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On February 8, 2019 the Atmosphere-Space Interaction Monitor (ASIM) passed a thunderstorm system north east of Puerto Rico and observed a TGF and an Elve from the same lightning stroke at the very beginning of a lightning flash. A second Elve was observed 456 ms later but without any signature of a TGF about 300 km south-east of the first Elve. The strokes associated with the two Elve events were detected by WWLLN and Vaisala, which allows for an absolute timing accuracy of the ASIM measurements of at least 100 us. Images of the lighting strokes support the source locations for the Elves and TGF. Both the rise time of the UV pulse by ASIM MMIA photometer and radio measurements from Puerto Rico indicate that the first stroke was an intracloud positive while the latter was a cloud-to-ground stroke. The UV emissions from the Elves preceded the optical emissions in 777 nm by50 us and 90 us, respectively. This can partly be explained by the scattering of 777 nm within the cloud. Current moments derived from radio measurements at Puerto Rico and Duke University indicate a fast (30 us) and large (200 kA) current pulse emitting an electromagnetic wave that produces an Elve and a slow (1-2 ms) current producing the optical signals.

Using the Modular X- and Gamma-ray Sensor (MXGS) and the Modular Multi-spectral Imaging Array (MMIA) of the Atmosphere-Space Interactions Monitor (ASIM), we investigate the time sequence of the Terrestrial gamma-ray flashes and the optical emissions from the associated lighting. A common observation in the ASIM data is that the TGFs are observed before or during a weak increase in the optical signals in 337 nm and 777.4 nm, and prior to- or at the onset of the main optical pulse. Using data from the MXGS and MMIA instruments for the period from April 2019, we assess the time sequence and the relationship between the observed TGF duration and the time between the onset of the TGF and the onset of the main optical pulse, with a relative timeing uncertainty of +/- 5 µs. The data prior to April 2019 is presented in Bjørge-Engeland et al.

In a genome alignment of five individuals of the ascomycete fungus Zymoseptoria pseudotritici , a close relative of the wheat pathogen Z. tritici (synonym Mycosphaerella graminicola ), we observed peculiar diversity patterns. Long regions up to 100 kb without variation alternate with similarly long regions of high variability. The variable segments in the genome alignment are organized into two main haplotype groups that have diverged ∼3% from each other. The genome patterns in Z. pseudotritici are consistent with a hybrid speciation event resulting from a cross between two divergent haploid individuals. The resulting hybrids formed the new species without backcrossing to the parents. We observe no variation in 54% of the genome in the five individuals and estimate a complete loss of variation for at least 30% of the genome in the entire species. A strong population bottleneck following the hybridization event caused this loss of variation. Variable segments in the Z. pseudotritici genome exhibit the two haplotypes contributed by the parental individuals. From our previously estimated recombination map of Z. tritici and the size distribution of variable chromosome blocks untouched by recombination we estimate that the hybridization occurred ∼380 sexual generations ago. We show that the amount of lost variation is explained by genetic drift during the bottleneck and by natural selection, as evidenced by the correlation of presence/absence of variation with gene density and recombination rate. The successful spread of this unique reproductively isolated pathogen highlights the strong potential of hybridization in the emergence of pathogen species with sexual reproduction.

Isoenzymes of adenosine deaminase (ADA) in the eelpout, Zoarces viviparus, are determined by an autosomal locus, Ada. In a population at Hov near Aarhus two common alleles and two rare variant alleles, mutually codominant, were found. Data on the segregation of alleles at locus Ada were obtained from samples of mother-offspring combinations. Free recombination between Ada and previously described enzyme loci was verified.

Inference in fitness in natural populations is discussed emphasizing the key role of the concept of selection components. The applicability of observational designs including different kinds of population samples of mother-offspring combinations is reviewed and analysed. It is shown that complete information about all selection components is obtainable for one-locus polymorphism by using mother-offspring combinations with three or more offspring per mother in organisms where every brood has only one sire.

