Approaches to the Hardy-Weinberg manifold

Let fertilities and death rates be additive, let fertilities be positive, and let mating be random in the Nagylaki-Crow continuous model of selection at a multiallelic locus in a monoecious population. Then polymorphisms are in Hardy-Weinberg proportions. If some fertilities vanish, there is an example of a diallelic polymorphism that is not in Hardy-Weinberg proportions. If the fertilities are larger, in one sense or another, than the difference between any two death rates, then convergence to the Hardy-Weinberg manifold is shown. If, in addition, the Malthusian parameters are constant, and only a finite number of equilibria exist, then global convergence to equilibria is proved.     

Equilibria and dynamics of selection at a diallelic autosomal locus in the Nagylaki-Crow continuous model of a monoecious population

Let birth rates and death rates be constant, birth rates positive, fertilities additive, and each birth rate not larger than twice any other birth rate. Global convergence to equilibria is proved for the model in the title. There is at most one polymorphic equilibrium or there are a continuum of equilibria. The phase portraits are given. If there is a polymorphic equilibrium, then the largest negatively invariant set in the state space is a continuous curve connecting the two fixation equilibria. This curve coincides with the Hardy-Weinberg manifold exactly when the death rate is additive. Disregarding extinction, the polymorphic equilibria are the same for the continuous model as for the corresponding discrete model exactly when the death rate is additive.     

The role of recombination and selection in the modifier theory of sex-ratio distortion

The equilibrium configurations for a two-locus multialle model of sex-linked meiotic drive are studied with regard to the recombination fraction:limit cycles can occur in the case of small recombination while stable equilibrium points associated with linkage equilibrium can exist for an intermediate range of recombination values depending on the equilibrium sex ratio, linkage disequilibrium at nearby equilibrium points taking turn with loser linkage. The evolutionary dynamics in two-locus sex-ratio distortion systems is enlightened: while equilibria with a sex ratio closer to 1/2 are more likely to be stable with respect to perturbations on the frequencies of sex-ratio distorters that are represented at equilibrium, such equilibria are also more vulnerable to the invasion of mutant distorters when there is some degree of linkage with the sex-determining locus. For X-linked multimodifier systems of sex-ratio distortion, differential fertilities and viabilities are incorporated and a maximum principle is suggested.     

A Symmetric Two-Locus Fertility Model

A model in which selection is mediated by differential fertilities among the genotypes at two diallelic loci is proposed. Fertility depends only on the number of heterozygous loci participating in the mating. Classes analogous to symmetric equilibria in symmetric viability models are determined explicitly and shown to exhibit stability behavior very different from the viability results. Linkage equilibrium is shown to occur in a relatively asymmetric fashion and to overlap in stability with linkage disequilibrium. In many cases single-locus or two-locus polymorphism is shown to be stable simultaneously with chromosome fixation even under very tight linkage. It is suggested that historical effects may be of great significance in the evolution of systems in which fertility is the primary agent of natural selection.     

Selection in plant populations of effectively infinite size II. Protectedness of a biallelic polymorphism

The biologically important problem of protectedness of genetic polymorphisms in monoecious plant populations exhibiting genotypically determined variation in rates of self-fertilization and sexually asymmetrical fertilities has hitherto escaped exact, analytical treatment for the reason that appropriate mathematical techniques relying on allelic frequencies do not seem to exist. For the particular case of one locus and two alleles it was possible to develop such a technique which provides conditions of high precision for protectedness of an allele. A comparison of the results with those already known from models that appear to be specializations of the present model showed that some of the earlier conclusions can be generalized, while others have to be handled with great care or should even be rejected. Above all, this concerns the role of self-fertilization, which is frequently considered to counteract the establishment of genetic polymorphisms. However, it turned out that increasing the heterozygote selfing rate also increases protectedness for both alleles in all situations. Moreover, even if the amount of self-fertilization is the same for all genotypes, asymmetry in the production of ovules and pollen, which is more the rule than an exception, may imply protectedness only for comparatively large selfing rates. The probably most outstanding finding is that, depending on the ovule and pollen fertilities, protectedness may be realized only within small ranges of selfing rates, and these ranges may vary from arbitrarily low to arbitrarily high rates. On the other hand, if the ovule fertilities show strong overdominance for the heterozygote—more precisely, if the heterozygote produces more than twice as many ovules as either of the homozygotes—both alleles are protected irrespective of the pollen fertilities and rates of self-fertilization; this generalizes earlier results obtained for more specific models.     

Genetics of the polycross

Summary Seeds from polycrosses with Norway spruce, in which the same sixteen male parents were crossed to a number of female parents in each of two years, were analysed electrophoretically to detect departures from male gamete frequencies expected assuming equal male fertilities, and to detect heterogeneity among female parents in male gamete frequencies in seeds. The data were also used to estimate the fertilities of the polycross trees used as male parents. Significant departures from male gamete frequencies expected assuming equal male fertilities were found in the seed pooled from all crosses. Male fertilities estimated from male gamete frequencies in seed from all crosses also departed significantly from expectation. The results are discussed with respect to assumptions made when estimating general combining abilities and expected response to selection in polycrosses.     

