Habitat-specific demography: evidence for source-sink population structure in a mammal,the pika

Theory suggests that populations may persist in sink habitats that cannot support replacement-level birth rates. Although it is commonly believed that organisms that can actively select habitat should rarely occur in sinks, the frequency of use of sinks in free-ranging species is not well-documented. We found that a population of American pikas ( Ochotona princeps, Lagomorpha) inhabiting distinct alpine habitats (meadow and snowbed) in Wyoming, USA, had habitat-specific demographic rates that produced a source-sink population structure. Population size increased in both habitats in summer and declined in both habitats in winter, with populations in snowbeds increasing more during summer and decreasing more over winter. Birth rates were consistently higher in meadows and populations in meadows had a consistently higher finite rate of increase (lambda, from life tables) than did those in snowbeds, for which lambda was far below that needed for replacement. Patterns of immigration, population structure, and temporal variation in population size were as expected if meadows were functional sources and snowbeds functional sinks. Patterns of snowmelt differed between habitats, predicted the critical difference in birth rates between habitats, and are a likely primary cause of the differences in habitat-specific birth rates that we observed. This study provides a clear example of source-sink population structure for a mammal.     

Wealth as a source of density dependence in human population growth

In modern industrialized countries, human birth rates have been declining persistently for decades. In many cases they have now fallen below the replacement threshold. However, unlike in natural populations where population growth is constrained by limited resources, birth rates in modern industrialized countries are negatively correlated with resource availability. Here, declining birth rates in human populations are shown to be a manifestation of density‐dependent population growth brought on by socioeconomic development. This is demonstrated by combining empirical power law relations between population size, gross domestic product (GDP) per capita, and fertility in a simple theoretical model describing population dynamics in developed countries. For a closed population, the model exhibits growth to a globally stable equilibrium population size, for both national and city populations. A version of the model that is open with respect to immigration and the influence of foreign technology and capital exhibits a good fit to long‐term time series data on population size, GDP per capita, and birth rates for the United States, France and Japan.     

A mechanistic model based analysis of cotton aphid population dynamics data

1 A 2‐year field study was conducted to generate data on seasonal abundance patterns of cotton aphids Aphis gossypii Glover and to develop a mechanistic model based on cumulative population size. The treatments consisted of three irrigation levels (Low, Medium and High) with 65%, 75% and 85% evapotranspiration replacement and three nitrogen fertility treatments (blanket‐rate‐N, variable‐rate‐N and no nitrogen). 2 A nonlinear regression equation, the analytical solution of a cumulative size mechanistic model, was fitted to each of the 27 individual data sets collected in 2003 and in 2004. The size and time of the peak, the cumulative aphid density, and the birth and death rates were estimated for each population, and each of these five variables was analyzed as a response variable in the analysis of variance. 3 For 2003 (a dry year), the Water (irrigation) main effect was found to be significant for the time of peak, the death rate and the cumulative density. The lower aphid death rate at low water levels might be due to the water stress in plants. 4 For 2004 (a year with moderate precipitation), the Nitrogen main effect was significant for both the birth and death rates. As nitrogen applications were increased, the decrease in both the aphid birth and death rates translates into a decrease in crowding and an increase in aphid survival. 5 The fact that treatment effects may be manifested through birth and death rate parameters in the new mechanistic model opens up new avenues for analyzing population size data of this kind.     

Effective size of fluctuating populations with two sexes and overlapping generations

We derive formulas that can be applied to estimate the effective population size N(e) for organisms with two sexes reproducing once a year and having constant adult mean vital rates independent of age. Temporal fluctuations in population size are generated by demographic and environmental stochasticity. For populations with even sex ratio at birth, no deterministic population growth and identical mean vital rates for both sexes, the key parameter determining N(e) is simply the mean value of the demographic variance for males and females considered separately. In this case Crow and Kimura's generalization of Wright's formula for N(e) with two sexes, in terms of the effective population sizes for each sex, is applicable even for fluctuating populations with different stochasticity in vital rates for males and females. If the mean vital rates are different for the sexes then a simple linear combination of the demographic variances determines N(e), further extending Wright's formula. For long-lived species an expression is derived for N(e) involving the generation times for both sexes. In the general case with nonzero population growth and uneven sex ratio of newborns, we use the model to investigate numerically the effects of different population parameters on N(e). We also estimate the ratio of effective to actual population size in six populations of house sparrows on islands off the coast of northern Norway. This ratio showed large interisland variation because of demographic differences among the populations. Finally, we calculate how N(e) in a growing house sparrow population will change over time.     

Estimating the ratio of effective to actual size of an age‐structured population from individual demographic data

The effective population size is a central concept for understanding evolutionary processes in a finite population. We employ Fisher's reproductive value to estimate the ratio of effective to actual population size for an age‐structured population with two sexes using random samples of individual vital rates. The population may be subject to environmental stochasticity affecting the vital rates. When the mean sex ratio at birth is known, improved efficiency is obtained by utilizing the records of total number of offspring rather than considering separately female and male offspring. We also show how to incorporate uncertain paternity.     