The eelpout population in Kald Cove has been sampled in five consecutive years, from 1969-1974. Each winter a sample including mother-offspring combinations was procured and analyzed for Est/I/ . The samples in the years 1969-1971 have been published. This paper reports on the samples in 1972, 1973 and 1974. The total data of the five year period analyzed here comprised 12,607 adults (5935 motheroffspring combinations) and 253 adolescents. The adults have been sorted into five age classes, which permits the observation of seven discrete cohorts. The data have been subjected to a refined selection component analysis, introducing analysis of the time stability of the population and the adult zygotic selection component, i.e. the viability of differences between the genotypes during their mature life. The new data support the previously found deficit of heterozygotes among adults and indicate that it arises from juvenile zygotic selection, i.e. selection occurring by differential elimination of the zygotes before maturity. The previous preliminary indications of reproductive selection have not been confirmed in this more extensive analysis.

It is not generally realized that genetics has finally solved the age-old problem of the reason for the existence (i.e. the function) of sexuality and sex, and that only geneticists can properly answer the question, “Is sex necessary?” ▪ Abstract Optimality arguments and modifier theory are reviewed as paradigms for the study of the evolution of recombination. Optimality criteria (such as maximization of mean fitness) may agree with results from models developed in terms of the evolution of recombination at modifier loci. Modifier models demonstrate, however, that equilibrium mean fitness can decrease during the evolution of recombination rates and is not always maximized. Therefore, optimality arguments do not successfully predict the conditions under which increased or decreased recombination will evolve. The results from modifier models indicate that decreased recombination rates are usually favored when the population is initially near a polymorphic equilibrium with linkage disequilibrium. When the population is subject to directional selection or to deleterious mutations, increased recombination may be favored under certain conditions, provided that there is negative epistasis among alleles.

The genetic mechanism underlying an ‘Optimal outcrossing distance’ in plants (sensu Price and Waser) is discussed. Monte Carlo simulations of a population subdivided as a one-dimensional stepping-stone are used to evaluate possible genetic mechanisms. A simple genetic model with two types of unlinked loci, underdominant and partially dominant, with multiplicative effects on fitness, is found to create an ‘Optimal outcrossing distance’ under a wide range of parameter values. The results are compared to, and discussed in relation to a two-locus epistatic model, previously investigated by Campbell and Waser, and found to give very similar results.

Familial aggregation of complex diseases may have many causes in addition to and apart from genetic predisposition due to common ancestry. For example, exposure to an environment that induces susceptibility to a disease may produce similar familial aggregations when the environment is shared by family members. In general, according to the principles of (Johannsen 1903), the emergence of a disease phenotype is the result of the combined effects of the genotype of the individual and the environment that it experiences during development. The heritability of a disease is a measure of familial aggregation in terms of the covariances among family members relative to the variance in disease state in the general population. Thus heritability expresses the within-family resemblance, observed by Darwin and inferred by him to reflect inheritance, which formed the core of his (Darwin 1859) theory of evolution. Darwin’s inspiration originated from the practical use of family resemblance in animal breeding. Animal breeders have long known that a major obstacle to progress in genetic improvement is the interaction between familial aggregation of environments and the effects of similar genetics within families. The potential importance of this interaction, recognized in classical studies of the genetic epidemiology of complex diseases and other quantitative characters, has reemerged in studies of the effects of epigenetic modifications, their variation, and their transmission between generations.

The introgression of genes carried by a small group of immigrants is studied. The recipient and the donor populations differ at several autosomal loci subject to weak selection, and two allelic forms of each gene are considered. Fitness variation is determined by additive allelic effects, by dominance effects, and by two-locus additive-by-additive epistatic interaction of the effects of the alleles. The fate of the group of immigrants is quantified by the selection barrier that describes the cumulative mean fitness of the hybrids and hybrid descendants relative to the fitness of the resident population. The monomorphic and the polymorphic loci of the recipient population contribute differently to the selection barrier. If the genetic difference between recipient and donor population is small, then the contribution of the monomorphic loci is dominated by a positive term dependent on the difference in gene frequencies. The contribution of the polymorphic loci depends only on the difference of the leading order in the pairwise linkage disequilibria between the two populations. This contribution may be positive or negative; and, thus, polymorphic loci may either contribute to the barrier or inflate the introgression.

