A new change-in-ratio procedure robust to unequal catchability of types of animal

The change-in-ratio technique is a useful practical procedure for the estimation of game animal population sizes. The major problem with this technique is failure of the assumption that both types of animals are captured or sighted with equal probabilities. Here we extend the change-in-ratio technique to the case where there are two removals with emphasis on the special situation where there are two consecutive single-type removals. The advantage of this extension is that it allows an estimation procedure which is robust to unequal capture or sighting probabilities. It is also possible to test the assumption of equal sighting probabilities. Some numerical results on mean squared error of the population size estimator for the new design and the traditional design are given. The procedure is illustrated on some juvenile grass carp data collected in a small pond where the population size is known. We believe this technique is potentially useful to wildlife and fisheries biologists and that more statistical research would be beneficial.     

Estimating selective advantage of two alleles in discrete time

A class of estimators for the selective advantage, s, in a Wright-Fisher model with two alleles, variable population size, and genic selection is derived via martingale theory. Explicit expressions are given for these estimators which only involve simple computation. The optimal estimate among this class of estimators is obtained. Asymptotic results are readily established by an application of a martingale central limit theorem. The performance of this optimal estimator is compared to known estimators by means of a simulation study.     

Distribution and dispersal in populations capable of resource depletion

Summary A simulation model has been used to investigate the influence of animal (insect) distribution and dispersal among exhaustable resource units (food plants). Population size and stability were used as measures of success. The results showed that population size and stability are highest when egg batch size is as large as can be supported by the average food plant or slightly larger if larval dispersal occurs. Clumping of egg batches of food plants increases population stability when egg batches are small by insuring that some food plants will not be overcrowded. Increasing the proportion of larval dispersers or the success of dispersers can increase or decrease population size and stability depending on the original egg batch distribution, but individuals which produce offspring some of which disperse, generally have a selective advantage. Density dependent larval dispersal decreases population stability. Finally, individuals with lower reproductive capacities can have a selective advantage over those with higher reproductive capacities under certain conditions of egg batch size and larval dispersal.     

Maximum-likelihood estimation of gene location by linkage disequilibrium.

Linkage disequilibrium, D, between a polymorphic disease and mapped markers can, in principle, be used to help find the map position of the disease gene. Likelihoods are therefore derived for the value of D conditional on the observed number of haplotypes in the sample and on the population parameter Nc, where N is the effective population size and c the recombination fraction between the disease and marker loci. The likelihood is computed explicitly for the case of two loci with heterozygote superiority and, more generally, by computer simulations assuming a steady state of constant population size and selective pressures or neutrality. It is found that the likelihood is, in general, not very dependent on the degree of selection at the loci and is very flat. This suggests that precise information on map position will not be obtained from estimates of linkage disequilibrium.     

The fixation probability of two competing beneficial mutations

Suppose that a beneficial mutation is undergoing a selective sweep when another beneficial mutation arises at a linked locus. We study the fixation probability of the double mutant, i.e., one (produced by recombination) that carries both mutations. Previous analysis works well for the case where the earlier beneficial mutation confers a greater selective advantage than the later mutation, but not so well in the opposite case. We present an approach to approximating the fixation probability in the case where the later mutation confers a greater selective advantage.     

The fixed-size Luria-Delbruck model with a nonzero death rate

What is the expected number of mutants in a stochastically growing colony once it reaches a given size, N? This is a variant of the famous Luria-Delbruck model which studies the distribution of mutants after a given time-lapse. Instead of fixing the time-lapse, we assume that the colony size is a measurable quantity, which is the case in many in-vivo oncological and other applications. We study the mean number of mutants for an arbitrary cell death rate, and give partial results for the variance. For a restricted set of parameters we provide analytical results; we also design a very efficient computational method to calculate the mean, which works for most of the parameter values, and any colony size, no matter how large. We find that a cellular population with a higher death rate will contain a larger number of mutants than a population of equal size with a smaller death rate. Also, a very large population will contain a larger percentage of mutants; that is, irreversible mutations act like a force of selection, even though here the mutants are assumed to have no selective advantage. Finally, we investigate the applicability of the traditional, 'fixed-time' approach and find that it approximates the 'fixed-size' problem whenever stochastic effects are negligible.     

