The expected number of heterozygous sites in a subdivided population.

A simple, exact formula is derived for the expected number of heterozygous sites per individual at equilibrium in a subdivided population. The model of infinitely many neutral sites is posited; the linkage map is arbitrary. The monoecious, diploid population is subdivided into a finite number of panmictic colonies that exchange gametes. The backward migration matrix is arbitrary, but time independent and ergodic (i.e., irreducible and aperiodic). With suitable weighting, the expected number of heterozygous sites is 4Neu, where Ne denotes the migration effective population number and u designates the total mutation rate per gene (or DNA sequence). For diploid migration, this formula is a good approximation if Ne >> 1.     

The time sequence of the relation between mean crowding and mean density with some population processes

Summary The theories of the stochastic processes are applied to construct mathematical models for describing the processes of population change as an ever changing the distribution of individuals in a space. These models consist of two mathematical expressions which are named the spatial distribution probability function (Q _n (t)) and the transition probability function (P _i,n (t)), respectively. The former gives the spatial distribution at any future time. Given an actual spatial distribution at any time, the latter function converts it to the spatial distribution at any future time. According to these models, we discussed the time sequence of the mean crowding-mean density relation (Iwao andKuno, 1971) in some population processes such as mortality, birth, immigration, growth, and their combined processes.     

The dentition of the Indian Knoll skeletal population: odontometrics and cusp number.

Data on the permanent dentition of 153 individuals from the well known Indian Knoll skeletal population are presented. Mesiodistal and buccolingual measurements were taken with a Helios dial caliper. Cusp number of maxillary and mandibular molars are recorded. The Indian Knoll dentition is larger than many modern groups but smaller than Australoid or Mesolithic groups. With the exception of maxillary 12, males have larger teeth than females in both dimensions. The lower canine is the most dimorphic tooth. Through rank order correlation, an association was shown between the sexual dimorphism of the mesiodistal and buccolingual dimensions. Compared to modern groups, the Indian Knoll population displays a moderate degree of sexual dimorphism in tooth size. In general, the coefficients of variation were greater for the more distal teeth within morphological classes. Amounts of size variability did not differ significantly between the sexes; moreover, rank order correlations indicated that patterns of variability in both dimensions were similar for males and females. The predominant cusp number pattern for upper molars is 4-3-3 and for lowers 5-5(4)-5. No sex differences were shown for cusp occurrence or bilateral asymmetry in cusp number.     

The mean and variance of the number of segregating sites since the last hitchhiking event

 Tight linkage may cause a reduction of nucleotide diversity in a chromosomal region if an advantageous mutation appears in that region which is driven to fixation by directional selection. This process is usually called genetic hitchhiking. If selection is strong, the entire process takes place during a time period of length 2s ln (2N) that is very short relative to 2N generations [s is the selection coefficient of the advantageous mutation and N the effective diploid population size]. On the time scale of 2N generations, which is characteristic for neutral evolution, we may therefore call this process a hitchhiking event. Using coalescent methods, we analyzed a model in which a hitchhiking event occurred in a chromosomal region of zero-recombination in the past at time x. Such a hitchhiking “catastrophe” wipes out completely genetic variation that existed in a population before that time. Standing variation observed at present must therefore be due to mutations that have arisen since time point x. Assuming that all newly arising mutations are neutral, we derived expressions for the expectation, variance and also for the higher moments of the number of nucleotide sites segregating in a sample of n genes as a function of x. The result for the first moment is then used to estimate the time back to the last hitchhiking event based on DNA polymorphism data from Drosophila. Assuming that directional selection is the sole determinant of the level of genetic variation in the gene regions surveyed, we obtained estimates of x that were typically in the order of 0.1N generations. Received 14 May 1996; received in revised form 26 August 1996     

On the genealogy of a population of biparental individuals

If one goes backward in time, the number of ancestors of an individual doubles at each generation. This exponential growth very quickly exceeds the population size, when this size is finite. As a consequence, the ancestors of a given individual cannot be all different and most remote ancestors are repeated many times in any genealogical tree. The statistical properties of these repetitions in genealogical trees of individuals for a panmictic closed population of constant size N can be calculated. We show that the distribution of the repetitions of ancestors reaches a stationary shape after a small number G(c) approximately log N of generations in the past, that only about 80% of the ancestral population belongs to the tree (due to coalescence of branches), and that two trees for individuals in the same population become identical after G(c)generations have elapsed. Our analysis is easy to extend to the case of exponentially growing population.     

The effective number of a population that varies cyclically in size. I. Discrete generations

We consider a dioecious population having numbers of males and females that vary over time in cycles of length k. It is shown that if k is small in comparison with the numbers of males and females in any generation of the cycle, the effective population number (or size), N(e), is approximately equal to the harmonic mean of the effective population sizes during any given cycle. This result holds whether the locus under consideration is autosomal or sex-linked and whether inbreeding effective population numbers or variance effective population numbers are involved in the calculation of N(e). If, however, only two successive generations in the cycle are considered and the population changes in size between these generations, the inbreeding effective population number, N(eI), differs from the variance effective population number, N(eV). The mutation effective population number turns out to be the same as the number derived using calculations involving probabilities of identity by descent. It is also shown that, at least in one special case, the eigenvalue effective population number is the same as N(eV).     

The effective number of a population that varies cyclically in size. II. Overlapping generations

We consider haploid and dioecious age-structured populations that vary over time in cycles of length k. Results are obtained for both autosomal and sex-linked loci if the population is dioecious. It is assumed that k is small in comparison with numbers of haploid individuals (or of numbers of males and females) in any generation of a cycle. The inbreeding effective population size N(e) is then approximately given by the expression [T summation operator (k-1)(j=0)1/[N(e)(j)T(j)]](-1), where N(e)(j) and T(j) are, respectively, the effective population size and generation interval that would hold if the population was at all times generated in the same way as at time j. The constant T, which is the effective overall generation interval, is defined to be k times the harmonic mean of the quantities T(j). Our expressions for T and N(e), in terms of N(e)(j) and T(j), are general, but the N(e)(j)s are derived under the assumption that offspring are produced according to Poisson distributions.     

