Corridors for migration between large subdivided populations, and the structured coalescent

We study the ancestral genetic process for samples from two large, subdivided populations that are connected by migration to, from, and within a small set of subpopulations, or demes. We consider convergence to an ancestral limit process as the numbers of demes in the two large, subdivided populations tend to infinity. We show that the ancestral limit process for a sample includes a recent instantaneous adjustment to the sample size and structure followed by a more ancient process that is identical to the usual structured coalescent, but with different scaled parameters. This justifies the application of a modified structured coalescent to some hierarchically structured populations.     

Population structure in the Connecticut Valley. I. Marital migration

This study reports on an analysis of marital migration among 12 communities in the Connecticut River Valley of Massachusetts during the years 1790-1849. Genetic inferences are drawn, and the requisite assumptions considered. The effect of geographic distance on genetic kinship is predicted using Malécot's isolation-by-distance model. The resulting estimates are discussed in terms of geographic and historical factors. The configuration of communities as predicted by kinship values approximates closely their actual geographic locations. Estimated genetic heterogeneity was low for the historical Connecticut Valley population, and community isolation breaks down rapidly over time. The region thus assumes its place among a number of sedentary, agricultural populations for which the isolation-by-distance model provides an adequate representation. A regression analysis which includes variables in addition to distance indicates that historical and economic factors contribute some additional explanatory power to the distribution of mating frequencies.     

On the non-existence of an optimal migration rate

 We show that an optimal migration rate may not exist in a population distributed over an infinite number of individual living sites if empty sites occur. This is the case when the mean number of offspring per individual μ is finite. We make the assumption of uniform migration to other sites whose rate is determined by the parent’s genotype or the offspring’s genotype at a single locus in a diploid hermaphrodite population undergoing random mating. In both cases, for μ small enough, any population at fixation would go to extinction. Moreover, in the latter case, for intermediate values of μ, the only fixation state that could resist the invasion of any mutant would lead the population to extinction. These are the two conditions for the non-existence of an optimal migration rate. They become less stringent as the cost for migration expressed by a coefficient of selection 1−β becomes larger, that is, closer to 1. The results are obtained assuming that the allele at fixation is either nondominant or dominant. Although the optimal migration rate is the same in both cases when it exists, the optimality properties may differ. Received 14 December 1995; received in revised form 5 April 1996     

Population structure in the Connecticut Valley: II. A comparison of multidimensional scaling solutions of migration matrices and isonymy

In a previous paper (Swedlund et al., 1984) we have described the population structure of the historical Connecticut River Valley of Massachusetts in terms of matrimonial migration matrices. Using procedures described by Morton (1973), Harpending and Jenkins (1974), Jorde (1980), and others the exchanges between subdivisions which make up the matrices are made column stochastic and analyzed to predict genetic kinship. Subsequently the kinship estimates within and between subdivisions can be interpreted as genetic covariance and compared to the actual geographic distances between the respective subdivisions using a principal components analysis. In the present paper we extend these results by applying nonmetric multidimensional scaling to the migration matrices, and to isonymy matrices based on the same communities. We demonstrate that the multidimensional scaling configurations of marital migration represent the actual geographic relationships between the communities quite effectively for this particular case study from historical Massachusetts. Moreover, we argue that while these migration data may provide good estimates of social and genetic exchange between the subdivisions, surname analysis may also be informative of processes not revealed in the migration matrices alone.     

Density-dependent migration and human population structure in historical Massachusetts

Studies of population structure often focus on the effects of population size and migration rates on genetic variation. Few studies, however, have investigated the relationship between these two factors. The purpose of this paper is to determine the extent to which migration (and gene flow) is density-dependent (that is, affected by population size) for populations in historical Massachusetts. Data from 4,859 marriage records were analyzed from four populations in north-central Massachusetts during the time period 1741 to 1849. These data were placed into 29 samples defined in terms of population and time cohort. Within each cohort the overall exogamy rate was computed along with three estimates of gene flow based on marital migration: local migration (k), long-distance migration (m), and effective migration rate (me). Three samples show unusually low rates that reflect the history of settlement. Regression analyses were used with the remaining samples, and they show nonlinear density-dependent migration that is unrelated to temporal trends. Migration is highest in samples with small population sizes (less than 800) and large population sizes (greater than 1,600). Migration is lowest in medium-sized populations. Two processes are suggested to explain this curvilinear relationship of migration and population size. In small populations, the lack of suitable potential mates and/or availability of settled land leads to an increase in migration into the population. As population size increases, this migration decreases. After populations reach a certain size, migration increases again, most likely reflecting the economic pull of larger populations. These patterns could act to enhance, or counter, genetic drift, depending on the direction of density dependence.     