Population Dynamics of the Segregation Distorter Polymorphism of DROSOPHILA MELANOGASTER

Two two-locus models of the population dynamics of the segregation distortion (SD) polymorphism of Drosophila melanogaster are described. One model is appropriate for understanding the population genetics of SD in nature, whereas the other is a special case appropriate for understanding an artificial population that has been extensively analysed. The models incorporate the general features of the Sd and Rsp loci which form the core of the SD system. It is shown that the SD polymorphism can be established only when there is sufficiently tight linkage between Sd and Rsp. An approximate treatment, valid for tight linkage, is given of all the equilibria of the system and their stabilities. It is shown that the observed composition of natural and artificial populations with respect to the Sd and Rsp loci is predicted well by the model, provided that restrictions are imposed on the fertilities of certain genotypes. Highly oscillatory paths towards equilibrium are usually to be expected on the basis of this model. The selection pressures on inversions introduced into this system are also investigated.     

Variation in Male Fertilities and Pairwise Mating Probabilities in Picea glauca

Frequencies of multilocus male gametes in seeds collected from clones in several blocks of a white spruce seed orchard were analyzed as part of a 2-yr study of mating system variation in this species. Observed frequencies of male gamete types departed significantly from those expected assuming equal male fertilities among clones. Male gamete frequencies in seed crops were significantly heterogeneous among clones within blocks, and among blocks within clones. Clonal male fertilities were estimated from male gamete frequency data. These estimates were highly skewed, with a small proportion of the clones contributing male gametes to the majority of the seed. The estimates were significantly heterogeneous among clones within blocks, and among blocks within clones. Between-year variation in clonal male fertilities was also pronounced, with male fertilities of some clones changing by as much as three orders of magnitude. Clonal male fertility was significantly correlated with clonal male cone production in both years. These results are important with regard to assumptions made for the estimation of general combining ability, average genetic correlation among progeny from single parents, and expected response to selection in open-pollinated plant populations.     

Selection with gene-cytoplasm interactions. I. Maintenance of cytoplasm polymorphisms

General conditions for the protectedness of gene-cytoplasm polymorphisms are considered for a biallelic model with two cytoplasm types and under the assumption that nuclear polymorphisms cannot be maintained in the presence of only one cytoplasm type. Analytical results involving male fertilities, female fertilities, viabilities and selfing rates are obtained, and numerical results show spiral and cyclic behavior of population trajectories. It is shown that a maternally inherited cytoplasmic polymorphism cannot be maintained in the absence of a nuclear polymorphism, and that a gene-cytoplasm polymorphism can only be maintained if the population shows sexual asymmetry, i.e. , if the ratio of male to female fertility varies among genotypes. Thus, the classical viability selection model does not allow gene-cytoplasm polymorphisms.     

Global analysis of a two-allele mating system

The global analysis of a two-allele mating system is provided. One allele is dominant to the other. Mating is random, but mating between unlike phenotypes has a lower fertility than that between like phenotypes. The generations are discrete and non-overlapping. The population is considered to be infinite, and the model is deterministic. There are three equilibria of the difference equations. Two of the equilibria are the (two) homozygous states, and these are asymptotically stable. The third equilibrium is polymorphic but is unstable. It is proven that almost all populations converge to one of the two homozygous states. The remaining populations converge to the unstable equilibrium, and lie on a curve that separates the basins of attraction for the other two equilibria. This curve is shown to satisfy some simple properties.     

Evolution under Fertility and Viability Selection

Evolution at a single multiallelic locus under arbitrary weak selection on both fertility and viability is investigated. Discrete, nonoverlapping generations are posited for autosomal and X-linked loci in dioecious populations, but monoecious populations are studied in both discrete and continuous time. Mating is random. The results hold after several generations have elapsed. With an error of order s [i.e., O(s)], where s represents the selection intensity, the population evolves in Hardy-Weinberg proportions. Provided the change per generation of the fertilities and viabilities due to their explicit time dependence (if any) is O(s2), the rate of change of the deviation from Hardy-Weinberg proportions is O(s2). If the change per generation of the viabilities and genotypic fertilities is smaller than second order [i.e., o(s2)], then to O(s2) the rate of change of the mean fitness is equal to the genic variance. The mean fitness is the product of the mean fertility and the mean viability; in dioecious populations, the latter is the unweighted geometric mean of the mean viabilities of the two sexes. Hence, as long as there is significant gene frequency change, the mean fitness increases. If it is the fertilities of matings that change slowly [at rate o(s2)], the above conclusions apply to a modified mean fitness, defined as the product of the mean viability and the square root of the mean fertility.     

紫色土造林树种苗期生态条件的研究Ⅱ

本文讨论了19种树苗在几种不同性质和肥力水平紫色土上的生态反应和适应性,利用方差分析的方法阐明了不同土壤条件幼苗生长差异的显著性,利用簇群分析揭示了不同树种的生态相似性,根据幼苗的相对高度评价了土壤的肥力。     

Evolutionarily stable strategies of mutual help between relatives having unequal fertilities

The evolutionarily stable strategy of mutual help between relatives having unequal fertilities is studied in a kin selection model, which also takes into account competition between kins and the possibility of reciprocation. It turns out that competition and reciprocation can establish ESSs which are completely different from those expected by Hamilton's basic theory.     

