Information Criterion (AICc). The AICc‐selected model (model 2), with testosterone, body condition index, and home range size as predictor variables, showed that male mating success was positively correlated with testosterone. To our knowledge, this is the first report to show the relationship of testosterone and individual mating success in any snake species. A parallel study conducted on male fitness in A. contortrix of the same population used microsatellite markers to assign parentage of fathers (known mothers). Unlike our study, they found that snout‐vent length was positively correlated with reproductive success and that males were experiencing greater sexual selection. This relationship has been detected under natural conditions in other species of snakes. Although behavioural data are important in any mating system analysis, they should not stand alone to infer parentage, relationships or selection metrics (e.g. Bateman gradients). Long‐term sperm storage by females, female cryptic choice, and other factors contribute to the complexity of mating success of males. Accordingly, we thus conclude that estimates of reproductive success and fitness in cryptic species, such as copperheads and other snakes, require robust molecular methods to draw accurate conclusions regarding proximate and evolutionary responses. © 2015 The Linnean Society of London, Biological Journal of the Linnean Society, 2015, 115 , 185–194.     

A Monte Carlo simulation model for studying evolution in age-structured populations

This model provides for any number of genotypes defined by age-specific survival and fecundity rates in a population with completely overlapping generations and growing under the control of density-governing functions affecting survival or fecundity. It is tested in situations involving two alleles at one locus. Nonselection populations at Hardy–Weinberg equilibrium obey the ecogenetic law; i.e., each genotype follows Lotka's law regarding rate of increase and stable age distribution as if it were an independent true-breeding population. Populations experiencing age- and density-independent selection approximate this situation, and the changes in gene frequency are predicted by relative fitnesses bases on λ, the finite rate of increase of the genotypes. Polymorphic gene equilibria occurring at steady-state population sizes are determined by fitnesses based on R, the net reproductive rate. In examples involving differences in generation time produced by age-dependent differences in fecundity, the allele associated with longer generation time may be favored or opposed by selection, depending on whether the density-governing factor controlling population size affects survival or fecundity. If such genotypes have similar R's, a genetic equilibrium may be established if the population is governed by a density function acting upon fecundity. Received: August 23, 1999 / Accepted: July 13, 2000     

Emergence and loss of assortative mating in sympatric speciation

We have studied an agent model which presents the emergence of sexual barriers through the onset of assortative mating, a condition that might lead to sympatric speciation. In the model, individuals are characterized by two traits, each determined by a single locus A or B. Heterozygotes on A are penalized by introducing an adaptive difference from homozygotes. Two niches are available. Each A homozygote is adapted to one of the niches. The second trait, called the marker trait has no bearing on the fitness. The model includes mating preferences, which are inherited from the mother and subject to random variations. A parameter controlling recombination probabilities of the two loci is also introduced. We study the phase diagram by means of simulations, in the space of parameters (adaptive difference, carrying capacity, recombination probability). Three phases are found, characterized by (i) assortative mating, (ii) extinction of one of the A alleles and (iii) Hardy-Weinberg like equilibrium. We also make perturbations of these phases to see how robust they are. Assortative mating can be gained or lost with changes that present hysteresis loops, showing the resulting equilibrium to have partial memory of the initial state and that the process of going from a polymorphic panmictic phase to a phase where assortative mating acts as sexual barrier can be described as a first-order transition.     

LONG-TERM EXPERIMENTAL EVOLUTION IN ESCHERICHIA COLI. III. VARIATION AMONG REPLICATE POPULATIONS IN CORRELATED RESPONSES TO NOVEL ENVIRONMENTS

Twelve populations of Escherichia coli were founded from a single clone and propagated for 2000 generations in identical glucose-limited environments. During this time, the mean fitnesses of the evolving populations relative to their common ancestor improved greatly, but their fitnesses relative to one another diverged only slightly. Although the populations showed similar fitness increases, they may have done so by different underlying adaptations, or they may have diverged in other respects by random genetic drift. Therefore, we examined the relative fitnesses of independently derived genotypes in two other sugars, maltose and lactose, to determine whether they were homogeneous or heterogeneous in these environments. The genetic variation among the derived lines in fitness on maltose and lactose was more than 100-times greater than their variation in fitness on glucose. Moreover, the glucose-adapted genotypes, on average, showed significant adaptation to lactose, but not to maltose. That pathways for use of maltose and glucose are virtually identical in E. coli, except for their distinct mechanisms of uptake, suggests that the derived genotypes have adapted primarily by improved glucose transport. From consideration of the number of generations of divergence, the mutation rate in E. coli, and the proportion of its genome required for growth on maltose (but not glucose), we hypothesize that pleiotropy involving the selected alleles, rather than random genetic drift of alleles at other loci, was the major cause of the variation among the derived genotypes in fitness on these other sugars.     

Current Sperm Competition Determines Sperm Allocation in a Tephritid Fruit Fly

Sperm competition (SC) occurs when the sperm of two or more males compete for the same set of ova. Theoretical models and experimental observations indicate that the presence of rival males causes focal males to adjust sperm allocation in a given copulation. Males allocate more sperm when they perceive the presence of one rival male (SC risk), either before or during mating, or when they perceive the presence of multiple rival males before mating (previous SC intensity). Conversely, males are expected to allocate fewer sperm when they perceive the presence of rival males during mating (current SC intensity). Here, we varied male perception of SC by manipulating the number of rival males, both before mating (from emergence to mating) and during mating (at the time of mating) to examine their effects on mating latency, copulation duration, and sperm allocation in the South American fruit fly Anastrepha fraterculus. We showed that exposure to rival males at the time of mating decreased mating latency. However, in contrast to the theory, exposure to multiple rivals at the time of mating increased sperm allocation. Female and male size were significant predictors of mating latency, copulation duration, and sperm allocation. Our results showed that there is a plastic response of males to the level of perceived SC through the number of rival males. Current levels of SC intensity are important in shaping male responses to SC, although the patterns in this species are opposite to predictions from the existing theory. We propose that female preference for males forming leks could explain lower sperm counts when encountering only one or two males.