The estimation of population size,migration rates and survival in a stratified population

Summary Estimates of survival, migration rates, and population size are developed for a triple catch marking experiment onn (n>-2) areas with migration among all areas and death in all areas occurring, but no recruitment (birth). This repressents the extension to three sampling times of the method ofChapman andJunge (1956) for estimates in a stratified population. The method is further extented to allow for ‘losses on capture’.     

The distribution of the coalescence time and the number of pairwise nucleotide differences in the "isolation with migration" model

This paper is concerned with the “isolation with migration” model, where a panmictic ancestral population gave rise to a symmetric n-island model, time τ ago. Explicit analytical expressions are derived for the probability density function of the coalescence time of a pair of genes sampled at random from the same subpopulation or from different subpopulations, and for the probability distribution of the number of pairwise nucleotide differences.     

Invasive advance of an advantageous mutation: nucleation theory

For sedentary organisms with localized reproduction, spatially clustered growth drives the invasive advance of a favorable mutation. We model competition between two alleles where recurrent mutation introduces a genotype with a rate of local propagation exceeding the resident's rate. We capture ecologically important properties of the rare invader's stochastic dynamics by assuming discrete individuals and local neighborhood interactions. To understand how individual-level processes may govern population patterns, we invoke the physical theory for nucleation of spatial systems. Nucleation theory discriminates between single-cluster and multi-cluster dynamics. A sufficiently low mutation rate, or a sufficiently small environment, generates single-cluster dynamics, an inherently stochastic process; a favorable mutation advances only if the invader cluster reaches a critical radius. For this mode of invasion, we identify the probability distribution of waiting times until the favored allele advances to competitive dominance, and we ask how the critical cluster size varies as propagation or mortality rates vary. Increasing the mutation rate or system size generates multi-cluster invasion, where spatial averaging produces nearly deterministic global dynamics. For this process, an analytical approximation from nucleation theory, called Avrami's Law, describes the time-dependent behavior of the genotype densities with remarkable accuracy.     

Seed migration and the structure of plant populations

Summary We use chemical typing to compare sub-populations of thyme (Thymus vulgaris L.) growing in the channels of several ravines where seed migration is expected from sub-populations on the plateu to associated basin subpopulations. The results indicate that seed migration does occur. However, there is little effective gene flow between sub-populations. We discuss the implications of restricted gene flow for population dynamics and structure.     

Cross-cultural analysis of migration rates: effects of geographic distance and population size.

A model is developed that treats migration rates among populations as a function of the geographic distance between them and the size of both sources and recipient population. Specifically, mij/mjj = a(Ni/Nj)pe-bd, where mij/mjj is the relative migration rate into population j from population i, Ni is the size of the source population, Nj is the size of the recipient population, d is the geographic distance between populations i and j, p is a measure of differential density-dependence, b is a measure of distance decay, and a is an adjustment parameter with little demographic meaning. Methods of parameter estimation and hypothesis testing using maximum likelihood are outlined. These methods are applied to migration matrix data from 13 samples obtained from the literature representing a wide range of ecological settings. All samples show a significant effect of geographic distance on migration, and all but one show a significant effect of differential population size. All but one sample show an overall tendency for migration to be negative density-dependent; that is, the relative migration rate is greater from larger populations to smaller populations than the reverse.     

Polymorphism with selection and genotype-dependent migration

For a single autosomal locus with multiple alleles both an island and a multiple-niche model with discrete nonoverlapping generations are formulated for the maintenance of genetic variability. Both models incorporate viability selection in an arbitrary way and allow for genotypic differences in the pertinent migration structure. Random drift is ignored, and mating is at random. A global analysis is given for the island model in the neutral case. For a subdivided population, conditions are derived for the existence of a protected polymorphism, and the model is examined in some special two-niche cases. Of particular consideration is the loss of neutral alleles due solely to population regulation and genotype-dependent migration, and the possible existence of equilibrium clines without selection.M. M. was supported by USPHS Pre-doctoral training grant No. GM 7197 to the University of Chicago; this work represents part of the author's Doctoral dissertation.     

Phenotypic consequences of nonrandom migration in the Jirels of Nepal

The phenotypic structure of human populations is shaped by a number of factors such as population size and marital migration. This paper examines the impact of migration on the between-village phenotypic differentiation of the Jirels, a tribal group of eastern Nepal. Data on stature and five cranial measurements for 526 individuals (males and females) are utilized to illustrate the patterns of phenotypic variation. A permutation method is used to generate the phenotypic consequences of random migration constrained to observed levels of movement. The results suggest that Jirel migration is nonrandom and that it produces higher levels of phenotypic differentiation than would result from a random migration process.     