Iterated birth and death process as a model of radiation cell survival

The iterated birth and death process is defined as an n-fold iteration of a stochastic process consisting of the combination of instantaneous random killing of individuals in a certain population with a given survival probability s with a Markov birth and death process describing subsequent population dynamics. A long standing problem of computing the distribution of the number of clonogenic tumor cells surviving a fractionated radiation schedule consisting of n equal doses separated by equal time intervals tau is solved within the framework of iterated birth and death processes. For any initial tumor size i, an explicit formula for the distribution of the number M of surviving clonogens at moment tau after the end of treatment is found. It is shown that if i-->infinity and s-->0 so that is(n) tends to a finite positive limit, the distribution of random variable M converges to a probability distribution, and a formula for the latter is obtained. This result generalizes the classical theorem about the Poisson limit of a sequence of binomial distributions. The exact and limiting distributions are also found for the number of surviving clonogens immediately after the nth exposure. In this case, the limiting distribution turns out to be a Poisson distribution.     

Effect of epistasis and linkage on fixation probability in three-locus models: an ancestral recombination-selection graph approach

We study the probability of ultimate fixation of a single new mutant arising in an individual chosen at random at a locus linked to two other loci carrying previously arisen mutations. This is done using the Ancestral Recombination-Selection Graph (ARSG) in a finite population in the limit of a large population size, which is also known as the Ancestral Influence Graph (AIG). An analytical expansion of the fixation probability with respect to population-scaled recombination rates and selection intensities is obtained. The coefficients of the expansion are expressed in terms of the initial state of the population and the epistatic interactions among the selected loci. Under the assumption of weak selection at tightly linked loci, the sign of the leading term, which depends on the signs of epistasis and initial linkage disequilibrium, determines whether an increase in recombination rates increases the chance of ultimate fixation of the new mutant. If mutants are advantageous, this is the case when epistasis is positive or null and the initial linkage disequilibrium is negative, which is an expected state in a finite population under directional selection. Moreover, this is also the case for a neutral mutant modifier coding for higher recombination rates if the same conditions hold at the selected loci. Under the same conditions, deleterious mutants are disfavored for ultimate fixation and neutral modifiers for higher recombination rates still favored. The recombination rates between the modifier locus and the selected loci do not come into play in the leading terms of the approximation for the fixation probability, but they do in higher-order terms.     

Estimating cladoceran birth rate: use of the egg age distribution to estimate mortality of ovigerous females and eggs

The birth rate of natural cladoceran populations can change rapidly (during 2–3 days), reflecting rapid changes in their environment. If the egg ratio is calculated on the basis of egg age distribution, the birth rate can be estimated at short sampling intervals (shorter than egg stage duration) by modified Paloheimo's (1974) formula. When female size structure and age of eggs in clutches at the beginning and the end of a sampling interval are known, death rates of ovigerous females and eggs in separate size classes can be determined and incorporated in birth rate estimates. All these methods have been employed using the data on the population of Diaphanosoma brahyurum from the lake Obsterno (North-Western Belarus) in July–August, 1992. The birth rate values computed by the proposed methods and Poloheimo's formula differed significantly in many cases. The accuracy of birth rate estimations from various calculation methods was tested using a computer simulation. The model contains the essential features of cladoceran life history: distinct egg, juvenile and adult stages, development of eggs and reproduction. The population was divided into 25 age classes, each of 1 day duration. Durations of the egg, juvenile and adult stages were set at 3, 6 and 20 days, respectively. The embryogenesis was divided into three egg stages, each of 1 day duration. Survivorship was set from 0.2 up to 1.0 for each age class. The survivorship and brood size were changed through each of five time intervals (days) that allowed to simulate an increase or reduction of population density. Fecundity, survivorship and egg stage duration remained constant during each of 5 days that assumed stability of an environment (this does not occur in nature). Nevertheless, the egg ratio, proportion of juveniles and birth rates were variable even under these circumstances. Computer simulations showed that Poloheimo's formula evaluates birth rate with the relative error of 62% and usually overestimates its values. We propose methods to decrease errors of birth rate estimations by 3.5–5.5 times.     