Variation in the size of foxes in Scotland

Foxes become larger from south to north in Scotland, independently of the climate, the prey taken or the productivity of the areas in which each population lives. In areas where fluctuating vole populations are important as food, foxes born in high vole years are no larger than those born in poor years. However foxes became smaller after a severe winter in north-east Scotland, followed by a gradual increase in the size of each year class through a series of years with mild winters and expanding rabbit populations. It is suggested that food availability is determining the average size of foxes through selection, and the north/south cline in size is the result of increased hunting hours at higher latitudes during winter. The selective advantage of different size animals under conditions of different food availability is discussed.     

Genetic variability due to geographical inhomogeneity

The frequency of one of two alleles is studied as a function of position and time in a one, two, or three dimensional region. A nonlinear diffusion equation is employed. Each allele is assumed to have a selective advantage in some part of the region. An asymptotic solution is constructed for the case when the selection coefficient is large compared to the diffusion coefficient, i.e. when selection acts more rapidly than diffusion. Then as time increases, the solution tends to a cline, i.e. an equilibrium distribution in which both alleles are present everywhere, each predominating where it has the advantage. In a narrow region around the boundary where the selective advantage switches from one allele to the other, both alleles are present with comparable frequencies. Along a line normal to this boundary, the frequency varies as in a one dimensional habitat with a simple variation in selective advantage. The asymptotic solution is compared with the numerical solution for a special two dimensional case, and the agreement is found to be good.Research supported by the National Science Foundation.     

The Effect of Change in Population Size on DNA Polymorphism

The expected number of segregating sites and the expectation of the average number of nucleotide differences among DNA sequences randomly sampled from a population, which is not in equilibrium, have been developed. The results obtained indicate that, in the case where the population size has changed drastically, the number of segregating sites is influenced by the size of the current population more strongly than is the average number of nucleotide differences, while the average number of nucleotide differences is affected by the size of the original population more severely than is the number of segregating sites. The results also indicate that the average number of nucleotide differences is affected by a population bottleneck more strongly than is the number of segregating sites.     

Effect of population structure on the amount of polymorphism and the fixation probability under overdominant selection

Under overdominant selection, mutants substantially contribute to increase the amount of polymorphism. It is also known that under neutrality as the migration rates among demes decrease in a subdivided population, the amount of polymorphism increases along with the increase of the effective population size, N(e). In this study, under overdominant selection the effect of population subdivision on the amount of polymorphism was investigated using the diffusion approximation and the low migration approximation. It was shown that if selection is medium or strong (e.g., N(T)s > 1, where N(T) is the population size and s is the selective advantage of heterozygotes), the nucleotide diversity, pi, decreases along with the decrease of Nm against the increase of N(e), where N is the size of demes and m is the migration rate per deme. In addition, the ratio of the nucleotide diversity to the evolutionary rate also decreases along with the decrease of Nm. In some cases the ratio becomes smaller than that expected under neutrality as Nm decreases.     

A neutral model with fluctuating population size and its effective size

We consider a diffusion model with neutral alleles whose population size is fluctuating randomly. For this model, the effects of fluctuation of population size on the effective size are investigated. The effective size defined by the equilibrium average heterozygosity is larger than the harmonic mean of population size but smaller than the arithmetic mean of population size. To see explicitly the effects of fluctuation of population size on the effective size, we investigate a special case where population size fluctuates between two distinct states. In some cases, the effective size is very different from the harmonic mean. For this concrete model, we also obtain the stationary distribution of the average heterozygosity. Asymptotic behavior of the effective size is obtained when the population size is large and/or autocorrelation of the fluctuation is weak or strong.     

The origin of subfunctions and modular gene regulation

Evolutionary explanations for the origin of modularity in genetic and developmental pathways generally assume that modularity confers a selective advantage. However, our results suggest that even in the absence of any direct selective advantage, genotypic modularity may increase through the formation of new subfunctions under near-neutral processes. Two subfunctions may be formed from a single ancestral subfunction by the process of fission. Subfunction fission occurs when multiple functions under unified genetic control become subdivided into more restricted functions under independent genetic control. Provided that population size is sufficiently small, random genetic drift and mutation can conspire to produce changes in the number of subfunctions in the genome of a species without necessarily altering the phenotype. Extensive genotypic modularity may then accrue in a near-neutral fashion in permissive population-genetic environments, potentially opening novel pathways to morphological evolution. Many aspects of gene complexity in multicellular eukaryotes may have arisen passively as population size reductions accompanied increases in organism size, with the adaptive exploitation of such complexity occurring secondarily.     