Exact moment calculations for genetic models with migration,mutation, and drift

Using properties of moment stationarity we develop exact expressions for the mean and covariance of allele frequencies at a single locus for a set of populations subject to drift, mutation, and migration. Some general results can be obtained even for arbitrary mutation and migration matrices, for example: (1) Under quite general conditions, the mean vector depends only on mutation rates, not on migration rates or the number of populations. (2) Allele frequencies covary among all pairs of populations connected by migration. As a result, the drift, mutation, migration process is not ergodic when any finite number of populations is exchanging genes. In addition, we provide closed-form expressions for the mean and covariance of allele frequencies in Wright's finite-island model of migration under several simple models of mutation, and we show that the correlation in allele frequencies among populations can be very large for realistic rates of mutation unless an enormous number of populations are exchanging genes. As a result, the traditional diffusion approximation provides a poor approximation of the stationary distribution of allele frequencies among populations. Finally, we discuss some implications of our results for measures of population structure based on Wright's F-statistics.     

Demography and social organization of free-ranging Lemur catta in the Beza Mahafaly Reserve,Madagascar

In 1987, a long-term study of the demography of Lemur catta was begun in southern Madagascar. Eighty-five ringtailed lemurs were captured, marked, and released. Adult age classes were estimated using patterns of dental attrition. Including young, 155 individuals from nine groups were identified and monitored over 18 months. The study population of the reserve remained stable, with a growth rate of 0.98. Group sizes ranged from nine to 22 individuals (mean 14). Home ranges were larger (32 ha) and population densities lower (135/km²) than those for previously studied populations, and there was a relationship between habitat quality (e.g., no. of large trees) and these factors. At the beginning of the study, there were more adult males than females, but the sex ratio reached 1.00 by the last census. Females first gave birth at 3 years of age, and 80% or more of the females gave birth in 2 consecutive years. Fifty-two percent of the infants died in the first year and, given preliminary findings, only 40% of those born reach adulthood. Age-specific fertility patterns were similar to those reported for anthropoid primates. Forty-seven percent of the adult males migrated or were missing within a year. This included 78% of the 3–4 year olds and 38% of older age classes. No females were observed to migrate. One group split during the study. Demographic patterns are discussed and related to patterns in other populations of ringtailed lemurs as well as in anthropoids.     

Statistical analysis of the migration component of genetic drift

Statistical methods are introduced for analysis of the migration component of genetic drift, i.e., of the stochastic changes that affect allele frequencies during migration between local groups. Attention focuses on alpha M, a parameter that measures the extent to which this component of drift departs from the ideal of independent random sampling, and which can be interpreted as a measure of the extent to which migration is kin-structured. It is shown that alpha M can be estimated from genetic data, even in the absence of information about the genealogical relationships of migrants, and Monte-Carlo simulations are used to approximate the sampling distribution of the estimator under the null hypothesis of independent random sampling. Application of these methods to data from the Aland Islands, Finland, shows that the migration pattern there is consistent with the hypothesis of independent random sampling.     

The distribution of the coalescence time and the number of pairwise nucleotide differences in a model of population divergence or speciation with an initial period of gene flow

This paper is concerned with a model of “isolation with an initial period of migration”, where a panmictic ancestral population split into n descendant populations which exchanged migrants symmetrically at a constant rate for a period of time and subsequently became completely isolated. In the limit as the population split occurred an infinitely long time ago, the model becomes an “isolation after migration” model, describing completely isolated descendant populations which arose from a subdivided ancestral population. The probability density function of the coalescence time of a pair of genes and the probability distribution of the number of pairwise nucleotide differences are derived for both models. Whilst these are theoretical results of interest in their own right, they also give an exact analytical expression for the likelihood, for data consisting of the numbers of nucleotide differences between pairs of DNA sequences where each pair is at a different, independent locus. The behaviour of the distribution of the number of pairwise nucleotide differences under these models is illustrated and compared to the corresponding distributions under the “isolation with migration” and “complete isolation” models. It is shown that the distribution of the number of nucleotide differences between a pair of DNA sequences from different descendant populations in the model of “isolation with an initial period of migration” can be quite different from that under the “isolation with migration model”, even if the average migration rate over time (and hence the total number of migrants) is the same in both scenarios. It is also illustrated how the results can be extended to other demographic scenarios that can be described by a combination of isolated panmictic populations and “symmetric island” models.