Measurement error and estimates of population extinction risk

It is common to estimate the extinction probability for a vulnerable population using methods that are based on the mean and variance of the long‐term population growth rate. The numerical values of these two parameters are estimated from time series of population censuses. However, the proportion of a population that is registered at each census is typically not constant but will vary among years because of stochastic factors such as weather conditions at the time of sampling. Here, we analyse how such sampling errors influence estimates of extinction risk and find sampling errors to produce two opposite effects. Measurement errors lead to an exaggerated overall variance, but also introduce negative autocorrelations in the time series (which means that estimates of annual growth rates tend to alternate in size). If time series data are treated properly these two effects exactly counter balance. We advocate routinely incorporating a measure of among year correlations in estimating population extinction risk.     

Selection against demographic stochasticity in age-structured populations

It has been shown that differences in fecundity variance can influence the probability of invasion of a genotype in a population; i.e., a genotype with lower variance in offspring number can be favored in finite populations even if it has a somewhat lower mean fitness than a competitor. In this article, Gillespie's results are extended to population genetic systems with explicit age structure, where the demographic variance (variance in growth rate) calculated in the work of Engen and colleagues is used as a generalization of "variance in offspring number" to predict the interaction between deterministic and random forces driving change in allele frequency. By calculating the variance from the life-history parameters, it is shown that selection against variance in the growth rate will favor a genotypes with lower stochasticity in age-specific survival and fertility rates. A diffusion approximation for selection and drift in a population with two genotypes with different life-history matrices (and therefore different mean growth rates and demographic variances) is derived and shown to be consistent with individual-based simulations. It is also argued that for finite populations, perturbation analyses of both the mean and the variance in growth rate may be necessary to determine the sensitivity of fitness to changes in the life-history parameters.     

Connection between stochastic and deterministic modelling of microbial growth

We present in this paper various links between individual and population cell growth. Deterministic models of the lag and subsequent growth of a bacterial population and their connection with stochastic models for the lag and subsequent generation times of individual cells are analysed. We derived the individual lag time distribution inherent in population growth models, which shows that the Baranyi model allows a wide range of shapes for individual lag time distribution. We demonstrate that individual cell lag time distributions cannot be retrieved from population growth data. We also present the results of our investigation on the effect of the mean and variance of the individual lag time and the initial cell number on the mean and variance of the population lag time. These relationships are analysed theoretically, and their consequence for predictive microbiology research is discussed.     

The effect of population history on the lengths of ancestral chromosome segments

An isolated population is a group of individuals who are descended from a founding population who lived some time ago. If the founding individuals are assumed to be noninbred and unrelated, a chromosome sampled from the population can be represented as a mosaic of segments of the original ancestral types. A population in which chromosomes are made up of a few long segments will exhibit linkage disequilibrium due to founder effect over longer distances than a population in which the chromosomes are made up of many short segments. We study the length of intact ancestral segments by obtaining the expected number of junctions (points where DNA of two distinct ancestral types meet) in a chromosome. Assuming random mating, we study analytically the effects of population age, growth patterns, and internal structure on the expected number of junctions in a chromosome. We demonstrate that the type of growth a population has experienced can influence the expected number of junctions, as can population subdivision. These effects are substantial only when population sizes are very small. We also develop an approximation to the variance of the number of junctions and show that the variance is large.     

Statistics for Microsatellite Variation Based on Coalescence

The stepwise mutation model, which was at one time chiefly of interest in studying the evolution of protein charge-states, has recently undergone a resurgence of interest with the new popularity of microsatellites as phylogenetic markers. In this paper we describe a method which makes it possible to transfer many population genetics results from the standard infinite sites model to the stepwise mutation model. We study in detail the properties of pairwise differences in microsatellite repeat number between randomly chosen alleles. We show that the problem of finding the expected squared distance between two individuals and finding the variance of the squared distance can be reduced for a wide range of population models to finding the mean and mean square coalescence times. In many cases the distributions of coalescence times have already been studied for infinite site problems. In this study we show how to calculate these quantities for several population models. We also calculate the variance in mean squared pairwise distance (an estimator of mutation rate × population size) for samples of arbitrary size and show that this variance does not approach zero as the sample size increases. We can also use our method to study alleles at linked microsatellite loci. We suggest a metric which quantifies the level of association between loci—effectively a measure of linkage disequilibrium. It is shown that there can be linkage disequilibrium between partially linked loci at mutation–drift equilibrium.     