A Markov Chain Model of Population Growth with an Application in Epidemiology

This paper is concerned with a class of population growth processes in discrete time; the simple epidemic process is considered as a specific example. A Markov chain model is constructed and standard Markov methods are used to study the main biological concepts. A simple and explicit formula is obtained for the transient distribution of the population size. Then, the cost of the process is defined and the joint probability generating function of its components is derived. Finally, the results are extended to the case where the inter-transition periods are bounded i.i.d. random variables.     

Soft Selective Sweeps in Complex Demographic Scenarios

Adaptation from de novo mutation can produce so-called soft selective sweeps, where adaptive alleles of independent mutational origin sweep through the population at the same time. Population genetic theory predicts that such soft sweeps should be likely if the product of the population size and the mutation rate toward the adaptive allele is sufficiently large, such that multiple adaptive mutations can establish before one has reached fixation; however, it remains unclear how demographic processes affect the probability of observing soft sweeps. Here we extend the theory of soft selective sweeps to realistic demographic scenarios that allow for changes in population size over time. We first show that population bottlenecks can lead to the removal of all but one adaptive lineage from an initially soft selective sweep. The parameter regime under which such “hardening” of soft selective sweeps is likely is determined by a simple heuristic condition. We further develop a generalized analytical framework, based on an extension of the coalescent process, for calculating the probability of soft sweeps under arbitrary demographic scenarios. Two important limits emerge within this analytical framework: In the limit where population-size fluctuations are fast compared to the duration of the sweep, the likelihood of soft sweeps is determined by the harmonic mean of the variance effective population size estimated over the duration of the sweep; in the opposing slow fluctuation limit, the likelihood of soft sweeps is determined by the instantaneous variance effective population size at the onset of the sweep. We show that as a consequence of this finding the probability of observing soft sweeps becomes a function of the strength of selection. Specifically, in species with sharply fluctuating population size, strong selection is more likely to produce soft sweeps than weak selection. Our results highlight the importance of accurate demographic estimates over short evolutionary timescales for understanding the population genetics of adaptation from de novo mutation.     

Discrete Clutch Sizes, Local Mate Competition, and the Evolution of Precise Sex Allocation

Optimal sex allocation under a population structure with local mate competition has been studied mainly in deterministic models that are based on the assumption of continuous clutch sizes; Hamilton's (1967) model is the classic example. When clutch sizes are small, however, this assumption is not appropriate. When taking the discrete nature of eggs into account it becomes critically important whether females control only the mean sex ratio (“binomial” females) or the variance as well (“precise” females). As both types of sex ratio control have been found, it is of interest to investigate their evolutionary stability. In particular, it may be questioned whether perfect control of the sex ratio is always favoured by natural selection when mating groups are small. Models based on discrete clutch sizes are developed to determine evolutionarily stable (ES) sex ratios. It is predicted that when all females are of the binomial type they should produce a lower proportion of daughters than predicted by Hamilton's model, especially when clutch size and foundress number are small. When all females are of the precise type, the ES number of sons should generally be either a stable mixed strategy or a pure strategy, but there are special cases (for two foundresses and particular clutch sizes) where the ES number of sons lies in a trajectory of neutrally stable mixed strategies; the predicted mean sex ratios can be either higher or lower than predicted by Hamilton's model. The existence of ES mixed strategies implies that individual females do not necessarily have to produce sex ratios with perfect precision; some level of imperfection can be tolerated (i.e., will not be selected against). When the population consists of both binomial and precise females, the latter always have a selective advantage. This advantage of precision does not disappear when precision approaches fixation in the population. The latter result contradicts the conclusions of Taylor and Sauer (1980) which is due to their way of expressing selective advantage; they define selective advantage as the between-generation increase per allele, which will always become vanishingly small when an allele reaches fixation, irrespective of fitness differences.     

