Biological growth and spread modeled by systems of recursions. II. Biological theory

The mathematical theory developed in Part I is applied to a selection-migration model in population genetics with sex-linked locus and to the host-vector or venereal disease epidemic model. In both models, a constant c*(xi) is found for each unit vector xi. The mathematical results imply that under certain initial conditions, the frequency of the advantageous gene in the male and female gametic outputs or the epidemic will spread at a speed c*(xi) in the direction xi as time goes to infinity. Time is measured in discrete nonoverlapping generations. In most cases, we can find a formula for c*(xi).     

Long-term response to selection I. Relation to breeding population size, intensity, and accuracy with additive gene action

Dual CDC-6500 computers were used to simulate the probabilistic aspects of genetic selection and reproduction in random mating populations with additive gene action. These simulations involved either 10 or 20 replicates of 200 consecutive nonoverlapping generations for 72 combinations of breeding population sizes, mating ratios, selection intensities, and accuracies of genetic determination of quantitative phenotypes. The results demonstrate that at least some long-term responses can be characterized by modified exponential functions that, with increasing generations, approach asymptotically to limits whose expectations increase linearly with the inverse tangent of multiples of the expected initial responses. The multiplicative constants are greater for populations with large effective breeding population size than for those of smaller size. Agreement with and discrepancies from past theoretical results are discussed. The supposition is made that the general form of these equations will be retained for broader situations than those simulated, but probably not for nonadditive gene action.     

The Dynamics of Finite Haploid Populations with Overlapping Generations. II. the Diffusion Approximation

The dynamics of a gene in a haploid population can be explained approximately by considering the average reproductive value of the gene. The dynamics of the average reproductive value are similar to those of a gene in a population with nonoverlapping generations with the following modifications: The effective population size, N_e, replaces N; the average mutation rates µ* and ν* replace µ and ν; the average overall selection r*+(T-1)s** replaces s; and time is measured in terms of generations, T. The implications of the average selection coefficient to adaptive life histories are discussed.     

A genetic model with mutation and selection

Two deterministic models of a multiallele population in which mutation and selection both operate are considered, and formulae for the gene frequencies are obtained. Both models are of a diploid population in which selection is additive and mutation is general; generations are discrete and nonoverlapping. In the first model, the stationary solution of the discrete equations is found. In the second, the discrete time process is approximated by a continuous time one, and the resulting differential equations are solved. The transient case for two alleles is solved explicitly, and the results are graphed. An application is given to sequences of sites.     

The effects of seasonality on discrete models of population growth

The effects of seasonality on the dynamics of a bivoltine population with discrete, nonoverlapping generations are examined. It is found that large seasonality is inevitably destabilizing but that mild seasonality may have a pronounced stabilizing effect. Seasonality also allows for the coexistence of alternative stable states (equilibria, cycles, chaos). These solutions may be seasonally in-phase, out-of-phase, or asynchronous. In-phase solutions correspond to winter regulation of population density, whereas out-of-phase solutions correspond to summer regulation. Analysis suggests that summer regulation is possible only in mildly seasonal habitats.     

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.     

Selection and mutation at a diallelic X-linked locus

Equilibria and convergence of gene frequencies are studied in the case of a diallelic X-linked locus under the influence of selection and mutation. The model used is that of an infinite diploid population with nonoverlapping discrete generations and random mating. It is proved that if the mutation rates and fitnesses are constant and the mutation rates are less than one-third, then global convergence of gene frequencies to equilibria occurs. The phase portraits of the dynamical system describing the change of allelic frequencies from one generation to the next are determined. Convergence of gene frequencies is monotone from a certain generation on if every other generation is skipped. In the case without mutation, our proof of this monotone convergence simplifies G. Palm's original proof [37].     

Biological populations obeying difference equations: stable points, stable cycles, and chaos.

For biological populations with nonoverlapping generations, population growth takes place in discrete time steps and is described by difference equations. Some of the simplest such nonlinear difference equations can exhibit a remarkable spectrum of dynamical behavior, from stable equilibrium points, to stable cyclic oscillations between two population points, to stable cycles with four points, then eight, 16, etc., points, through to a chaotic regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but bounded population fluctuations, can occur. This rich dynamical structure is overlooked in conventional linearized stability analyses; its existence in the simplest and fully deterministic nonlinear (“density dependent”) difference equations is a fact of considerable mathematical and ecological interest.     