Formation of the population composition of a stationary cell culture and the participation of nonproliferating cells with a doubled DNA content in regulating its density

The conditions of the cultivation of chick embryo diploid cells were alternated (prolonged maintenance with or without medium replacement, with or without consequent cell replating in fresh medium). In different times of culture growth, the cell DNA content was assessed by cytophotometry; the percentage of non-labeled mitoses after incubating the cells with 3H-thymidine and colcemide, as well as the cell density were determined. The phenomenon, detected earlier, of the accumulation of cells containing 4c DNA during the transition of the culture from logarithmic into the stationary phase of growth, was confirmed. These cells were shown to differ in their ability to survive in conditions of stationary culture and by proliferative potential. The fraction of cells reversibly arrested in G2-period was described, by which fraction the change of the cell population size is occurring after the decrease of its proliferation rate. The transitional stage is distinguished at the beginning of the stationary phase of culture growth. During this stage the stabilization of structural and numerical composition of the population is taking place.     

The Effect of an Upper Limit to Population Size on Persistence Time

For a population with density-independent vital rates in a randomly varying environment, previous authors have calculated the probability that population size will first drop to some specified (arbitrary) low level at a given time (the first passage time distribution (FPTD), which may be interpreted as a distribution of extinction times). In this paper, we study the FPTD For a stochastic model of density-independent population growth which includes a hard upper limit to population size. We discuss the conditions under which this distribution may be approximated by the FPTD of a Wiener process with a reflecting boundary condition, for which an exact calculation is presented in an appendix. We compare the FPTD of the new model with its counterpart in the model without an upper limit. The most important effects of introducing the upper limit are: (a) ultimate extinction becomes certain; (b) if the long run growth rate in the absence of the upper boundary was small but positive, extinction within ecologically significant times is likely; (c) for larger values of the long run growth rate, persistence over ecologically significant times is almost certain. We discuss the implications of result (b) for conservation. Result (c) establishes that "density-vague" regulation can produce persistent, but bounded, populations.     

Population dynamics in variable environments. III. Evolutionary dynamics of r-selection

The boundary dynamics of a genetic model for an age-structured population in a temporally fluctuating environment are analyzed. The condition for invasion by a new allele identifies the logarithmic growth rate a of each life history phenotype as the fitness measure relevant to “r-selection.” An analytical formula is obtained for fitness a when temporal variance in life history characters is small. This formula reveals the major qualitative and quantitative effects of the average life history, fluctuations, and temporal autocorrelation on fitness. A similar approximation is obtained for the log-variance of population number so that the statistical distribution of population size can be estimated.     

Interspecific differences in stochastic population dynamics explains variation in Taylor's temporal power law

Taylor’s power law, i.e. that the slope for the increase in variance with mean population size is between 1 and 2 at a logarithmic scale, provides one of the few quantitative relationships in population ecology, yet the underlying ecological mechanisms are only poorly understood. Stochastic theory of population dynamics predicts that demographic and environmental stochasticity will affect the slope differently. In a stable environment under the influence of demographic stochasticity alone the slope will be equal to 1. In large populations in which demographic variance will have a negligible effect on the dynamics the slope will approach 2. In addition, the slope will also be influenced by how the strength of density dependence is related to mean population size. To disentangle the relative contribution of these processes we estimate the mean‐variance relationship for a large number of populations of British birds. The variance in population size of most species decreased with the mean due to decreased influence of demographic stochasticity at larger population sizes. Interspecific differences in demographic stochasticity was the main factor influencing variation in slopes of Taylor’s power law among species through a significant negative relationship between the slope and demographic variance. In addition, slopes were influenced by interspecific variation in life history parameters such as adult survival and clutch size. These analyses show that Taylor’s power law is generated from an interplay between stochastic and density dependent factors, modulated by life history.     

Estimating Bacterial Growth Parameters by Means of Detection Times

We developed a new numerical method to estimate bacterial growth parameters by means of detection times generated by different initial counts. The observed detection times are subjected to a transformation involving the (unknown) maximum specific growth rate and the (known) ratios between the different inoculum sizes and the constant detectable level of counts. We present an analysis of variance (ANOVA) protocol based on a theoretical result according to which, if the specific rate used for the transformation is correct, the transformed values are scattered around the same mean irrespective of the original inoculum sizes. That mean, termed the physiological state of the inoculum, , and the maximum specific growth rate, μ, can be estimated by minimizing the variance ratio of the ANOVA procedure. The lag time of the population can be calculated as λ = −ln /μ; i.e. the lag is inversely proportional to the maximum specific growth rate and depends on the initial physiological state of the population. The more accurately the cell number at the detection level is known, the better the estimate for the variance of the lag times of the individual cells.     

