Dynamic regimes in a model of single-locus density-dependent selection

A model of density-dependent selection in a Mendelian single-locus population was analyzed in the case where the fitnesses of genotypic forms are exponential functions of the population size. Analytical and numerical studies of the model were performed for a diallelic locus, and parametric regions were established for different dynamic behaviors of the model. The diallelic model of density-dependent selection was generalized to a multiallelic locus; the results of its analysis are described.     

Manifestation of multimodality in a simple ecological-genetic model of population evolution

An investigation of the nature of dynamics of the population size and genetic structure is carried out for a homogeneous ecologically limited population influenced by density-dependent r-K selection in a single diallelic genetic locus. The detailed study of the results of basic types of natural selection is carried out with additional consideration of the influence of initial conditions. It is shown that coexistence of several different asymptotic dynamic modes (with their own attraction basins) is possible in numerous enough parametric domains which are meaningful biologically.     

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.     

A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

Polymorphism with migration and selection

Summary Two single-locus, deterministic models with discrete nonoverlapping generations are formulated for the maintenance of genetic variation in each of two distinct biological situations. The first two models are applicable to an autosomal locus in an hermaphroditic plant population with mixed selfing and random mating. They describe the interaction of migration and viability selection for, respectively, an island migration model and for a subdivided population. Pollen as well as seed may disperse. Sufficient conditions are derived and discussed for the existence of protected polymorphism in the diallelic case. The remaining two models are pertinent to migration and selection at a single X-linked locus. An island model is again considered as well as that of a subdivided population. Mating is at random, selection occurs only through viability differences, and the migration structure for males and females may differ. For a diallelic population, protection conditions are derived and discussed vis-à-vis the autosomal case.M.M. was supported by a U.S. Public Health Service training grant (Grant No. GM780).     

Population trajectories for single locus additive fecundity selection and related selection models

A selection model which comprises models of additive fecundities as well as models of viability, fecundity, or differential mating selection acting only in one sex, is investigated for an autosomal gene locus in a population reproducing in nonoverlapping generations. The recurrence equations and basic properties of the genotypic population trajectories and equilibrium points are formulated for the multiallelic case. For the diallelic case, the trajectory development is discussed in more detail, and it is proven that every population trajectory converges to a Hardy-Weinberg equilibrium point.     

Natural selection and density-dependent population growth

Natural selection was studied in the context of density-dependent population growth using a single locus, continuous time model for the rates of change of population size and allele frequency. The maximization principle of density-dependent selection was applied to a class of fitness expressions with explicit recruitment and mortality terms. Three general results were obtained: First, at low population densities, the genetic basis of selection is the difference between the mean recruitment rate and the mean mortality rate. Second, at densities much higher than the equilibrium population size, selection is expected to act to minimize the mean mortality rate. Third, as the population approaches its equilibrium density, selection is predicted to maximize the ratio of the mean recruitment rate to the mean mortality rate.     

A diffusion model for geographically structured populations

Summary A diffusion model is derived for the evolution of a diploid monoecious population under the influence of migration, mutation, selection, and random genetic drift. The population occupies an unbounded linear habitat; migration is independent of genotype, symmetric, and homogeneous. The treatment is restricted to a single diallelic locus without dominance. With the customary diffusion hypotheses for migration and the assumption that the mutation rates, selection coefficient, variance of the migrational displacement, and reciprocal of the population density are all small and of the same order of magnitude, a boundary value problem is deduced for the mean gene frequency and the covariance between the gene frequencies at any two points in the habitat. Supported by the National Science Foundation (Grant No. DEB77-21494).     

Migration-selection polymorphism in dioecious populations

Summary Conditions are derived for a protected polymorphism in a dioecious population subdivided into an arbitrary number of demes which exchange migrants. Generations are discrete and nonoverlapping; mutation and random drift are neglected. The analysis is restricted to a diallelic autosomal locus. In contrast to the monoecious case, the protection criteria depend on the order of migration and selection; they become identical for adult and juvenile migration if both the male and female backward migration matrices are symmetric, or the migration or selection patterns in the two sexes are the same. The protection conditions are presented explicitly for the Levene model. A recessive allele is protected in a panmictic dioecious population if the unweighted average of the recessive-to-dominant fitness ratios in the two sexes exceeds unity.Supported by the National Science Foundation (Grant No. DEB77-21494)     

The equilibrium and stability for n alleles under the density-dependent selection

The problem of the equilibrium under the density-dependent selection for n-alleles belonging to one locus is considered. The new “measure of quality” of the population φ is introduced and it is shown that the equilibrium points are the constrained stationary points of the function φ and all locally stable equilibriums are its local maximum points. The analoque of the Fisher theorem for density-dependent selection is considered.     

