Density-Dependent Selection Incorporating Intraspecific Competition. II. a Diploid Model

A diploid model is introduced and analyzed in which intraspecific competition is incorporated within the context of density-regulated selection. It is assumed that each genotype has a unique carrying capacity corresponding to the equilibrium population size when only that type is present. Each genotypic fitness at a single diallelic autosomal locus is a decreasing function of a distinctive effective population size perceived as a result of intraspecific competition. The resulting fitnesses are both density and frequency dependent with selective advantage determined by a balance between genotypic carrying capacity and sensitivity to intraspecific competition. A major finding is that intergenotypic interactions may allow genetic variation to be more easily maintained than in the corresponding model of purely density-dependent selection. In addition, numerical study confirms the possible existence of multiple interior equilibria and that neither overdominance in fitness nor carrying capacity is necessary for stability. The magnitude of the equilibrium population size and optimization principles are also discussed.     

Balanced polymorphisms of quasineutral alleles

When the average number of progeny born to the genotypes at a locus are linear functions of a random variable such that, on the average, the number of progeny produced by each genotype is the same (quasineutrality), then there will be a systematic pressure on gene frequency that tends to favor the genotype with the smallest variance in the number of progeny. In particular, there will be a stable equilibrium attained whenever the variance in the number of progeny produced by the heterozygous genotype is less than that of both homozygous genotypes. We assume that the random variable affecting fitness is some function of an underlying environmental variable that tends to smooth out the day-to-day fluctuations in the environmental variable, such as the mean or the range or the variance. The effect of the environment on fitness can, therefore, be assumed to be approximately constant for periods corresponding to one generation. If the logarithm of the average number of progeny is a linear function of a random variable, then in contrast to the foregoing conclusions, there will be no tendency for any change in gene frequency at the locus (Kimura, M. (1954), Process leading to quasi-fixation of genes in natural populations due to random fluctuations of selection intensities, Genetics39, 280–295; (1964), Diffusion models in population genetics, J. App. Prob.1, 177–232.).     

Activity Variants of Acid Phosphatase-3 among Chromosome 3 Inversions of DROSOPHILA PSEUDOOBSCURA

Sexual selection is measured between two strains of Drosophila melanogaster: a wild strain and a strain mutant at the sepia locus. Frequency-dependent male mating was found to be successful, whereas the female genotype exerted no influence. The rarer the male genotype becomes, the greater is its mating success. A selection model is built for this behavior characteristic in which selection operates differently in the two sexes. The genetic consequencies of this model upon the maintenance of genetic polymorphism at the sepia locus are compared to experimental data from previous population cage studies. The fit obtained with this sexual selection model is compared to that of the larvel selection model previously investigated. A model composed of both sexual and larval components of fitness is presented. The role that each major selection component is expected to play in experimental populations as the gene frequency changes is discussed. Sexual selection leads to an equilibrium level higher than larval selection, and the combined model is very close to the experimental values.     

Density-dependent selection migration model with non-monotone fitness functions

This paper studies the classical single locus, diallelic selection model with diffusion for a continuously reproducing population. The phase variables are population density and allele frequency (or allele density). The genotype fitness depend only on population density but include one-hump functions of the density variable. With mild assumptions on genotype fitnesses, we study the geometry of the nullclines and the asymptotic behavior of solutions of the selection model without diffusion. For the diffusion model with zero Neumann boundary conditions, we use this geometric information to show that if the initial data satisfy certain conditions then the corresponding solution to the reaction-diffusion equation converges to the spatially constant stable equilibrium which is closest to the initial data.Research partially supported by NSF grant DMS-8920597Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina     

Fluctuating Selection,Egg Banks and Population Genetic Structure in Cyclically Parthenogenetic Species

Associations between neutral genetic markers and genes under selection have been suggested to explain the population genetic structure of neutral genes in cyclically parthenogenetic freshwater invertebrates. A simulation model was constructed in order to analyse the extrapolated consequences of observed fluctuations in genotype frequencies in Daphnia, in the presence of egg banks. When of sufficient depth and magnitude, egg banks in combination with fluctuating selection were shown to maintain genetic variation within populations indefinitely. The level of equilibrium diversity increased with the depth and magnitude of the banks, and with intensity of selection. The same threshold was responsible for genetic differentiation between populations, which was independent of migration rate, and which was attained very rapidly following initial Hardy–Weinberg equilibrium. In the absence of selection, egg banks increased the effective size of local populations, thereby decreasing genetic differentiation at migration-drift equilibrium. These results suggest that egg banks are crucial to the genetic structure in the presence of fluctuating selective pressures, but more data are needed if this knowledge is to be used in an improved general understanding of the genetic structure of cyclically parthenogenetic species.     

