Effective size in management and conservation of subdivided populations

A numerical method for computing the eigenvalue variance effective size of a subdivided population connected by any fixed pattern of migration is described. Using specific examples it is shown that total effective size of a subdivided population can become less than the sum of the subpopulation sizes as a result of directionalities in the pattern of migration. For an extension of the model with threshold harvesting and local deterministic logistic population dynamic we consider the problem of maximizing the total harvesting yield with constraints on the total effective size. For some simple source-sink systems and more complicated population structures where subpopulations differ in their degree of isolation, it is shown to be optimal, for a given total effective size, to raise the harvesting thresholds relatively more in small and in isolated populations. Finally, we show how the method applies to populations which are supplemented, either intentionally or unintentionally. It is shown that the total effective size can be reduced by several orders of magnitude if the captive component of a population is much smaller than the wild component, even with symmetric backward migration.     

Effective population size,genetic diversity,and coalescence time in subdivided populations

A formula for the effective population size for the finite island model of subdivided populations is derived. The formula indicates that the effective size can be substantially greater than the actual number of individuals in the entire population when the migration rate among subpopulations is small. It is shown that the mean nucleotide diversity, coalescence time, and heterozygosity for genes sampled from the entire population can be predicted fairly well from the theory for randomly mating populations if the effective population size for the finite island model is used.     

Fixation indices in subdivided populations.

Without restricting the evolutionary forces that may be present, the theory of fixation indices, or F-statistics, in an arbitrarily subdivided population is developed systematically in terms of allelic and genotypic frequencies. The fixation indices for each homozygous genotype are expressed in terms of the fixation indices for the heterozygous genotypes. Therefore, together with the allelic frequencies, the latter suffice to describe population structure. Possible random fluctuations in the allelic frequencies (which may be caused, e.g., by finiteness of the subpopulations) are incorporated so that the fixation indices are parameters, rather than random variables, and these parameters are expressed in terms of ratios of evolutionary expectations of heterozygosities. The interpretation of some measures of population differentiation is also discussed. In particular, F(ST) is an appropriate index of gene-frequency differentiation if and only if the genetic diversity is low.     

Gamete frequencies at two sex-linked loci in random mating age-structured populations

A study is made of the change with time of frequencies of gametic types with one or two sex-linked loci in an infinite random mating age-structured population. Recurrence equations for these gamete frequencies are derived under the assumptions that all matings of adults are equally fertile and the number of matings at any time is proportional to the number of mature females at that time. These generalize others in the literature. It is shown that gamete frequencies approach their limiting values at geometric rates in the long run. This implies that the asymptotic behavior of the gamete frequencies is like what it is in populations with discrete generations if the unit of time is replaced by an appropriately chosen generation interval. With either one locus or two loci, the generation interval is bounded below by an analogous measure from standard demographic theory. This result also holds when there are two autosomal loci. In numerical examples from both this paper and a previous one by Pollak and Callanan, the lower bound is a good estimate of the generation interval.     

Effective size of fluctuating salmon populations

Pacific salmon are semelparous but have overlapping year classes, which presents special challenges for the application of standard population genetics theory to these species. This article examines the relationship between the effective number of breeders per year (N(b)) and single-generation and multigeneration effective population size (N(e)) in salmon populations that fluctuate in size. A simple analytical model is developed that allows calculation of N(e) on the basis of the number of spawners in individual years and their reproductive contribution (productivity) to the next generation. Application of the model to a 36-year time series of data for a threatened population of Snake River chinook salmon suggests that variation in population dynamic processes across years reduced the multigeneration N(e) by approximately 40-60%, and reductions may have been substantially greater within some generations. These reductions are comparable in magnitude to, and in addition to, reductions in N(b) within a year due to unequal sex ratio and nonrandom variation in reproductive success. Computer simulations suggest that the effects of variable population dynamics on N(e) observed in this dataset are not unexpected for species with a salmon life history, as random variation in productivity can lead to similar results.     

Effective size of harvested ungulate populations

The harvest of ungulate populations is often directed against certain sex or age classes to maximize the yield in terms of biomass, number of shot animals or number of trophies. Here we examine how such directional harvest affects the effective size of the population. We parameterize an age-specific model assumed to describe the dynamics of Fennoscandian moose. Based on expressions for the demographic variance     for a small subpopulation of heterozygotes Aa bearing a rare neutral allele a , we use this model to calculate how different harvest strategies influence the effective size of the population, given that the population remains stable after harvest. We show that the annual genetic drift, determined by     , increases with decreasing harvest rate of calves and increasing sex bias in the harvest towards bulls 1 year or older. The effective population size per generation decreased with reduced harvest of calves and increased harvest of bulls 1 year or older. The magnitude of these effects depends on the age-specific pattern of variation in reproductive success, which influences the demographic variance. This shows that the choice of harvest strategy strongly affects the genetic dynamics of harvested ungulate populations.     