A stochastic model of tumor response to fractionated radiation: limit theorems and rate of convergence

The iterated birth and death Markov process is defined as an n-fold iteration of a birth and death Markov process describing kinetics of certain population combined with random killing of individuals in the population at moments tau 1,...,tau n with given survival probabilities s1,...,sn. A long-standing problem of computing the distribution of the number of clonogenic tumor cells surviving an arbitrary fractionated radiation schedule is solved within the framework of iterated birth and death Markov process. It is shown that, for any initial population size iota, the distribution of the size N of the population at moment t > or = tau n is generalized negative binomial, and an explicit computationally feasible formula for the latter is found. It is shown that if i --> infinity and sn --> 0 so that the product iota s1...sn tends to a finite positive limit, the distribution of random variable N converges to a probability distribution, which for t = tau n turns out to be Poisson. In the latter case, an estimate of the rate of convergence in the total variation metric similar to the classical Law of Rare Events is obtained.     

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.     

Estimating population birth rates of zooplankton when rates of egg deposition and hatching are periodic

Summary I present a general method of computing finite birth and death rates of natural zooplankton populations from changes in the age distribution of eggs and changes in population size. The method is applicable to cases in which eggs hatch periodically owing to variable rates of oviposition. When morphological criteria are used to determine the age distribution of eggs at the beginning and end of a sampling interval, egg mortality can be incorporated in estimates of population birth rate. I raised laboratory populations of Asplanchna priodonta, a common planktonic rotifer, in semicontinuous culture to evaluate my method of computing finite birth rate. The Asplanchna population became synchronized to a daily addition of food but grew by the same amount each day once steady state was achieved. The steady-state rate of growth, which can be computed from the volume-specific dilution rate of the culture, was consistent with the finite birth rate predicted from the population's egg ratio and egg age distribution.     

Cladoceran birth and death rates estimates: experimental comparisons of egg-ratio methods

I. Birth and death rates of natural cladoceran populations cannot be measured directly. Estimates of these population parameters must be calculated using methods that make assumptions about the form of population growth. These methods generally assume that the population has a stable age distribution.
2. To assess the effect of variable age distributions, we tested six egg ratio methods for estimating birth and death rates with data from thirty-seven laboratory populations of Daphnia pulicaria. The populations were grown under constant conditions, but the initial age distributions and egg ratios of the populations varied. Actual death rates were virtually zero, so the difference between the estimated and actual death rates measured the error in both birth and death rate estimates.
3. The results demonstrate that unstable population structures may produce large errors in the birth and death rates estimated by any of these methods. Among the methods tested, Taylor and Slatkin's formula and Paloheimo's formula were most reliable for the experimental data.
4. Further analyses of three of the methods were made using computer simulations of growth of age-structured populations with initially unstable age distributions. These analyses show that the time interval between sampling strongly influences the reliability of birth and death rate estimates. At a sampling interval of 2.5 days (equal to the duration of the egg stage), Paloheimo's formula was most accurate. At longer intervals (7.5–10 days), Taylor and Slatkin's formula which includes information on population structure was most accurate.     

Absorbing phenomena and escaping time for Muller's ratchet in adaptive landscape

Background

The accumulation of deleterious mutations of a population directly contributes to the fate as to how long the population would exist, a process often described as Muller's ratchet with the absorbing phenomenon. The key to understand this absorbing phenomenon is to characterize the decaying time of the fittest class of the population. Adaptive landscape introduced by Wright, a re-emerging powerful concept in systems biology, is used as a tool to describe biological processes. To our knowledge, the dynamical behaviors for Muller's ratchet over the full parameter regimes are not studied from the point of the adaptive landscape. And the characterization of the absorbing phenomenon is not yet quantitatively obtained without extraneous assumptions as well.

Methods

We describe how Muller's ratchet can be mapped to the classical Wright-Fisher process in both discrete and continuous manners. Furthermore, we construct the adaptive landscape for the system analytically from the general diffusion equation. The constructed adaptive landscape is independent of the existence and normalization of the stationary distribution. We derive the formula of the single click time in finite and infinite potential barrier for all parameters regimes by mean first passage time.

Results

We describe the dynamical behavior of the population exposed to Muller's ratchet in all parameters regimes by adaptive landscape. The adaptive landscape has rich structures such as finite and infinite potential, real and imaginary fixed points. We give the formula about the single click time with finite and infinite potential. And we find the single click time increases with selection rates and population size increasing, decreases with mutation rates increasing. These results provide a new understanding of infinite potential. We analytically demonstrate the adaptive and unadaptive states for the whole parameters regimes. Interesting issues about the parameters regions with the imaginary fixed points is demonstrated. Most importantly, we find that the absorbing phenomenon is characterized by the adaptive landscape and the single click time without any extraneous assumptions. These results suggest a graphical and quantitative framework to study the absorbing phenomenon.

     

Neutral evolution in spatially continuous populations

We introduce a general recursion for the probability of identity in state of two individuals sampled from a population subject to mutation, migration, and random drift in a two-dimensional continuum. The recursion allows for the interactions induced by density-dependent regulation of the population, which are inevitable in a continuous population. We give explicit series expansions for large neighbourhood size and for low mutation rates respectively and investigate the accuracy of the classical Malécot formula for these general models. When neighbourhood size is small, this formula does not give the identity even over large scales. However, for large neighbourhood size, it is an accurate approximation which summarises the local population structure in terms of three quantities: the effective dispersal rate, sigma(e); the effective population density, rho(e); and a local scale, kappa, at which local interactions become significant. The results are illustrated by simulations.     