Density- and Frequency-Dependent Selection at the Mdh-2 Locus in DROSOPHILA PSEUDOOBSCURA

We have studied differences in the number of Drosophila pseudoobscura produced in a culture when the flies differ with respect to two alleles (F and S) at the Mdh-2 locus, which codes for a malate dehydrogenase enzyme. The studies were done at low and at high density in two- and three-genotype combinations (S/S, F/F and S/F), with one-genotype cultures as controls.——Density affects the fitness of the Mdh-2 genotypes. Different genotypes are differently affected, and the genotype of the competitors also makes a difference on the fitness of a given genotype. When three genotypes are present in a culture, particularly at high density, intergenotypic competition is less intense than intragenotypic competition at several frequency combinations. That is, there is "overcompensation": the three genotypes together exploit the environmental resources better than one genotype alone.—The fitness of the genotypes is frequency dependent in both two-genotype and three-genotype combinations. An inverse relationship between frequency and fitness is observed at high density. This may lead to a stable polymorphism, because the fitness of a genotype increases as its frequency decreases.—Forty independent strains, sampled from a natural population, were used in the experiments. This ensures that more than 95% of the variation present in the genome in the natural population is also present is the experimental cultures. It also ensures that the genetic background of the Mdh-2 alleles is randomized in the same way as it is in nature. However, the possibility remains that Mdh-2 alleles in nature are nonrandomly associated with alleles at closely linked loci. If linkage disequilibrium is present in the experiments because it exists in nature, then the observed effects (such as frequency-dependent selection) would affect the Mdh-2 locus in nature as well.     

Genetic polymorphism and evolution in parthenogenetic animals. I. Polyploid curculionidae

The genetic variability at enzyme loci in different triploid and tetraploid parthenogenetic weevil populations has been elucidated by starch gel electrophoresis. The overall genotype of individual weevils belonging to different populations has been determined for over 25 loci. The results are compared with those obtained for diploid bisexual races of either the same or closely related species. The variation within a parthenogenetic population differs from that in diploid, sexually reproducing populations, i.e. the allele frequencies are not in a Hardy-Weinberg equilibrium. The results indicate that apomictic parthenogenetic populations can differentiate genetically. The genotypes within a population resemble each other more than genotypes belonging to different populations. It is evident that evolution still continues—even if slowed down—in parthenogenetic weevils. A comparison between the allele relationships in geographically isolated polyploid parthenogenetic populations and related diploid bisexual forms does not support the hypothetical hybrid origin of parthenogenesis and polyploidy in weevils. Parthenogenesis within a parthenogenetic weevil species is evidently monophyletic.     

Natural selection and fitness entropy in a density-regulated population

The entropy H(p_o,p*) of a population with the initial allele frequency p_o given the equilibrium polymorphic frequency p* has been proposed as a measure of natural selection. In the present paper, we have extended this concept to include a particular aspect of density-dependent selection. We compared size trajectory of a population initially at genetic equilibrium, N(t), with the size trajectories of populations not initially at p*,N(t), but which do eventually converge to a common equilibrium allele frequency and equilibrium density, N*. The following experimentally testable hyopthesis was established. The total area defined by the difference between the trajectories of N(t) and N(t) as they converge to N* is directly proportional to the fitness entropy when population size is transformed using the density-dependent fitness value. Two properties of this relationship were noted. First, it is independent of the magnitude of natural selection and, secondly, it does not depend upon the initial population density as long as the equilibrium and nonequilibrium populations have the same initial numbers. This hypothesis was evaluated with experimental data on the flour beetle Tribolium castaneum.     

Changes in gene frequencies at the octanol dehydrogenase locus of Drosophila melanogaster imposed by environmental ethanol

Experiments have been performed to study the effect of selection at the Odh locus in Drosophila melanogaster populations using different alcohol concentrations in the medium. The data can be best interpreted by assuming frequency-dependent selection. When genotype frequencies are considered as independent variables and values of Wrightian fitness as dependent variables, it turns out that different functions describe the selection of the coexisting genotypes. A linear equation is used for the SF genotype and a hyperbolic function for the FF genotype. No function of good fit could be found for the SS genotype. Simulation experiments using these functions fit our data well.     