Fixation probability and time in subdivided populations

New alleles arising in a population by mutation ultimately are either fixed or lost. Either is possible, for both beneficial and deleterious alleles, because of stochastic changes in allele frequency due to genetic drift. Spatially structured populations differ from unstructured populations in the probability of fixation and the time that this fixation takes. Previous results have generally made many assumptions: that all demes contribute to the next generation in exact proportion to their current sizes, that new mutations are beneficial, and that new alleles have additive effects. In this article these assumptions are relaxed, allowing for an arbitrary distribution among demes of reproductive success, both beneficial and deleterious effects, and arbitrary dominance. The effects of population structure can be expressed with two summary statistics: the effective population size and a variant of Wright's F(ST). In general, the probability of fixation is strongly affected by population structure, as is the expected time to fixation or loss. Population structure changes the effective size of the species, often strongly downward; smaller effective size increases the probability of fixing deleterious alleles and decreases the probability of fixing beneficial alleles. On the other hand, population structure causes an increase in the homozygosity of alleles, which increases the probability of fixing beneficial alleles but somewhat decreases the probability of fixing deleterious alleles. The probability of fixing new beneficial alleles can be simply described by 2hs(1 - F(ST))N(e)/N(tot), where hs is the change in fitness of heterozygotes relative to the ancestral homozygote, F(ST) is a weighted version of Wright's measure of population subdivision, and N(e) and N(tot) are the effective and census sizes, respectively. These results are verified by simulation for a broad range of population structures, including the island model, the stepping-stone model, and a model with extinction and recolonization.     

Genetic identity in subdivided populations. I. Two equal-sized subpopulations

Moments of the steady state frequency spectrum (probabilities of identity of samples of genes) are obtained for a subdivided population by using standard recursive identity by state calculations. These moments are used to obtain variances for some measures of genetic identity, including Nei's normalized genetic identity (I) and genetic distance (?log_eI). The results are compared with those obtained from the corresponding undivided population theory, including adjustments to the effective number to try to account for subdivision. Undivided population approximations based on effective number are surprisingly accurate, regardless of the migration rate, when sampling exclusively from one subpopulation.     

Analysis of disruptive selection in subdivided populations

Background

Analytical methods have been proposed to determine whether there are evolutionarily stable strategies (ESS) for a trait of ecological significance, or whether there is disruptive selection in a population approaching a candidate ESS. These criteria do not take into account all consequences of small patch size in populations with limited dispersal.

Results

We derive local stability conditions which account for the consequences of small and constant patch size. All results are derived from considering R_m, the overall production of successful emigrants from a patch initially colonized by a single mutant immigrant. Further, the results are interpreted in term of concepts of inclusive fitness theory. The condition for convergence to an evolutionarily stable strategy is proportional to some previous expressions for inclusive fitness. The condition for evolutionary stability stricto sensu takes into account effects of selection on relatedness, which cannot be neglected. It is function of the relatedness between pairs of genes in a neutral model and also of a three-genes relationship. Based on these results, I analyze basic models of dispersal and of competition for resources. In the latter scenario there are cases of global instability despite local stability. The results are developed for haploid island models with constant patch size, but the techniques demonstrated here would apply to more general scenarios with an island mode of dispersal.

Conclusions

The results allow to identity and to analyze the relative importance of the different selective pressures involved. They bridge the gap between the modelling frameworks that have led to the R_m concept and to inclusive fitness.

     

Breakdown of gametophytic self-incompatibility in subdivided populations

In many hermaphroditic flowering plants, self-fertilization is prevented by self-incompatibility (SI), often controlled by a single locus, the S-locus. In single isolated populations, the maintenance of SI depends chiefly on inbreeding depression and the number of SI alleles at the S-locus. In subdivided populations, however, population subdivision has complicated effects on both the number of SI alleles and the level of inbreeding depression, rendering the maintenance of SI difficult to predict. Here, we explore the conditions for the invasion of a self-compatible mutant in a structured population. We find that the maintenance of SI is strongly compromised when a population becomes subdivided. We show that this effect is mainly caused by the decrease in the local diversity of SI alleles rather than by a change in the dynamics of inbreeding depression. Strikingly, we also find that the diversity of SI alleles at the whole population level is a poor predictor of the maintenance of SI. We discuss the implications of our results for the interpretation of empirical data on the loss of SI in natural populations.     