We study a population exposed to a lethal infectious disease. Host response is carried at one locus with two alleles while the pathogen occurs in two variants. Based on an Si-type epidemic model we derive explicit equations for the dynamics of each genotype. By assuming small variations in both host and disease, we obtain a separation in time scales between epidemic and evolutionary processes. This allows us to describe explicitly the changes in host and disease gene frequencies. The resulting model has a rich behaviour including multiple stable states and oscillations. However, in the oscillatory situation the model is degenerate excluding the possibility of limit cycles. We show that the degeneracy can only be removed by frequency dependent selection in the pathogen, for example by including direct interaction of virus in a free-living stage. The qualitative conclusions extend to an SIR-type epidemic model, where recovery with immunity from the disease is possible.

We survey the distribution of haplotypes at the self-incompatibility (SI) locus of Arabidopsis lyrata (Brassicaceae) at 12 locations spread over the species' natural distribution in Iceland. Previous investigations of the system have identified 34 functionally different S haplotypes maintained by frequency-dependent selection and arranged them into four classes of dominance in their phenotypic expression. On the basis of this model of dominance and the island model of population subdivision, we compare the distribution of S haplotypes with that expected from population genetic theory. We observe 18 different S haplotypes, recessive haplotypes being more common than dominant ones, and dominant ones being shared by fewer populations. As expected, differentiation, although significant, is very low at the S locus even over distances of up to 300 km. The frequency of the most recessive haplotype is slightly larger than expected for a panmictic population, but consistent with a subdivided population with the observed differentiation. Frequencies in nature reflect effects of segregation distortion previously observed in controlled crosses. The dynamics of the S-locus variation are, however, well represented by a 12-island model and our simplified model of dominance interactions.

Identification and characterization of the self-incompatibility genes in Brassicaceae species now allow typing of self-incompatibility haplotypes in natural populations. In this study we sampled and mapped all 88 individuals in a small population of Arabidopsis lyrata from Iceland. The self-incompatibility haplotypes at the SRK gene were typed for all the plants and some of their progeny and used to investigate the realized mating patterns in the population. The observed frequencies of haplotypes were found to change considerably from the parent generation to the offspring generation around their deterministic equilibria as determined from the known dominance relations among haplotypes. We provide direct evidence that the incompatibility system discriminates against matings among adjacent individuals. Multiple paternity is very common, causing mate availability among progeny of a single mother to be much larger than expected for single paternity.

Influenza in humans is characterised by strongly annual dynamics and antigenic evolution leading to partial escape from prior host immunity. The variability of new epidemic strains depends on the amount of virus currently circulating. In this paper, the amount of antigenic variation produced each year is dependent on the epidemic size. Our model reduces to a one-dimensional map and a full mathematical analysis is presented. This simple system suggests some basic principles which may be more generally applicable. In particular, for diseases with antigenic drift, vaccination may be doubly beneficial. Not only does it protect the population through classical herd immunity, but the overall case reduction reduces the chance of new variants being produced; hence, subsequent epidemics may be milder as a result of this positive feedback. Also, a disease with a high innate rate of antigenic variation will always be able to invade a susceptible population, whereas a disease with less potential for variation may require several introduction events to become endemic.

Evolutionary games have been applied as simple mathematical models of populations where interactions between individuals control the dynamics. Recently, it has been proposed to use this type of model to describe the evolution of tumour cell populations with interactions between cells. We extent the analysis to allow for synergistic effects between cells. A mathematical model of a tumour cell population is presented in which population-level synergy is assumed to originate through the interaction of triplets of cells. A threshold of two cooperating cells is assumed to be required to produce a proliferative advantage. The mathematical behaviour of this model is explored. Even this simple synergism (minor clustering effect) is sufficient to generate qualitatively different cell-population dynamics from the models published previously. The most notable feature of the model is the existence of an unstable internal equilibrium separating two stable equilibria. Thus, cells of a malignant phenotype can exist in a stable polymorphism, but may be driven to extinction by relatively modest perturbations of their relative frequency. The proposed model has some features that may be of interest to biological interpretations of gene therapy. Two prototypical strategies for gene therapy are suggested, both of them leading to extinction of the malignant phenotype: one approach would be to reduce the relative proportion of the cooperating malignant cell type below a certain critical value. Another approach would be to increase the critical threshold value without reducing the relative frequency of cells of the malignant phenotype.