RARER NEED NOT BE BETTER IF COMMONER IS WORSE: FREQUENCY-DEPENDENT SELECTION FOR DEVELOPMENTAL TIME AT THE ALCOHOL DEHYDROGENASE LOCUS OF THE OLIVE FRUIT FLY,BACTROCERA OLEAE

Whereas the importance of frequency-dependent selection in life-history traits, behavioral characters and source allocation patterns is widely accepted, its role in governing biochemical and molecular polymorphisms remains poorly understood. Here we demonstrate a case of allozyme frequency-dependent selection. When olive fruit flies (Bactrocera oleae) are reared on an artificial larval medium, an allele at the alcohol dehydrogenase locus that is present in very low frequency in natural populations increases to about one-third in less than five generations. We show here that the time from the hatching of the egg to the eclosion of the adult is affected by the genotype composition of the larval population that grows in the same cup of food. Cultures consisting of one genotype only have the longest developmental time, and two-allele cultures in which the two homozygotes and the heterozygote occur in a 1:1:2 ratio show the shortest developmental time. Cultures with intermediate genotypic compositions show intermediate levels of developmental time. The results can be explained by assuming that the developmental time of a genotype depends on the frequency array of all genotypes in the larval population and is not merely a function of its own frequency. It is even possible that the developmental time of a genotype becomes longer as the genotype becomes rarer, yet the genotype will be favored because the developmental times of the competing genotypes become even longer owing to the associated increase of their frequencies. Given that developmental time is inversely related to fitness, this generates a frequency-dependent selection, with developmental times changing progressively until the population arrives at an equilibrium. One optimum population composition that provides a satisfactory fit to allele frequency changes in our experimental populations is when the two alleles occur in equal frequencies and genotypes are in Hardy-Weinberg proportions. We argue that this type of selection is consistent with the role of alcohol dehydrogenase as a detoxifying enzyme in a medium that undergoes continuous chemical changes during its use by the feeding larvae.     

Frequency-Dependent Selection at the PGM-1 Locus of DROSOPHILA PSEUDOOBSCURA

Frequency-dependent fitness was studied at the Pgm-1 locus of Drosophila pseudoobscura with respect to two fitness components: rate of development and larva-to-adult survival. The Pgm-1 locus is very polymorphic with only two alleles, Pgm-1¹⁰⁰ and Pgm-1¹⁰⁴, occurring at high frequencies. For each of these two alleles, 20 homozygous strains were obtained from a sample of 1,140 wild-inseminated females. First-instar larvae of the two genotypes were combined in a set of eight different frequencies: 0.0, 0.10, 0.25, 0.40, 0.60, 0.75, 0.90, and 1.0. Frequency-dependent fitness effects were observed for the two survival-related fitness components examined: larvae of the less common genotype develop faster and have a higher probability of survival than larvae of the more common genotype. The rate of survival at intermediate genotypic frequencies is similar to that in pure cultures. If selection acted solely as frequency-dependent effects on survival-related components of fitness, the equilibrium frequency of the Pgm-1¹⁰⁰ allele would be 0.615 for a two-genotype system, which fits an observed frequency range for this allele in nature between 0.55 and 0.71. Experimentally created linkage disequilibrium was excluded from the experiment by using a large number of independent strains. It is nevertheless possible that the frequency-dependent selection may not affect the Pgm-1 locus per se, but may reflect a linkage disequilibrium present in the natural population. Even if this were the case, the frequency-dependent selection could affect the frequency of the Pgm-1 alleles in nature.     

Dynamical behavior for population genetics models of differential and difference equations with nonmonotone fitnesses

This paper studies the dynamical behavior of classical 2-dimensional models of continuously and discretely reproducing diploid populations with two alleles at one locus. The phase variables are allele frequency and population density. The genotype fitnesses are not assumed to be monotonically decreasing functions of density. Hence the mean fitness curve is more complicated than in the monotonic case. If genotype fitnesses are only density dependent, results concerning equilibrium stability are obtained similar to those for the monotonic case, and periodic solutions are precluded in the differential equation model. An example with one-hump genotype fitnesses is presented and analyzed.Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina     

Esterase-6 polymorphism inDrosophila melanogaster:Effects of temperature and methyl malonate on genotypic trajectories in polymorphic populations set up with highly inbred lines