An exact sampling formula for the Wright-Fisher model and a solution to a conjecture about the finite-island model

An exact sampling formula for a Wright-Fisher population of fixed size N under the infinitely many neutral alleles model is deduced. This extends the Ewens formula for the configuration of a random sample to the case where the sample is drawn from a population of small size, that is, without the usual large-N and small-mutation-rate assumption. The formula is used to prove a conjecture ascertaining the validity of a diffusion approximation for the frequency of a mutant-type allele under weak selection in segregation with a wild-type allele in the limit finite-island model, namely, a population that is subdivided into a finite number of demes of size N and that receives an expected fraction m of migrants from a common migrant pool each generation, as the number of demes goes to infinity. This is done by applying the formula to the migrant ancestors of a single deme and sampling their types at random. The proof of the conjecture confirms an analogy between the island model and a random-mating population, but with a different timescale that has implications for estimation procedures.     

Fixation of Strategies for an Evolutionary Game in Finite Populations

A stochastic evolutionary dynamics of two strategies given by 2x 2 matrix games is studied in finite populations. We focus on stochastic properties of fixation: how a strategy represented by a single individual wins over the entire population. The process is discussed in the framework of a random walk with site dependent hopping rates. The time of fixation is found to be identical for both strategies in any particular game. The asymptotic behavior of the fixation time and fixation probabilities in the large population size limit is also discussed. We show that fixation is fast when there is at least one pure evolutionary stable strategy (ESS) in the infinite population size limit, while fixation is slow when the ESS is the coexistence of the two strategies.     

Determinants of human population growth

The 20th century has seen unprecedented growth of the human population on this planet. While at the beginning of the century the Earth had an estimated 1.6 billion inhabitants, this number grew to 6.1 billion by the end of the century, and further significant growth is a near certainty. This paper tries to summarize what factors lie behind this extraordinary expansion of the human population and what population growth we can expect for the future. It discusses the concept of demographic transition and the preconditions for a lasting secular fertility decline. Recent fertility declines in all parts of the world now make it likely that human population growth will come to an end over the course of this century, but in parts of the developing world significant population growth is still to be expected over the coming decades. The slowing of population growth through declining birth rates, together with still increasing life expectancy, will result in a strong ageing of population age structure. Finally, this paper presents a global level systematic analysis of the relationship between population density on the one hand, and growth and fertility rates on the other. This analysis indicates that in addition to the well-studied social and economic determinants, population density also presents a significant factor for the levels and trends of human birth rates.     

Determinants of birth interval in a rural Mediterranean population (La Alpujarra, Spain)

The fertility pattern, in terms of birth intervals, in a rural population not practicing contraception belonging to La Alta Alpujarra Oriental (southeast Spain) is analyzed. During the first half of the 20th century, this population experienced a considerable degree of geographical and cultural isolation. Because of this population's high variability in fertility and therefore in birth intervals, the analysis was limited to a homogenous subsample of 154 families, each with at least five pregnancies. This limitation allowed us to analyze, among and within families, effects of a set of variables on the interbirth pattern, and to avoid possible problems of pseudoreplication. Information on birth date of the mother, age at marriage, children's birth date and death date, birth order, and frequency of miscarriages was collected. Our results indicate that interbirth intervals depend on an exponential effect of maternal age, especially significant after the age of 35. This effect is probably related to the biological degenerative processes of female fertility with age. A linear increase of birth intervals with birth order within families was found as well as a reduction of intervals among families experiencing an infant death. Our sample size was insufficient to detect a possible replacement behavior in the case of infant death. High natality and mortality rates, a secular decrease of natality rates, a log-normal birth interval, and family-size distributions suggest that La Alpujarra has been a natural fertility population following a demographic transition process.     

Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models

How growth, mortality, and dispersal in a species affect the species' spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species' spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results.     

The time to extinction for a stochastic SIS-household-epidemic model

We analyse a Markovian SIS epidemic amongst a finite population partitioned into households. Since the population is finite, the epidemic will eventually go extinct, i.e., have no more infectives in the population. We study the effects of population size and within household transmission upon the time to extinction. This is done through two approximations. The first approximation is suitable for all levels of within household transmission and is based upon an Ornstein-Uhlenbeck process approximation for the diseases fluctuations about an endemic level relying on a large population. The second approximation is suitable for high levels of within household transmission and approximates the number of infectious households by a simple homogeneously mixing SIS model with the households replaced by individuals. The analysis, supported by a simulation study, shows that the mean time to extinction is minimized by moderate levels of within household transmission.