Expectations for the time scale and structure of allelic genealogies in finite populations are formed under three models of sporophytic self-incompatibility. The models differ in the dominance interactions among the alleles that determine the self-incompatibility phenotype: In the SSIcod model, alleles act codominantly in both pollen and style, in the SSIdom model, alleles form a dominance hierarchy, and in SSIdomcod, alleles are codominant in the style and show a dominance hierarchy in the pollen. Coalescence times of alleles rarely differ more than threefold from those under gametophytic self-incompatibility, and transspecific polymorphism is therefore expected to be equally common. The previously reported directional turnover process of alleles in the SSIdomcod model results in coalescence times lower and substitution rates higher than those in the other models. The SSIdom model assumes strong asymmetries in allelic action, and the most recessive extant allele is likely to be the most recent common ancestor. Despite these asymmetries, the expected shape of the allele genealogies does not deviate markedly from the shape of a neutral gene genealogy. The application of the results to sequence surveys of alleles, including interspecific comparisons, is discussed.

We study a hybrid zone between two populations of a diploid organism. The populations differ at one locus. Homozygotes have equal fitnesses and the heterozygote fitness is reduced by β + δ (β is the birth rate deviation and δ is the death rate deviation). The populations extend along a one dimensional continuous habitat, and migration occurs by diffusion of individuals. The model is formulated as a set of simple continuous time demographic models without age structure for the three genotypes, and the system is transformed into three new variables, the total population size N, the gene frequency p, and the deviation from Hardy-Weinberg proportions F. The gene frequency in a steady state cline always follows a hyperbolic tangent closely. Analysis of the asymptotic behavior of the cline far from the hybrid zone suggests a qualitative prediction of the shape of N, p and F over the zone. For weak selection the shape is determined by a central steepness of $$\sqrt {\beta + \delta /4D} $$ , as observed by Bazykin in 1969, where D is the diffusion coefficient. For strong selection the cline is less steep than the Bazykin cline, and the form is dominated by the migration process. The steepness at the center of the cline is close to $$\sqrt {b/4D} $$ where b is the birth rate of homozygotes.

A model of determination of sex by an individual′s genotype at n loci is discussed. The parameters that determine the probability that an individual is male (or female) depend only on the loci at which the individual is heterozygous, extending the two-gene theory of Feldman et al . (1991, Genetics 129 , 297-312). The analysis uses a set indexation technique that allows a compact expression of the recombination process, and a transformation of the chromosome frequencies that induces a natural subgroup structure on hyperplanes in the frequency simplex. It is shown that the polymorphic equilibria correspond to these subgroups. Conditions are determined under which equilibria that exhibit the even sex ratio are stable. How recombination affects the stability of other equilibria, including those with linkage disequilibrium, are also explored.

Recombination, including chromosomal segregation, shuffles together the genetic material carried by different members of a sexual species. This genetic mixing unties the evolutionary fate of alleles at one locus from the fate of alleles at neighboring loci and can increase the amount of genetic variation found within a population. In the process, however, recombination separates advantageous gene combinations, the very gene combinations that enabled the parents to survive and reproduce. Whether or not the adaptation of a population to an environment is more rapid in the presence of recombination, that is, whether or not recombination speeds up the evolutionary process, depends critically on the ways in which this process is modelled. As we shall see, the effect of recombination depends on the population size, the initial population composition, and the selection regime under consideration.

The two conditions for stability of an evolutionary equilibrium, the m-stability and the δ-stability conditions, are discussed. The m-stability condition is a condition for the convergence of the population toward the equilibrium, and the δ-stability condition coresponds to a local version of the classic evolutionarily stable strategy (ESS) condition. Together the two conditions provide the condition for a continuous stable strategy. The convergence stability condition corresponds to the requirement for convergence due to initial increase of rare alleles in a monomorphic population, and the local ESS stability condition corresponds to the stability of a monomorphic population at the evolutionary equilibrium against the increase of rare alleles. In this way, an evolutionary equilibrium that is convergence stable, but not local ESS stable, will tend to become polymorphic. The local ESS stability condition therefore contributes more to a description of the dynamics of variation at an evolutionary equilibrium...