Frequency- and density-dependent selection on a quantitative character

The equilibrium distribution of a quantitative character subject to frequency- and density-dependent selection is found under different assumptions about the genetical basis of the character that lead to a normal distribution in a population. Three types of models are considered: (1) one-locus models, in which a single locus has an additive effect on the character, (2) continuous genotype models, in which one locus or several loci contribute additively to a character, and there is an effectively infinite range of values of the genotypic contributions from each locus, and (3) correlation models, in which the mean and variance of the character can change only through selection at modifier loci. It is shown that the second and third models lead to the same equilibrium values of the total population size and the mean and variance of the character. One-locus models lead to different equilibrium values because of constraints on the relationship between the mean and variance imposed by the assumptions of those models.——The main conclusion is that, at the equilibrium reached under frequency- and density-dependent selection, the distribution of a normally distributed quantitative character does not depend on the underlying genetic model as long as the model imposes no constraints on the mean and variance.     

Interactive use of computer models in teaching population genetics

A technique using a computer with a graphical display unit to teach students the effects of genetic drift, selection and migration is described. Both diallelic and triallelic loci are discussed. The Fokker-Planck equation is used as the mathematical model of the genetic system, and its validity as an approximation in this context is demonstrated by an investigation into selection at the ABO locus. An appendix contains a derivation from the Fokker-Planck equation of the formula used in the paper for the gene frequency distribution at a multiallelic locus in equilibrium under selection and migration.     

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     

Frequency-dependent selection in logistic growth models

This paper describes the dynamics of a continuously reproducing diploid population with two alleles at one locus. The dependent variables are allele frequency and population density. We modify the basic density-dependent logistic growth model by inserting three possible types of frequency dependence in the fitness functions. These models are analyzed and contrasted with the purely density-dependent situation. Examples are given of periodic fluctuations in allele frequency and population density, which would be impossible for purely density-dependent fitness functions.     

Further properties of Gavrilets' one-locus two-allele model of maternal selection

I derive several properties of the model proposed by Gavrilets for maternal selection at a single diallelic locus. Most notably, (i) stable oscillations of genotype frequencies (i.e., cycling) can occur and (ii) in the special case in which maternal effects and standard viability selection act multiplicatively, maternal selection effectively acts on maternally derived alleles only.     

Heterozygote advantage and the maintenance of polymorphism for multilocus traits

Explaining how polymorphism is maintained in the face of selection remains a puzzle since selection tends to erode genetic variation. Provided an infinitely large unsubdivided population and no frequency-dependance of selective values, heterozygote advantage is the text book explanation for the maintenance of polymorphism when selection acts at a diallelic locus. Here, we investigate whether this remains true when selection acts at multiple diallelic loci. We use five different definitions of heterozygote advantage that largely cover this concept for multiple loci. Using extensive numerical simulations, we found no clear associations between the presence of any of the five definitions of heterozygote advantage and the maintenance of polymorphism at all loci. The strength of the association decreases as the number of loci increases or as recombination decreases. We conclude that heterozygote advantage cannot be a general mechanism for the maintenance of genetic polymorphism at multiple loci. These findings suggest that a correlation between the number of heterozygote loci and fitness is not warranted on theoretical ground.     

Dynamical analysis of density-dependent selection in a discrete one-island migration model

A system of non-linear difference equations is used to model the effects of density-dependent selection and migration in a population characterized by two alleles at a single gene locus. Results for the existence and stability of polymorphic equilibria are established. Properties for a genetically important class of equilibria associated with complete dominance in fitness are described. The birth of an unusual chaotic attractor is also illustrated. This attractor is produced when migration causes chaotic dynamics on a boundary of phase space to bifurcate into the interior of phase space, resulting in bistable genetic polymorphic behavior.     

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.     

Adaptive topography of fluctuating selection in a Mendelian population

An adaptive topography is derived for a large randomly mating diploid population under weak density-independent selection in a fluctuating environment. Assuming a stationary distribution of environmental states with no temporal autocorrelation, a diffusion approximation for population size and allele frequency, p, reveals that the expected change in p involves the gradient with respect to p of the stochastic intrinsic rate of increase (the density-independent long-run growth rate), r = r - sigma 2 e/2, where r is the mean Malthusian fitness in the average environment and is the environmental variance in population growth rate. The expected relative fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its fitness with population growth rate. The influence of fitness correlation between genotypes is illustrated by an analysis of the Haldane-Jayakar model of fluctuating selection on a single diallelic locus, and on two loci with additive effects on a quantitative character.     

Density-dependent demographic variation determines extinction rate of experimental populations

Understanding population extinctions is a chief goal of ecological theory. While stochastic theories of population growth are commonly used to forecast extinction, models used for prediction have not been adequately tested with experimental data. In a previously published experiment, variation in available food was experimentally manipulated in 281 laboratory populations of Daphnia magna to test hypothesized effects of environmental variation on population persistence. Here, half of those data were used to select and fit a stochastic model of population growth to predict extinctions of populations in the other half. When density-dependent demographic stochasticity was detected and incorporated in simple stochastic models, rates of population extinction were accurately predicted or only slightly biased. However, when density-dependent demographic stochasticity was not accounted for, as is usual when forecasting extinction of threatened and endangered species, predicted extinction rates were severely biased. Thus, an experimental demonstration shows that reliable estimates of extinction risk may be obtained for populations in variable environments if high-quality data are available for model selection and if density-dependent demographic stochasticity is accounted for. These results suggest that further consideration of density-dependent demographic stochasticity is required if predicted extinction rates are to be relied upon for conservation planning.