Selection-mutation at a diallelic autosomal locus in a dioecious population

The population is assumed to be infinite dioecious with nonoverlapping discrete generations and random mating. It is assumed that the fitnesses and mutation rates are constant, heterozygotes are viable and the mutation rates are less than one-half. It is proved that the allelic frequencies converge to equilibria as the number of generations tends to infinity. The a priori types of phase portraits are determined. The method employed is elementary. The results extend those of [1, 2, 5, 8] to the case of selection-mutation rather than pure selection and those of [7] to the case of an autosomal rather than a sex-linked locus.     

A note on difference-delay equations.

Populations that obey differential-delay equations, and those that obey ordinary first-order difference equations, may be understood within a common general framework, wherein excessive time lags lead to stable limit cycle behavior. This note extends the analysis to include difference-delay equations (i.e., nonoverlapping generations with explicit time lags in the density dependent regulatory mechanisms).     

Neutral Genetic Patterns for Expanding Populations with Nonoverlapping Generations

We investigate the inside dynamics of solutions to integrodifference equations to understand the genetic consequences of a population with nonoverlapping generations undergoing range expansion. To obtain the inside dynamics, we decompose the solution into neutral genetic components. The inside dynamics are given by the spatiotemporal evolution of the neutral genetic components. We consider thin-tailed dispersal kernels and a variety of per capita growth rate functions to classify the traveling wave solutions as either pushed or pulled fronts. We find that pulled fronts are synonymous with the founder effect in population genetics. Adding overcompensation to the dynamics of these fronts has no impact on genetic diversity in the expanding population. However, growth functions with a strong Allee effect cause the traveling wave solution to be a pushed front preserving the genetic variation in the population. In this case, the contribution of each neutral fraction can be computed by a simple formula dependent on the initial distribution of the neutral fractions, the traveling wave solution, and the asymptotic spreading speed.     

Estimation of population divergence times from non-overlapping genomic sequences: examples from dogs and wolves

Despite recent technological advances in DNA sequencing, incomplete coverage remains to be an issue in population genomics, in particular for studies that include ancient samples. Here, we describe an approach to estimate population divergence times for non-overlapping sequence data that is based on probabilities of different genealogical topologies under a structured coalescent model. We show that the approach can be adapted to accommodate common problems such as sequencing errors and postmortem nucleotide misincorporations, and we use simulations to investigate biases involved with estimating genealogical topologies from empirical data. The approach relies on three reference genomes and should be particularly useful for future analysis of genomic data that comprise of nonoverlapping sets of sequences, potentially from different points in time. We applied the method to shotgun sequence data from an ancient wolf together with extant dogs and wolves and found striking resemblance to previously described fine-scale population structure among dog breeds. When comparing modern dogs to four geographically distinct wolves, we find that the divergence time between dogs and an Indian wolf is smallest, followed by the divergence times to a Chinese wolf and a Spanish wolf, and a relatively long divergence time to an Alaskan wolf, suggesting that the origin of modern dogs is somewhere in Eurasia, potentially southern Asia. We find that less than two-thirds of all loci in the boxer and poodle genomes are more similar to each other than to a modern gray wolf and that--assuming complete isolation without gene flow--the divergence time between gray wolves and modern European dogs extends to 3,500 generations before the present, corresponding to approximately 10,000 years ago (95% confidence interval [CI]: 9,000-13,000). We explicitly study the effect of gene flow between dogs and wolves on our estimates and show that a low rate of gene flow is compatible with an even earlier domestication date ～30,000 years ago (95% CI: 15,000-90,000). This observation is in agreement with recent archaeological findings and indicates that human behavior necessary for domestication of wild animals could have appeared much earlier than the development of agriculture.     

The Power of QTL Mapping with RILs

QTL (quantitative trait loci) mapping is commonly used to identify genetic regions responsible to important phenotype variation. A common strategy of QTL mapping is to use recombinant inbred lines (RILs), which are usually established by several generations of inbreeding of an F1 population (usually up to F6 or F7 populations). As this inbreeding process involves a large amount of labor, we are particularly interested in the effect of the number of inbreeding generations on the power of QTL mapping; a part of the labor could be saved if a smaller number of inbreeding provides sufficient power. By using simulations, we investigated the performance of QTL mapping with recombinant inbred lines (RILs). As expected, we found that the power of F4 population could be almost comparable to that of F6 and F7 populations. A potential problem in using F4 population is that a large proportion of RILs are heterozygotes. We here introduced a new method to partly relax this problem. The performance of this method was verified by simulations with a wide range of parameters including the size of the segregation population, recombination rate, genome size and the density of markers. We found our method works better than the commonly used standard method especially when there are a number of heterozygous markers. Our results imply that in most cases, QTL mapping does not necessarily require RILs at F6 or F7 generations; rather, F4 (or even F3) populations would be almost as useful as F6 or F7 populations. Because the cost to establish a number of RILs for many generations is enormous, this finding will cause a reduction in the cost of QTL mapping, thereby accelerating gene mapping in many species.     