Selection due to differential viability is studied in an n-locus two-allele model using a set indexation that allows the simplicity of the one-locus two-allele model to be carried to multi-locus models. The existence condition is analyzed for polymorphic equilibria with linkage equilibrium: Robbins' equilibria. The local stability condition is given for the Robbins' equilibria on the boundaries in the generalized non-epistatic selection regimes of Karlin and Liberman (1979). These generalized non-epistatic regimes include the additive selection model, the multiplicative selection model and the multiplicative interaction model, and their symmetric versions cover all the symmetric viability models.

The lectures focus on the development of selection component analyses based on population samples including mother-offspring combinations. The need for separation of selection into components is discussed and exemplified by experimental data. The genetic analysis of samples of both simple and complete mother-offspring combinations is developed for one-locus polymorphisms. This is subsequently used to develop an analysis of selection components using population samples including mother-offspring combinations, and this analysis is exemplified by data on natural populations of Zoarces viviparus (simple mother-offspring data) and Gammarus oceanicus (complete mother-offspring data). Finally, the mother- offspring analyses are generalized to cover polymorphisms at multiple loci, and as an introduction to this some population theory for multiple loci is discussed. Index set notation for multi-locus gamete types is introduced and applied to problems concerning population mixing. In addition, the effects of recurrent immigration into a population are investigated in a generalization of Wright’s island model and in the stepping-stone cline model.

The study of very simple selection models has formed a basis to advance our knowledge of simultaneous evolution at two or more loci. Detailed theoretical investigations of the dynamical properties of these models have elucidated many phenomena concerning fitness interactions among loci and the interplay between selection and recombination. The approach commenced with the analysis of the two-locus symmetric viability model by Lewontin and Kojima (1960) and expanded to analyses of models with an additive or a multiplicative structure in fitness. These simplified models posess a unique equilibrium where every locus is polymorphic and where the alleles at the various loci occur independently in the gametes of the population. This equilibrium is a totally polymorphic Robbins equilibrium, and in general, its existence places strict requirements on the fitness matrix (Bodmer and Felsenstein, 1967) because it must be an equilibrium of the corresponding multiple allele model obtained for absolute linkage of the loci. The Robbins equilibrium point may be viewed as representing an equilibrium descending from a totally polymorphic equilibrium for absolute linkage as long as the property of linkage equilibrium is not emphasized. The symmetric viability models have been generalized to three loci by Feldman, Franklin, and Thomson (1974) and to multiple loci by Karlin (1977) and Karlin and Avni (1981). Multi-locus versions of the additive and the multiplicative models of selection have been considered by Roux (1974) and Karlin and Liberman (1978b). Karlin and Liberman (1979a, b) suggested and analyzed selection models with a general combination of additive and multiplicative structures. These models are formulated and analyzed as multiple allele models, but the analysis simplifies considerably when each locus is assumed to posess only two alleles. The description of genotypes and gamete types in two allele models is aided by a set indexation (Christiansen, 1987, 1989a) which retains much of the simplicity of arguments that is characteristic of classical one-locus population genetics.

Single gene models for the evolution of rates of selfing and sib mating in the presence of an inbreeding depression of fitness are treated. In models in which the genetic constitution of the male gamete pool available for random mating is not affected by prior inbreeding, an allele which increases the rate of inbreeding will succeed, provided the inbreeding depression is not too large. The opposite result holds when the proportion of male genotypes available for random mating are reduced according to their rate of prior inbreeding.

The transmission dynamics of a communicable disease in a subdivided population where the spread among groups follows the proportionate mixing model while the within-group transmission can correspond to prefered mixing, proportionate mixing among subgroups, or mixing between social and nonsocial subgroups, is analyzed. It is shown that the threshold condition for the disease to persist is that either (i) the disease can persist within at least one group through intragroup contacts, or—if (i) does not hold—(ii) the intergroup transmission is sufficiently high. The among-group transmission is computed as an average where each subgroup's reproductive number is weighted according to its intragroup activity level squared and the total number of cases that one infectious individual will cause through intragroup contacts. The model thus allows for a study of the relative importance of communitywide disease transmission and of disease transmission within geographically or socially separate groups.