Environmental stochasticity and extinction risk in a population of a small songbird, the great tit

Using a long-term demographic data set, we estimated the separate effects of demographic and environmental stochasticity in the growth rate of the great tit population in Wytham Wood, United Kingdom. Assuming logistic density regulation, both the demographic (sigma2d = 0.569) and environmental (sigma2e = 0.0793) variance, with interactions included, were significantly greater than zero. The estimates of the demographic variance seemed to be relatively insensitive to the length of the study period, whereas reliable estimates of the environmental variance required long time series (at least 15 yr of data). The demographic variance decreased significantly with increasing population density. These estimates are used in a quantitative analysis of the demographic factors affecting the risk of extinction of this population. The very long expected time to extinction of this population (approximately 10(19) yr) was related to a relatively large population size (>/=120 pairs during the study period). However, for a given population size, the expected time to extinction was sensitive to both variation in population growth rate and environmental stochasticity. Furthermore, the form of the density regulation strongly affected the expected time to extinction. Time to extinction decreased when the maximum density regulation approached K. This suggests that estimates of viability of small populations should be given both with and without inclusion of density dependence.     

Frequency- and density-dependent selection on a quantitative character

The equilibrium distribution of a quantitative character subject to frequency- and density-dependent selection is found under different assumptions about the genetical basis of the character that lead to a normal distribution in a population. Three types of models are considered: (1) one-locus models, in which a single locus has an additive effect on the character, (2) continuous genotype models, in which one locus or several loci contribute additively to a character, and there is an effectively infinite range of values of the genotypic contributions from each locus, and (3) correlation models, in which the mean and variance of the character can change only through selection at modifier loci. It is shown that the second and third models lead to the same equilibrium values of the total population size and the mean and variance of the character. One-locus models lead to different equilibrium values because of constraints on the relationship between the mean and variance imposed by the assumptions of those models.——The main conclusion is that, at the equilibrium reached under frequency- and density-dependent selection, the distribution of a normally distributed quantitative character does not depend on the underlying genetic model as long as the model imposes no constraints on the mean and variance.     

Dynamics of proteins in a culture broth growing beta-hemolytic streptococci group C

The dynamic study of the protein spectrum of culture fluid during the growth of beta-hemolytic streptococcal strain H46A has been carried out by the methods of electrophoresis and isoelectrofocusing in polyacrylamide gel. Changes in the protein spectrum have phasic character and, on the whole, reflect the state of the microbial population, the presence of fractions corresponding to streptokinase and streptolysins being detected at all phases of growth. The electrophoretic mobility of streptokinase perceptibly changes at the end of the logarithmic phase; as shown by electrofocusing, at all stages of the population growth the heterogeneity of streptokinase is observed.     

Evolution of genetic variability and the advantage of sex and recombination in changing environments.

The role of recombination and sexual reproduction in enhancing adaptation and population persistence in temporally varying environments is investigated on the basis of a quantitative-genetic multilocus model. Populations are finite, subject to density-dependent regulation with a finite growth rate, diploid, and either asexual or randomly mating and sexual with or without recombination. A quantitative trait is determined by a finite number of loci at which mutation generates genetic variability. The trait is under stabilizing selection with an optimum that either changes at a constant rate in one direction, exhibits periodic cycling, or fluctuates randomly. It is shown by Monte Carlo simulations that if the directional-selection component prevails, then freely recombining populations gain a substantial evolutionary advantage over nonrecombining and asexual populations that goes far beyond that recognized in previous studies. The reason is that in such populations, the genetic variance can increase substantially and thus enhance the rate of adaptation. In nonrecombining and asexual populations, no or much less increase of variance occurs. It is explored by simulation and mathematical analysis when, why, and by how much genetic variance increases in response to environmental change. In particular, it is elucidated how this change in genetic variance depends on the reproductive system, the population size, and the selective regime, and what the consequences for population persistence are.     

The context-dependent effect of multiple paternity on effective population size

Effective population size (N(e)) is important because it describes how evolutionary forces will affect a population. The effect of multiple sires per female on N(e) has been the subject of some debate, at the crux of which is the effects of monandry and multiple-paternity (MP) on male variance in reproductive success. In both mating systems, females mate with several males over their lifetimes, but sire offspring with one male at a time in the former and have several sires per clutch in the latter. First, I theoretically show that whether the annual male variance in reproductive success in an MP population is greater or less than that of a monandrous population depends on the distributions of within-clutch paternity. Then, I simulated different distributions of within-clutch paternity under a range of parameters that characterize natural populations to show that an MP population can have an N(e) smaller or larger than that of a monandrous population with otherwise equal dynamics. The N(e(MP)):N(e(Monandry)) ratio increased with mating frequency and female variance in reproductive success, was equalized by long generation times, and was affected by the distribution of within-clutch paternities. The results of this model provide a unifying framework for the debate.     