Intraindividual variability in behavior shapes fitness landscapes

Although intraindividual variability (IIV) in behavior is fundamental to ecological dynamics, the factors that contribute to the expression of IIV are poorly understood. Using an individual‐based model, this study examined the effects of stochasticity on the evolution of IIV represented by the residual variability of behavior. The model describes a population of prey with nonoverlapping generations, in which prey take refuge upon encountering a predator. The strategy of a prey is characterized by the mean and IIV (i.e., standard deviation) of hiding duration. Prey with no IIV will spend the same duration hiding in a refuge at each predator encounter, while prey with IIV will have variable hiding durations among encounters. For the sources of stochasticity, within‐generation stochasticity (represented by random predator encounters) and between‐generation stochasticity (represented by random resource availability) were considered. Analysis of the model indicates that individuals with high levels of IIV are maintained in a population in the presence of between‐generation stochasticity even though the optimal strategy in each generation is a strategy with no IIV, regardless of the presence or absence of within‐generation stochasticity. This contradictory pattern emerges because the mean behavioral trait and IIV do not independently influence fitness (e.g., the sign of the selection gradient with respect to IIV depends on the mean trait). Consequently, even when evolution eventually leads toward a strategy with no IIV (i.e., the optimal strategy), greater IIV may be transiently selected. Between‐generation stochasticity consistently imposes such transient selection and maintain high levels of IIV in a population.     

Adaptation at Specific Loci. I. Natural Selection on Phosphoglucose Isomerase of Colias Butterflies: Biochemical and Population Aspects

Electrophoretic variants of phosphoglucose isomerase (PGI) in Colias butterflies have been studied from field and laboratory viewpoints. The transmission pattern is that of a dimeric enzyme controlled by one structural gene locus. Populations usually harbor four to six allelic mobility classes. These mobility classes are shared among species complexes, though their frequencies differ widely. Preliminary Ferguson plot analysis of the variants has been carried out. Purified preparations of Colias PGI alleles are more effective in standardizing Ferguson plots than heterologous proteins, such as ferritin. Variation of Ferguson plot parameters is not an infallible guide to electrophoretically "cryptic alleles," as one putative case proved to be due to nonallele-specific effects. S, M, and F mobility classes in two Colias semispecies show the same retardation coefficients in Ferguson plots. Adults early in the flight periods of their nonoverlapping generations show genotype frequencies in Hardy-Weinberg equilibrium, but heterozygote excess develops as the insects age. Simple directional selection and large-scale population mixing are unlikely to be causes of this, although several other selection modes remain possible. Identical-by-descent lines of the four frequent-to-common alleles in C. eurytheme have been set up in culture, and enzyme has been purified from these for study of functional properties. Major differenecs in heat stability and in various kinetic parameters are found among the ten possible genotypes. In some cases, heterosis for kinetic parameters is seen; in other cases, opposing trends in kinetic function and heat stability create potential for net heterosis in function. Possible interpretations of these results in an adaptive metabolic context are discussed, and directions for further work are stated.     

Cluster size polymorphism of active human ribosomal genes and simulation of the conditions of its stability through generations

Based on selective silver nitrate staining of active ribosomal gene (AcRG) clusters in nucleolus organizer regions (NORs) of human metaphase chromosomes, a technique was developed earlier to estimate the AcRG dosage in individual genomes as a sum of arbitrary units (0–3) ascribed to the silver precipitate (AgNOR) on ten NORs. The AcRG dosage was considered to be an additive quantitative trait determined by five polymorphic autosomal loci (with four allelic forms for each locus). A database was created to contain the data on AcRG cluster variants for more than 1000 individual human genomes. In this study, the population frequencies of AcRG cluster variants were determined. The results agreed with the hypothesis that stabilizing selection acts at the zygotic and/or early embryogenetic stage to restrain the AcRG genomic dosage (copy number) within a range from 14.9 to 23.7 arbitrary units (the cell is unviable when the trait is beyond this range). The average zygotic losses due to selection were estimated at 9.1–9.9% for a real population. A computer model where the AcRG dosage of a progeny results from a random combination of the AgNORs of the five acrocentric chromosome pairs of the parents was developed and used to simulate the formation of a certain AcRG genomic dosage through generations in a human panmictic population with nonoverlapping generations. A combination of stabilizing selection by total AcRG copy number and a certain spontaneous mutation rate (the probability of changes in the cluster size of a NOR as a result of unequal crossingover in meiotic prophase) was shown to be a sufficient condition for the restrain of equilibrium population frequencies of AgNOR size variants in a human panmictic population. Using the model, the most probable spontaneous mutation frequency was predicted to be (2.1–2.3) × 10⁻² per NOR per generation for human AgNORs. The predicted frequency was within the 95% confidence interval of the experimental rate, which was determined by studying the inheritance of AgNOR variants in real families.     