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.     

Modifiers of mutation rate: a general reduction principle

A deterministic two-locus population genetic model with random mating is studied. The first locus, with two alleles, is subject to mutation and arbitrary viability selection. The second locus, with an arbitrary number of alleles, controls the mutation at the first locus. A class of viability-analogous Hardy-Weinberg equilibria is analyzed in which the selected gene and the modifier locus are in linkage equilibrium. It is shown that at these equilibria a reduction principle for the success of new mutation-modifying alleles is valid. A new allele at the modifier locus succeeds if its marginal average mutation rate is less than the mean mutation rate of the resident modifier allele evaluated at the equilibrium. Internal stability properties of these equilibria are also described.     

Competitive exclusion and coexistence of pathogens in a homosexually-transmitted disease model

A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population has been studied completely by Jiang and Chai in (J Math Biol 56:373-390, 2008). In this paper, we give a analysis for a SIS STD with two competing strains, where populations are divided into three differential groups based on their susceptibility to two distinct pathogenic strains. We investigate the existence and stability of the boundary equilibria that characterizes competitive exclusion of the two competing strains; we also investigate the existence and stability of the positive coexistence equilibrium, which characterizes the possibility of coexistence of the two strains. We obtain sufficient and necessary conditions for the existence and global stability about these equilibria under some assumptions. We verify that there is a strong connection between the stability of the boundary equilibria and the existence of the coexistence equilibrium, that is, there exists a unique coexistence equilibrium if and only if the boundary equilibria both exist and have the same stability, the coexistence equilibrium is globally stable or unstable if and only if the two boundary equilibria are both unstable or both stable.     

Multiple stable equilibria in a predator-prey system

Necessary and sufficient conditions are given for three equilibria to occur in a predatorprey model and conditions are given for two of these to be stable. The existence of two stable equilibria requires predator intraspecific competition for either space or food, and the lower the prey growth rate the stronger this predator self-regulation must be. A prey growth rate that is skewed to the right, the ability of a few predators to survive at low prey densities, and predators with high searching effectiveness, long handling times, and large maximum per capita rate of increase all make two stable equilibria more likely.     

严重退化生态系统不同恢复和重建措施的植物多样性与地力差异研究

对花岗岩区土壤严重退化生态系统（对照）及４种不同恢复和重建措施建筑多样性和地力研究结果表明,花岗岩区红壤严重退化生态系统（对照）植物多样地性极低,土壤肥力极差,生态环境极为恶劣,改为杨梅果园（措施Ａ）或多树种混交（措施Ｂ）后,植物多样性明显增大,林地土壤肥力得到一定程度恢复,生态系统朝着良性循环方向发展,采取封山育林（措施Ｃ）方法,林下植被层和群落多样性恢复最快,林地土壤肥力亦得到较快的恢复,保留     

General analysis of frequency-dependent sexual selection at a multi-allelic locus

A general model is analyzed in which arbitrarily frequency-dependent selection acts on one sex of a diploid population with several alleles at one locus, as a result of viability or mating-success differences. The existence of boundary and polymorphic equilibria is examined, and conditions for local stability, internal and external, are obtained. The status of Hardy-Weinberg approximations in studying stability and approach to equilibria is also considered. The general principles are then applied to two specific models: one where genotypes fall into two phenotypic classes; and one with a hierarchy of dominance where viability and sexual selection are opposed. In the latter case it is found that, of all the equilibria present, there is one and only one which could possibly be stable: the existence of a unique globally stable equilibrium might then be inferred.     

Equilibrium properties of a multi-locus, haploid-selection, symmetric-viability model

Under haploid selection, a multi-locus, diallelic, two-niche Levene (1953) model is studied. Viability coefficients with symmetrically opposing directional selection in each niche are assumed, and with a further simplification that the most and least favored haplotype in each niche shares no alleles in common, and that the selection coefficients monotonically increase or decrease with the number of alleles shared. This model always admits a fully polymorphic symmetric equilibrium, which may or may not be stable.We show that a stable symmetric equilibrium can become unstable via either a supercritical or subcritical pitchfork bifurcation. In the supercritical bifurcation, the symmetric equilibrium bifurcates to a pair of stable fully polymorphic asymmetric equilibria; in the subcritical bifurcation, the symmetric equilibrium bifurcates to a pair of unstable fully polymorphic asymmetric equilibria, which then connect to either another pair of stable fully polymorphic asymmetric equilibria through saddle-node bifurcations, or to a pair of monomorphic equilibria through transcritical bifurcations. As many as three fully polymorphic stable equilibria can coexist, and jump bifurcations can occur between these equilibria when model parameters are varied.In our Levene model, increasing recombination can act to either increase or decrease the genetic diversity of a population. By generating more hybrid offspring from the mating of purebreds, recombination can act to increase genetic diversity provided the symmetric equilibrium remains stable. But by destabilizing the symmetric equilibrium, recombination can ultimately act to decrease genetic diversity.