The balance between the creation of associations between alleles at different loci by immigration and the convergence to linkage equilibrium due to the recombination process is studied in a theoretical model. The geographical structure of the model is a stepping-stone chain of populations linking two genetically constant source populations. The model assumes an arbitrary number of autosomal loci and considers genetic variation (two alleles at each locus) that is not subject to natural selection. The gene frequencies at each locus will then show a linear cline through the stepping-stone chain of populations. The deviation from linkage equilibrium through the stepping-stone cline is characterized by an equation for linear measures that provide the linkage disequilibrium measures for a given set of loci in terms of the gene frequencies and the linkage disequilibria in the source populations and in terms of the linkage disequilibrium measures through the cline for lower numbers of loci. Numerical examples of this iterative solution are given, and it is shown that the build-up of the higher order Bennett-disequilibria through the cline is considerably more pronounced than the build-up of two-locus disequilibria.

Sufficient conditions for protected polymorphism at an autosomal locus in a diploid organism are developed for a Levene model with explicit modeling of the population dynamics. Levene's condition for protected polymorphism, formulated in terms of weighted harmonic means of relative genotypic fitnesses, is only strictly valid for components of selection that are not immediately relevant for the regulation of the population size in the subpopulations. Global regulation of the population size produces Dempster's condition for protected polymorphism in terms of weighted arithmetic means of absolute genotypic fitnesses. The general condition for protection of an allele may be written in the same form as Levene's condition, but then the weights of the subpopulations depend on the fitnesses in the various habitats. The model allows for marginal habitats that cannot sustain a population if left isolated.

Evolution is considered at a single locus with multiple alleles. The alleles have additive effects on a character which is related to the exploitation of a food resource with a multidimensional quality description. Competition is described using the exploitative competition model of MacArthur and Levins. Directional selection favoring genotypes using resources closest to the resource mode will prevail as long as all genotypes on the average utilize resources far from this mode. When the utilization modes for all genotypes are close to the resource mode, then polymorphic genetic equilibria may exist. These equilibria can at most segregate m + 1 alleles, where m is the dimension of the resource quality description, except when special relations exist among the utilization modes of the genotypes. In these exceptional and degenerate cases curves of neutrally stable equilibria may exist, but in the general and nondegenerate cases the system converges to a globally stable equilibrium. The significance of these results for similar multiple locus models is discussed.

The theory of evolution at a selectively neutral locus that controls the recombination between two major loci that are under selection is studied. If the major loci are at a stable equilibrium in linkage disequilibrium under selection and recombination, then a mutation at the modifier locus will increase in frequency when rare if and only if it decreases the recombination fraction. If the major loci are in disequilibrium at a balance between selection against deleterious alleles and mutation towards them, then two new phenomena are observed. First, a recombination increasing mutation will succeed if the disequilibrium is negative and the modifier is sufficiently tightly linked to the major loci. Second, depending on the strength of selection, even if the disequilibrium is negative, recombination reduction may occur for looser linkage between the major and modifier loci.

The deterministic properties of a two-locus model with mutation and selection have been investigated. The mutation process is unidirectional, and the model is so constructed that the genetic variation at one locus is selectively neutral in the absence of a mutant allele at the other locus. All genotypes with three or four mutant alleles are deleterious, while the double heterozygotes may have the same fitness as the standard genotype. If one of the mutant alleles becomes fixed in the population, then the other locus will show a regular one-locus mutation-selection balance. Such a boundary equilibrium may be unstable or stable in the full two-locus setting. In the symmetric case, which is analyzed in details, the population will either go to one of the two boundary equilibria, or to a fully polymorphic equilibrium at which both the mutant alleles are rare. The origin of reproductive separation between two populations via the fixation of complementary deleterious mutants at different loci, and the fixation of nonfunctional alleles at duplicated loci, are two biological processes which both can be studied with the present model. In the last part of the paper we show how the results from the deterministic analysis can be used to predict how different factors will influence the rates of evolution in these systems.

The geographical variation in genotypic and morphological composition is compared for populations of the eelpout, Zoarces viviparus (L.), in Mariager Fjord, Denmark. The morphological characters are number of vertebrae, dorsal fin hard rays, pectoral fin rays, and pigment spots of dorsal fin. The geographical variation in these traits agrees well with that described by Johs. Schmidt half a century ago. The electrophoretical variation at five enzyme loci: Ada, EstIII, Got, PgmI and PgmII is described. The scale of differentiation is comparable for the two classes of traits, and differentiations between local populations down to a distance of about 10 km are shown.