The Genealogy of Samples in Models with Selection

We introduce the genealogy of a random sample of genes taken from a large haploid population that evolves according to random reproduction with selection and mutation. Without selection, the genealogy is described by Kingman''s well-known coalescent process. In the selective case, the genealogy of the sample is embedded in a graph with a coalescing and branching structure. We describe this graph, called the ancestral selection graph, and point out differences and similarities with Kingman''s coalescent. We present simulations for a two-allele model with symmetric mutation in which one of the alleles has a selective advantage over the other. We find that when the allele frequencies in the population are already in equilibrium, then the genealogy does not differ much from the neutral case. This is supported by rigorous results. Furthermore, we describe the ancestral selection graph for other selective models with finitely many selection classes, such as the K-allele models, infinitely-many-alleles models, DNA sequence models, and infinitely-many-sites models, and briefly discuss the diploid case.     

Evolution of senescence: Natural increase of populations displaying gompertz- or power-law death rates and constant or age-dependent maternity rates

Several hypothetical populations which differ in degrees of senescence are compared with respect to their rates of natural increase. The rate of natural increase is employed as a measure of selective advantage. The populations are characterized by their maternity and death rates, expressed as functions of age. Maternity rates are described by constant or quasi-human, age-dependent functions. Death rates are described by constant, Gompertzian (exponential) or power functions. Longevity functions, representing the probability of survival to a specific age, are obtained by integrating the death rate functions. The degree of senescence of a population is measured by the rapidity of ascent of its death-rate function or by the rectangularity of its longevity function. The increase in death rate late in life which constitutes senescence is compensated by a decrease in death rate early in life. The balance between the two changes in rate is, by assumption, such that the mean value of the longevity function is independent of the degree of senescence. This assumption makes it possible to separate the effects produced by the evolution of senescence from those caused by changes in longevity.The rate of natural increase is obtained by numerical solution of an integral characteristic equation. The results show that senescence is advantageous in all populations except those in which the maternity function is constant and the size is declining at a rapid rate. When the parameters entering into the longevity functions have values such that the functions approximate human longevity data, the improvement in the rate of natural increase resulting from senescence closely approaches limiting values obtained with the use of a precisely rectangular longevity function. Other results support the observation that reproduction at an early age confers greater selective advantage than equivalent reproduction later in life.     

Climate change: is the dark Soay sheep endangered?

It was recently reported that the proportion of dark-coloured Soay sheep (Ovis aries) in the Hebrides has decreased, despite the fact that dark sheep tend to be larger than lighter sheep, and there exists a selective advantage to large body size. It was concluded that an apparent genetic linkage between loci for the coat colour polymorphism and loci with antagonistic effects on body size explained the decrease. Those results explain why the proportion of dark animals is not increasing, but not why it is decreasing. Between 1985 and 2005 there was a significant increase in mean ambient temperature near the islands. We suggest that, while in the past a dark coat has offset the metabolic costs of thermoregulation by absorbing solar radiation, the selective advantage of a dark coat may be waning as the climate warms in the North Atlantic. In parallel, Bergman''s rule may be operating, reducing the selective advantage of large body size in the cold. Either or both of these mechanisms can explain the decrease in the proportion of dark-coloured larger sheep in this population in which smaller (and light-coloured) sheep should be favoured by their lower gross energy demand. If environmental effects are the cause of the decline, then we can expect the proportion of dark-coloured Soay sheep to decrease further.     

The role of weak selection and high mutation rates in nearly neutral evolution

Neutral dynamics occur in evolution if all types are ‘effectively equal’ in their reproductive success, where the definition of ‘effectively equal’ depends on the population size and the details of mutations. Empirically observed neutral genetic evolution in extremely large clonal populations can only be explained under current models if selection is completely absent. Such models typically consider the case where population dynamics occurs on a different timescale to evolution. However, this assumption is invalid when mutations are not rare in a whole population. We show that this has important consequences for the occurrence of neutral evolution in clonal populations. In highly connected type spaces, neutral dynamics can occur for all population sizes despite significant selective differences, via the forming of effectively neutral networks connecting rare neutral types. Biological implications include an explanation for the high diversity of rare types that survive in large clonal populations, and a theoretical justification for the use of neutral null models.