The genetic mating structure of subdivided populations I. Open-mating model

The genetic mating structure of a subdivided population can describe how parental genotypes gave rise to zygotes. When parents of the same genotype are considered together as one class (“open-mating”), three independent parameters of inbreeding and mating structure are needed to describe this structure at a diallelic locus. One is Wright's fixation index F. The other two are mating structure parameters, derived herein and termed the “effective selfing” rate E and the “inbreeding assortative selfing” rate D. E is the genetically equivalent proportion of self-fertilization at a single locus, and is given by standardized second and third central moments of gene frequencies of mates. E is a summary measure of inbreeding that includes effects due to self-fertilization and mating to relatives, as well as correlations between mates induced by Wahlund effects and/or selective diversification among neighborhoods. The second parameter D measures the tendency of inbred or more homozygous individuals to effectively self more (or less) than outbred or more heterozygous individuals. D is related to the maintenance of variation of inbreeding among individuals and/or to the prevalence of spatial variation of selection. D is independent of E, but together with E controls the generational change of inbreeding, ΔF. Extensions of the model to unequal allele frequencies in male vs female mates, and to multi-allelic loci, are also examined.     

Effective size of populations with heritable variation in fitness

The effective size of monogamous populations with heritable variation in fitness is formulated, and the expression obtained is compared with a published equation. It is shown that the published equation for dioecious populations is inappropriate for most animal and human populations, because the derivation is implicitly based on the assumption that zygotes are produced by random union of gametes, each from conceptual male and female gametic pools. A convenient equation for practical use is proposed, and the application is illustrated with the estimation of the effective size of a rural human community in Japan.     

Characterization and conservation of genetic diversity in subdivided populations

We review the available tools for analysing genetic diversity in conservation programmes of subdivided populations. Ways for establishing conservation priorities have been developed in the context of livestock populations, both from the classical population genetic analysis and from the more recent Weitzman's approach. We discuss different reasons to emphasize either within or between-population variation in conservation decisions and the methodology to establish some compromise. The comparison between neutral and quantitative variation is reviewed from both theoretical and empirical points of view, and the different procedures for the dynamic management of conserved subdivided populations are discussed.     

Stability properties of underdominance in finite subdivided populations

IN ISOLATED populations underdominance leads to bistable evolutionary dynamics: below a certain mutant allele frequency the wildtype succeeds. Above this point, the potentially underdominant mutant allele fixes. In subdivided populations with gene flow there can be stable states with coexistence of wildtypes and mutants: polymorphism can be maintained because of a migration-selection equilibrium, i.e., selection against rare recent immigrant alleles that tend to be heterozygous. We focus on the stochastic evolutionary dynamics of systems where demographic fluctuations in the coupled populations are the main source of internal noise. We discuss the influence of fitness, migration rate, and the relative sizes of two interacting populations on the mean extinction times of a group of potentially underdominant mutant alleles. We classify realistic initial conditions according to their impact on the stochastic extinction process. Even in small populations, where demographic fluctuations are large, stability properties predicted from deterministic dynamics show remarkable robustness. Fixation of the mutant allele becomes unlikely but the time to its extinction can be long.     

Effective population size and population subdivision in demographically structured populations

A fast-timescale approximation is applied to the coalescent process in a single population, which is demographically structured by sex and/or age. This provides a general expression for the probability that a pair of alleles sampled from the population coalesce in the previous time interval. The effective population size is defined as the reciprocal of twice the product of generation time and the coalescence probability. Biologically explicit formulas for effective population size with discrete generations and separate sexes are derived for a variety of different modes of inheritance. The method is also applied to a nuclear gene in a population of partially self-fertilizing hermaphrodites. The effects of population subdivision on a demographically structured population are analyzed, using a matrix of net rates of movement of genes between different local populations. This involves weighting the migration probabilities of individuals of a given age/sex class by the contribution of this class to the leading left eigenvector of the matrix describing the movements of genes between age/sex classes. The effects of sex-specific migration and nonrandom distributions of offspring number on levels of genetic variability and among-population differentiation are described for different modes of inheritance in an island model. Data on DNA sequence variability in human and plant populations are discussed in the light of the results.     

Analysis of genetic diversity for the management of conserved subdivided populations

Recent studies in the literature have appliedphylogenetic methods based on genetic distancesto set priorities for conservation of domesticanimal breeds. While these methods may beappropriate for between-species conservation,they are clearly inappropriate forwithin-species breed conservation, because theyignore within-breed variation. In this paper weshow the basic tools to analyse geneticdiversity in subdivided populations withinspecies, and illustrate the errors incurred byapplying methods based exclusively on geneticdistances. We also show that maximisation ofgenetic diversity (minimisation of coancestryor kinship) is equivalent to maximisation ofeffective population size, as in undividedpopulations, and derive a generalisation ofprevious equations for the prediction ofeffective size. Finally, we discuss thestrategies for conservation in the light of thetheory.