Genetic quality of individuals impacts population dynamics

Ample evidence exists that an increase in the inbreeding level of a population reduces the value of fitness components such as fecundity and survival. It does not follow, however, that these decreases in the components of fitness impact population dynamics in a way that increases extinction risk, because virtually all species produce far more offspring than can actually survive. We analyzed the effects of the genetic quality (mean fitness) of individuals on the population growth rate of seven natural populations in each of two species of wolf spider in the genus Rabidosa , statistically controlling for environmental factors. We show that populations of different sizes, and different inbreeding levels, differ in population dynamics for both species. Differences in population growth rates are especially pronounced during stressful environmental conditions (low food availability) and the stressful environment affects smaller populations (<500 individuals) disproportionately. Thus, even in an invertebrate with an extremely high potential growth rate and strong density-dependent mortality rates, genetic factors contribute directly to population dynamics and, therefore, to extinction risk. This is only the second study to demonstrate an impact of the genetic quality of individual genotypes on population dynamics in a wild population and the first to document strong inbreeding–environment interactions for fitness among populations. Endangered species typically exist at sizes of a few hundred individuals and human activities degrade habitats making them innately more stressful (e.g. global climate change). Therefore, the interaction between genetic factors and environmental stress has important implications for efforts aimed at conserving the Earth's biodiversity.     

A minimum on the mean number of steps taken in adaptive walks

I consider the adaptation of a DNA sequence when mutant fitnesses are drawn randomly from a probability distribution. I focus on "gradient" adaptation in which the population jumps to the best mutant sequence available at each substitution. Given a random starting point, I derive the distribution of the number of substitutions that occur during adaptive walks to a locally optimal sequence. I show that the mean walk length is a constant:L = e-1, where e approximately 2.72. I argue that this result represents a limit on what is possible under any form of adaptation. No adaptive algorithm on any fitness landscape can arrive at a local optimum in fewer than a mean of L = e-1 steps when starting from a random sequence. Put differently, evolution must try out at least e wild-type sequences during an average bout of adaptation.     

The mathematical modelling of population dynamics taking into account the adaptive behavior of individuals

The general approach for modelling of abundance dynamic of biological populations and communities is offered. The mechanisms of individual adaptation in changing environment are considered. The approach is detailed for population models without structure and with age structure. The property of solutions are investigated. As examples the author studies the concrete definitions of general models by analogy with models of Ricker and May. Theoretical analysis and calculations shows that survival of model population in extreme situation increases if adaptive behaviour is taking into account.     

The impact of using related individuals for haplotype reconstruction in population studies

Recent studies have highlighted the dangers of using haplotypes reconstructed directly from population data for a fine-scale mapping analysis. Family data may help resolve ambiguity, yet can be costly to obtain. This study is concerned with the following question: How much family data (if any) should be used to facilitate haplotype reconstruction in a population study? We conduct a simulation study to evaluate how changes in family information can affect the accuracy of haplotype frequency estimates and phase reconstruction. To reconstruct haplotypes, we introduce an EM-based algorithm that can efficiently accommodate unrelated individuals, parent-child trios, and arbitrarily large half-sib pedigrees. Simulations are conducted for a diverse set of haplotype frequency distributions, all of which have been previously published in empirical studies. A wide variety of important results regarding the effectiveness of using pedigree data in a population study are presented in a coherent, unified framework. Insight into the different properties of the haplotype frequency distribution that can influence experimental design is provided. We show that a preliminary estimate of the haplotype frequency distribution can be valuable in large population studies with fixed resources.     

On the number of independent cultural traits carried by individuals and populations

In species subject to individual and social learning, each individual is likely to express a certain number of different cultural traits acquired during its lifetime. If the process of trait innovation and transmission reaches a steady state in the population, the number of different cultural traits carried by an individual converges to some stationary distribution. We call this the trait-number distribution. In this paper, we derive the trait-number distributions for both individuals and populations when cultural traits are independent of each other. Our results suggest that as the number of cultural traits becomes large, the trait-number distributions approach Poisson distributions so that their means characterize cultural diversity in the population. We then analyse how the mean trait number varies at both the individual and population levels as a function of various demographic features, such as population size and subdivision, and social learning rules, such as conformism and anti-conformism. Diversity at the individual and population levels, as well as at the level of cultural homogeneity within groups, depends critically on the details of population demography and the individual and social learning rules.     

A sharp minimum on the mean number of steps taken in adaptive walks

It was recently conjectured by H.A. Orr that from a random initial point on a random fitness landscape of alphabetic sequences with one-mutation adjacency, chosen from a larger class of landscapes, no adaptive algorithm can arrive at a local optimum in fewer than on average e-1 steps. Here, using an example in which the mean number of steps to a local optimum equals (A-1)/A, where A is the number of distinct "letters" in the "alphabet" from which sequences are constructed, it is shown that as originally stated, the conjecture does not hold. It is also demonstrated that (A-1)/A is a sharp minimum on the mean number of steps taken in adaptive walks on fitness landscapes of alphabetic sequences with one-mutation adjacency. As the example that achieves the new lower bound has properties that are not often considered as potential attributes for fitness landscapes-non-identically distributed fitnesses and negative fitness correlations for adjacent points-a weaker set of conditions characteristic of more commonly studied fitness landscapes is proposed under which the lower bound on the mean length of adaptive walks is conjectured to equal e-1.