Effects of immigration on some stochastic logistic models: a cumulant truncation analysis.

This paper uses a new cumulant truncation methodology to investigate the stochastic power law logistic model with immigration, and illustrates the model with parameter values used to describe the growth of muskrat populations in the Netherlands. This model has a stable equilibrium distribution. The incorporation of immigration into the model, therefore, simplifies the qualitative nature of the stochastic solution. The (unconditional) cumulant functions for the transient and the equilibrium population size distributions are obtained, from which the distributions are shown to be near-normal at all times for the parameter values of interest. Approximating cumulant functions, which are relatively easy to find in practice, are derived and shown to be quite accurate, except for the case of massive immigration. As the level of immigration increases, the mean value rises more rapidly initially, as expected; however, the variance and the skewness of both the transient and the equilibrium distributions are reduced.     

A parity-structured matrix model for tsetse populations

A matrix model is used to describe the dynamics of a population of female tsetse flies structured by parity (i.e., by the number of larvae laid). For typical parameter values, the intrinsic growth rate of the population is zero when the adult daily survival rate is 0.970, corresponding to an adult life expectancy of 1/0.030 = 33.3 days. This value is plausible and consistent with results found earlier by others. The intrinsic growth rate is insensitive to the variance of the interlarval period. Temperature being a function of the time of the year, a known relationship between temperature and mean pupal and interlarval times was used to produce a time-varying version of the model which was fitted to temperature and (estimated) population data. With well-chosen parameter values, the modeled population replicated at least roughly the population data. This illustrates dynamically the abiotic effect of temperature on population growth. Given that tsetse flies are the vectors of trypanosomiasis ("sleeping sickness") the model provides a framework within which future transmission models can be developed in order to study the impact of altered temperatures on the spread of this deadly disease.     

Impact of developmental variance on behavior of models for insect populations I. Models for populations with unrestricted growth

The concept of developmental variance is discussed with reference to its use in models for insect populations. When included in a model, developmental variance is typically used to describe the variation of developmental periods among individuals. However, its presence in a model can also have indirect impact on survival and fertility schedules. This impact can lead to significant changes in population growth rates and generation times. These relationships between developmental variance and population growth in models are quantified and discussed.     

On stochastic logistic population growth models with immigration and multiple births

This paper develops a stochastic logistic population growth model with immigration and multiple births. The differential equations for the low-order cumulant functions (i.e., mean, variance, and skewness) of the single birth model are reviewed, and the corresponding equations for the multiple birth model are derived. Accurate approximate solutions for the cumulant functions are obtained using moment closure methods for two families of model parameterizations, one for badger and the other for fox population growth. For both model families, the equilibrium size distribution may be approximated well using the Normal approximation, and even more accurately using the saddlepoint approximation. It is shown that in comparison with the corresponding single birth model, the multiple birth mechanism increases the skewness and the variance of the equilibrium distribution, but slightly reduces its mean. Moreover, the type of density-dependent population control is shown to influence the sign of the skewness and the size of the variance.     

Differences in the Phenotypic Mean and Variance Between Two Geographically Separated Populations of Wood Frog (Lithobates sylvaticus)

Intraspecific phenotypic variation between populations separated by large geographic distances is common. Differences in the mean and variance of traits among populations can be used to infer the relative strength, direction, and type of selection on traits. Patterns in the mean provide information on the type of selection, and patterns in variance provide information on the strength of selection. However, interpretation of mean/variance patterns is difficult when two traits are linked and strongly correlated to fitness because it is unlikely that each trait will reach phenotypic optima. In amphibians time to metamorphosis and size at metamorphosis are positively related both phenotypically and genetically. Using a common-garden experiment we investigated whether selection favours shorter time to metamorphosis or increased mass at metamorphosis between two populations which differ in the length of the post-metamorphic growing season by 2–4 weeks. Animals from the population a shorter growing season took longer to reach and metamorphosed at a greater mass, while animals from the population with a longer period for post metamorphic growth reached metamorphosis faster, but at a smaller mass. Greater phenotypic variance was observed in both traits in the population with the shorter growing season. These data suggest that animals from the population with a restricted growth period maximise mass at metamorphosis at the expense of longer larval periods while animals from population with the longer post-metamorphic growth period sacrifice mass at metamorphosis to shorten the larval period and maximise larval survival. Differences in phenotypic variance among populations suggest either directional or diversifying selection has acted on both traits.