Population samples including mother-offspring combinations provide information on the selection components: zygotic selection, sexual selection, gametic seletion and fecundity selection, on the mating pattern, and on the deviation from linkage equilibrium among the loci studied. The theory for the analysis of a sample consisting of adult individuals including combinations of a mother and one randomly chosen offspring in the case of n polymorphic autosomal loci with ma alleles at the ath locus, a = 1, 2,…, n, is developed for an organism with discrete non-overlapping generations. In this general analysis complete determination of the genotypes is assumed. The modifications in the analysis for the case n = 2 and m1 = m2 = 2 when the two double heterozygotes cannot be separated are discussed, and the analysis is illustrated by data from a population of Zoarces viviparus (a marine teleost).

The convergence to linkage equilibrium due to the recombination process is studied in a theoretical model of mixed selfing and random mating. The model assumes an arbitrary number of autosomal loci and considers genetic variation (two alleles at each locus) that is not subject to natural selection. The deviation from linkage equilibrium is given in terms of Bennett disequilibria that measure the deviation from Robbins proportions in the gametes, and these are generalized to disequilibrium measures that describe the deviation of genotypic frequencies from Hardy-Weinberg proportions. Alternatively, the derivation from linkage equilibrium is given in terms of linear measures of gametic and of genotypic disequilibrium. The convergence to linkage equilibrium is dominated by the convergence of the gametic frequencies to Robbins proportions, in that the convergence to the genotypic frequencies characteristic of partial selfing is comparatively rapid. At linkage equilibrium the genotypic frequencies deviate from the Hardy-Weinberg proportions, and the generalized disequilibrium measures describing the derviation from Hardy-Weinberg proportions are closely related to joint inbreeding coefficients for the loci.

A symmetric viability model for two loci with two alleles at one locus and m alleles at the other is suggested and analyzed. The analysis of the equilibria is complete if the two loci are absolutely linked, while if recombination is allowed the analysis is incomplete. The dynamics of the mode! resemble those of the two locus two allele model, namely that for loose linkage there will be no correlation between the loci and for tight linkage there may be strong correlation. The major caveats to this are: 1. The equilibria stable for tight linkage may belong to an array of different structures dependent on the selection and the number of alleles. 2. If both loci are overdominant in viability, the stable equilibria always contain all alleles segregating in the population; otherwise, the stable equilibria may only be two locus two allele high complementarity equilibria for tight linkage. 3. For intermediate linkage values and special selection values the boundary two locus two allele high complementarity equilibria may be stable simultaneously with the totally polymorphic central point at which there is no association between the loci.

Using the model of exploitative competition of R. H. MacArthur and R. Levins (1967, Amer. Natur. 101, 377–385), evolution at a gene locus which influences the niche position is considered. The locus has multiple alleles, and the contributions of the alleles to the genotypic value are additive. The resource spectrum and the utilization functions of the genotypes are assumed to be Gaussian. Evolution will make the mode of the niche converge to the resource optimum, as long as the allele contributions are small compared to the distance between the mode of the niche and the resource optimum. When this distance is of the same order of magnitude as the allele contributions, then the globally stable equilibrium will maintain at most two alleles in the population, unless the allele contributions are large. Classical overdominance is not needed to maintain polymorphism. This result predicts high linkage disequilibrium in similar multilocus models. It is concluded that intraspecific competition can be a powerful force in maintaining two-allele polymorphisms, and that it can maintain high linkage disequilibrium among closely linked loci.

Two unlinked polymorphisms of a marine teleost, one in hemoglobin and one in esterase, show parallel clines through Danish waters, while two unlinked phosphoglucosemutase polymorphisms maintain constant gene frequencies. It is argued that the clinal and the constant polymorphisms cannot be accounted for simultaneously by random genetic drift of selectively neutral genes. It is therefore concluded that selection is responsible for at least one of the two classes of geographical patterns, be it the clinal, the constant, or both.