It is generally difficult to identify possible effects of selection at a specific locus because of the heterogeneity of the genetic background. Geographical patterns ofEst-6 gene frequencies suggest that there is selection at this locus but selection on loci closely linked to it cannot be excluded. Differences in catalytic properties between allozymes have been shownin vitro; further, several laboratory studies have shown apparent fitness differences between allozymes. Our study used inbred lines highly homogeneous in the genetic background. Four populations were set up fromEst-6s andEst-6F homozygous females inseminated by males of the same genotype at each combination of three factors: temperature (18 and 25°C); methyl malonate (presence or absence); input gene frequencies [p(S) = 0.2 and 0.8]. The populations were sampled periodically for about 28 generations. Methyl malonate was chosen to exert pressure in the enzymatic function of esterase-6. Statistical analyses show that: there are no sex differences; gene frequencies change from input values to those of the first sampling, when only individuals of the first generation are present at 18^oC or individuals of the second generation just begin to appear at 25°C; gene frequencies do not change thereafter and Hardy-Weinberg equilibrium is established. The changes in gene frequencies observed in the first generations suggest thatEst-6 can under certain conditions be a target of selection. Such conditions may not, however, occur in natural populations.     

An experimental check of fitness entropy vs. selective delay

A new macroparameter characterizing the movement of a non-equilibrium initial population structure toward the equilibrium structure under the natural selection process was suggested in a previous paper (Ginzburg, 1977). This parameter, named selective delay, is directly measurable from the population size growth curve without any knowledge of the frequencies of the different genotypes in the population. An equation was proposed describing the relationship between selective delay, average population fitness at the equilibrium point and the entropy distance between initial and equilibrium states of the population. The purpose of this paper is an experimental check of this equation. Experimental results with populations of flour beetle Tribolium castaneum segregating at the unsaturated fatty acid sensitive locus have shown a good correspondence to the theory.     

Evolutionary aspects and sensitivity studies of some major gene models

Extensive biometrical and statistically oriented studies in segregation and pedigree analyses reflect current efforts to demonstrate major gene factors playing a significant role for a whole hierarchy of multifactorial diseases and related risk factors exhibiting continuous variation. The evolutionary aspects of the changes in gene frequencies of some major gene one locus models admitting a broad range of genotype-phenotype associations and different forms of selection functions are investigated. The flexibility of differences among the genotypic-phenotypic distribution can take account of variable penetrance expressivity, complex multifarious heterogeneous background effects, or partial dominance concepts. The phenotype distribution and selection function are assumed to be time invariant such that the environments with which the population interacts do not depend on either the phenotypes or the genotypes present in the population of any particular generation. Viability selection optimizing or directional acts on the phenotypic level. We consider random mating, and concentrate mostly on evaluating the nature of the equilibrium structure for the cases of “strong” and “weak” selection. For weak stabilizing selection the determinants of superior genotypic fitness in the class of phenotypic symmetric distributions reside in minimizing a combination of the phenotypic variance and the deviation of the phenotypic mean from the optimal phenotype. With equal means of central phenotype values, a canalizing selection effect signifying fitness superiority for the genotype with minimal variance is in force. For strong stabilizing selection the genotype-phenotype density at the optimal value determines the relative genotype fitness value. For directional selection the determinants of the selection realizations depend on a “standardized” deviation of the mean phenotype distributional value relative to its total variance. The effects of symmetry as against asymmetry in the genotype distributions with prescribed means and variances were investigated by numerical computations.     

Heterozygosity in pin-thrum plants or with partial sex linkage

A two locus model is constructed for selection of a gene closely linked to the S locus in pin-thrum plants or to the sex determining part of the Y chromosome. Using this model, conditions for stability at the equilibrium point which is predicted by one-locus theory when there is heterozygotic superiority are derived. If the recombination value is small, it is found that this equilibrium point is unstable and that the gene frequencies go to a new stable equilibrium point at which the population has a higher average fitness. A few simple cases of selection and the implication of these to the theory of the evolution of the Y chromosome are discussed.     

Does genetic diversity reduce intraspecific competition in rotifer populations?

As a result of reduced intraspecific competition, genetically diverse populations may have higher relative fitness than genetically uniform populations. To test this hypothesis, we compared polyclonal (i.e., genetically diverse) versus monoclonal (i.e., composed of a single clonal genotype) experimental populations of the rotifer Brachionus plicatilis (Müller, 1786) growing separately and in competition. We estimated the following fitness components: intrinsic growth rate; carrying capacity; proportion of sexual females; diapausing egg production per sexual female and total egg production. Polyclonal populations showed similar dynamics to monoclonal populations and no statistical difference between their fitness components was detected. Therefore, results do not support the hypothesis that genetically diverse populations reduce competition through diversification in resource use. Instead, results suggest that B. plicatilis is a generalist consumer whose polyphagy does not depend on genetic differences, but on the broad diet of each genotype. However, clones showed significant differences in almost all fitness components demonstrating among-clone variation in life-history traits. We found a trade-off between sexual ratio and carrying capacity, highlighting the cost of sex in cyclical parthenogenetic rotifers. We discuss the mechanisms that could maintain the observed among-clone genetic variation in natural populations, and speculate on results implication for sex maintenance in rotifers.     