Population Genetics of Euphydryas Butterflies. I. Genetic Variation and the Neutrality Hypothesis

Twenty-one populations of the checkerspot butterfly, Euphydryas editha, and ten populations of Euphydryas chalcedona were sampled for genetic variation at eight polymorphic enzyme loci. Both species possessed loci that were highly variable from population to population and loci that were virtually identical across all populations sampled. Our data indicate that the neutrality hypothesis is untenable for the loci studied, and therefore selection is indicated as the major factor responsible for producing these patterns. Thorough ecological work allowed gene flow to be ruled out (in almost all instances) as a factor maintaining similar gene frequencies across populations. The Lewontin-Krakauer test indicated magnitudes of heterogeneity among standardized variances of gene frequencies inconsistent with the neutrality hypothesis. The question of whether or not to correct this statistic for sample size is discussed. Observed equitability of gene frequencies of multiple allelic loci was found to be greater than that predicted under the neutrality hypothesis. Genetic differentiation persisting through two generations was found between the one pair of populations known to exchange significant numbers of individuals per generation. Two matrices of genetic distance between populations, based on the eight loci sampled, were found to be significantly correlated with a matrix of environmental distance, based on measures of fourteen environmental parameters. Correlations between gene frequencies and environmental parameters, results of multiple regression analysis, and results of principle component analysis showed strong patterns of association and of "explained" variation. The correlation analyses suggest which factors might be further investigated as proximate selective agents.     

The Evolution of One- and Two-Locus Systems. II

Weak selection in a monoecious population is studied in two multiallelic panmictic models. In the first, a single locus is considered with continuous time and age-independent fertilities and mortalities. If the fertilities of the various matings and the genotypic mortalities may be expressed with an error at most of the second order in s (i.e., O(s ²)), where s is the intensity of selection, as sums of terms corresponding to the different genotypes and alleles, respectively, then after several generations the deviations from Hardy-Weinberg proportions are of O(s²). In the second model, two loci are treated with discrete nonoverlapping generations. It is shown that if the epistatic parameters are of O(s²), then after several generations the linkage disequilibria are reduced to O(s²). Assuming only weak selection, it is proved that in both models, after several generations, the total change in mean fitness is generally positive. It is likely that the exclusion of the initial period is usually unnecessary in natural populations. Exceptions are discussed.     

Genetic structure of an isolated native population group of northern Sibiria, the Nganasani (Tavgi) of the Taimyr. II. An analysis of intrapopulation variability

Chi-square contingency table analysis of phenotypic (genotypic) and gene frequencies of erythrocyte and blood serum groups and enzymes in a group of reindeer hunter and fishermen revealed heterogeneity within the population studied. Four out of twelve loci which have been compared were found to be involved in the process of differentiation into two local subgroups (subpopulations). No statistical differences have been observed between samples arbitrarily representing three generations. The data obtained support the hypothesis that the whole population still preserves the state of the stability. Traditional migration from adjacent populations close to nganasans by language and culture has made an important contribution into heterogeneity found in nganasans.     

Continuous-discrete models of the dynamics of an isolated population and of 2 competing species

The paper presents the analysis of various mathematical models for dynamics of isolated population and for competition between two species. It is assumed that mortality is continuous and birth of individuals of new generations takes place in certain fixed moments. Influence of winter upon the population dynamics and conditions of classic discrete model "deduction" of population dynamics (in particular, Moran-Ricker and Hassel's models) are investigated. Dynamic regimes of models under various assumptions about the birth and death rates upon the population states are also examined. Analysis of models of isolated population dynamics with nonoverlapping generations showed the density changes regularly if the birth rate is constant. Moreover, there exists a unique global stable level and population size stabilizes asymptotically at this equilibrium, i.e. cycle and chaotic regimes in various discrete models depend on correlation between individual productivity and population state in previous time. When the correlation is exponential upon mean population size the discrete Hassel model is realized. Modification of basis model, based on the assumption that during winter survival/death changes are constant, showed that population size at global level is stable. Generally, the dependence of population rate upon "winter parameters" has nonlinear character. Nonparametric models of competition between two species does not vary if the individual productivity is constant. In a phase space there are several stable stationary states and population stabilizes at one or other level asymptotically. So, in discrete models of competition between two species oscillation can be explained by dependence of population growth rate on the population size at previous times.