Selection due to variation in the fecundity among matings of genotypes with respect to many loci each with two alleles is studied. The fitness of a mating depends only on the genotypic distinction between homozygote and heterozygote at each locus in the two individuals, and differences among loci are allowed. This symmetric fertility model is therefore a generalization of the multiple-locus symmetric viability model. The phenomena seen in the two-locus symmetric fertility model generalize—e.g., the possibility of joint stability of equilibria with linkage equilibrium and with linkage disequilibrium, and the existence of different types of totally polymorphic equilibria with the gametic proportions in linkage equilibrium. The central equilibrium with genotypic frequencies in Hardy-Weinberg proportions and gametic frequencies in Robbins proportions exists for all symmetric fertility models. For some symmetric fertility regimes additional equilibria exist with gametic frequencies in linkage equilibrium and with genotypic frequencies in Hardy-Weinberg proportions at all except one locus. These equilibria may exist in the dioecious symmetric viability model, and then they will be locally stable. For free recombination the stable equilibria show linkage equilibrium, but several of these with different numbers of polymorphic loci may be stable simultaneously.

Adaptation through natural selection is one of the key elements of the Darwinian theory of evolution. This process has been the focus of much attention, partly because of its importance in the understanding of biological evolution, and partly because it appears as a nice directed deterministic process. The annoying aspect of Darwin’s theory of natural selection is, however, that the raw material of the process, the pheno- typic variation, originates as random errors in the hereditary transmission.

Thirteen enzyme loci have been resolved by starch gel electrophoresis from sera and red blood cells of samples of 52 polar bears from eastern Greenland. No variation in the mobility of electrophoretic bands was observed. The genetic variability is thus low as in other large mammals except man.

A two-locus model with three alleles at one locus and two at the other is studied. The viability system is such that all double heterozygotes have fitness unity, all single heterozygotes have fitness w < 1 and all double homozygotes have fitness w2. The following are the major findings: 1. There are more stable equilibria for tight linkage than in the corresponding three-locus model, even though the number of chromosomes is lower. 2. The equilibria stable for tight linkage do not belong to a unique high complementarity class, as is the case for two alleles at each locus. Instead the strength of selection determines the structure of the equilibrium. 3. The increase in number of alleles seems to reduce the possible extent of assocation between the loci. 4. The measure of this association is not well defined, although we have suggested a statistically standard way of getting over this. 5. A mutation introduced while a population is in linkage disequilibrium may, per medium only of the change in number of alleles, destroy the linkage disequilibrium.

The n-locus two-allele symmetric viability model is considered in terms of the parameters measuring the additive epistasis in fitness. The dynamics is analysed using a simple linear transformation of the gametic frequencies, and then the recurrence equations depend on the epistatic parameters and Geiringer's recombination distribution only. The model exhibits an equilibrium, the central equilibrium, where the 2n gametes are equally frequent. The transformation simplifies the stability analysis of the central point, and provides the stability conditions in terms of the existence conditions of other equilibria. For total negative epistasis (all epistatic parameters are negative) the central point is stable for all recombination distributions. For free recombination either a central point (segregating one, two, ... or n loci) or the n-locus fixation states are stable. For no recombination and some epistatic parameters positive the central point is unstable and several boundary equilibria may be locally stable. The sign structure of the additive epistasis is therefore an important determinant of the dynamics of the n-locus symmetric viability model. The non-symmetric multiple locus models previously analysed are dynamically related, and they all have an epistatic sign structure that resembles that of the multiplicative viability model. A non-symmetric model with total negative epistasis which share dynamical properties with the similar symmetric model is suggested.

Laboratory populations of Drosophila melanogaster bearing the Curly and Plum marked second chromosome inversions were observed in selection experiments for ten discrete generations. Maximum likelihood estimates of the relative fitnesses of Curly, Plum, Curly-Plum, and wild phenotypes were obtained from selection trajectories. Using these estimates, measures of multiplicative and additive epistasis were calculated. These were partitioned into pre-sampling and post-sampling components, and both were found to be significant. In several cases the sign of the epistasis of the two components was reversed, and the direction of net epistasis depended on the particular inversion. The significance of partitioning epistasis into components is discussed in the light of two locus population genetic theory.