Facultative sexual reproduction under frequency-dependent selection on a single locus

The evolution of a facultative sexual strategy that simultaneously produced sexual and asexual individuals was studied theoretically, under negative frequency-dependence of fitness. The organism was considered to be diploid, characterized by two loci concerning fitness and determining sexual strategy, between which a certain degree of linkage existed. The locus concerning fitness was assumed to involve two alleles, resulting in three genotypes, the relative fitness of an individual being defined by a decreasing function of frequency of its own genotype on this locus in the population. The sexual reproductive strategy was considered to be determined by three alleles; asexual, obligate sexual and facultative sexual. Simulations under various linkages between loci and level of frequency dependence of fitness showed that a facultative sexual strategy was generally able to invade and increase in the population. In particular, when the level of frequency dependence was high to some degree, the facultative strain producing many sexual individuals tended to exclusively occupy the population. Namely, the frequency-dependent selection resulted in a predominance of obligate sexual strategy over asexual strategy, simultaneously causing a subordination of the former to the facultative sexual strategy. This indicated that the evolution of sex should be considered carefully with respect to the possibility of invasion of facultative sex.     

QUANTITATIVE VARIATION IN FINITE PARTHENOGENETIC POPULATIONS: WHAT STOPS MULLER'S RATCHET IN THE ABSENCE OF RECOMBINATION?

Finite parthenogenetic populations with high genomic mutation rates accumulate deleterious mutations if back mutations are rare. This mechanism, known as Muller's ratchet, can explain the rarity of parthenogenetic species among so called higher organisms. However, estimates of genomic mutation rates for deleterious alleles and their average effect in the diploid condition in Drosophila suggest that Muller's ratchet should eliminate parthenogenetic insect populations within several hundred generations, provided all mutations are unconditionally deleterious. This fact is inconsistent with the existence of obligatory parthenogenetic insect species. In this paper an analysis of the extent to which compensatory mutations can counter Muller's ratchet is presented. Compensatory mutations are defined as all mutations that compensate for the phenotypic effects of a deleterious mutation. In the case of quantitative traits under stabilizing selection, the rate of compensatory mutations is easily predicted. It is shown that there is a strong analogy between the Muller's ratchet model of Felsenstein (1974) and the quantitative genetic model considered here, except for the frequency of compensatory mutations. If the intensity of stabilizing selection is too small or the mutation rate too high, the optimal genotype becomes extinct and the population mean drifts from the optimum but still reaches a stationary distribution. This distance is essentially the same as predicted for sexually reproducing populations under the same circumstances. Hence, at least in the short run, compensatory mutations for quantitative characters are as effective as recombination in halting the decline of mean fitness otherwise caused by Muller's ratchet. However, it is questionable whether compensatory mutations can prevent Muller's ratchet in the long run because there might be a limit to the capacity of the genome to provide compensatory mutations without eliminating deleterious mutations at least during occasional episodes of sex.     

Density dependent selection incorporating intraspecific competition 1. A haploid model

A haploid model is introduced and analyzed in which intraspecific competition is incorporated within a density dependent framework. It is assumed that each genotype has a unique carrying capacity corresponding to the equilibrium population size when fixed for that type. Each genotypic fitness at a single multi-allelic locus is a function of a distinctive effective population size formed by adding the numbers of each genotype present, weighted by an intraspecific competition coefficient. As a result, the fitnesses depend upon the relative frequencies of the various genotypes as well as the total population size. Intergenotypic interactions can have a profound effect upon the outcome of the population. In particular, when the density effect of one individual upon another depends upon their respective genotypes, a unique stable interior equilibrium is possible in which all alleles are present. This stands in contrast to the purely density dependent haploid system in which the only possible stable state corresponds to fixation for the type with the highest carrying capacity. In the present model selective advantage is determined by a balance between carrying capacity and sensitivity to density pressures from other genotypes. Fixation for the genotype with the highest carrying capacity, for instance, will not be stable if it exerts a sufficiently weak competitive effect upon the other genotypes. In the diallelic case, maintenance of both alleles at a stable equilibrium requires that the net intragenotypic competition between individuals of like genotype be stronger than that between unlike types. As for purely density regulated systems, there may be no stable equilibria and/or regular and chaotic cycling may occur. The results may also be interpreted in terms of a discrete time model of interspecific competition with each haplotype representing a different species.