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.     

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.     

Complex population dynamics and the coalescent under neutrality

Estimates of the coalescent effective population size N(e) can be poorly correlated with the true population size. The relationship between N(e) and the population size is sensitive to the way in which birth and death rates vary over time. The problem of inference is exacerbated when the mechanisms underlying population dynamics are complex and depend on many parameters. In instances where nonparametric estimators of N(e) such as the skyline struggle to reproduce the correct demographic history, model-based estimators that can draw on prior information about population size and growth rates may be more efficient. A coalescent model is developed for a large class of populations such that the demographic history is described by a deterministic nonlinear dynamical system of arbitrary dimension. This class of demographic model differs from those typically used in population genetics. Birth and death rates are not fixed, and no assumptions are made regarding the fraction of the population sampled. Furthermore, the population may be structured in such a way that gene copies reproduce both within and across demes. For this large class of models, it is shown how to derive the rate of coalescence, as well as the likelihood of a gene genealogy with heterochronous sampling and labeled taxa, and how to simulate a coalescent tree conditional on a complex demographic history. This theoretical framework encapsulates many of the models used by ecologists and epidemiologists and should facilitate the integration of population genetics with the study of mathematical population dynamics.     

The Dynamics of Territory Acquisition: A Model of Two Coexisting Strategies

Two potential strategies for acquiring territories are (1) fighting and taking over a territory from its owner, or (2) waiting until a territory owner dies and then taking its place. In this paper we explore territory acquisition using these two strategies, using a population dynamical model. Factors expected to affect the predominance of each strategy are injury rate, rate of successful territory takeover, birth and death rates on the territory, and birth and death rates while non-territorial. We explore the effects of these parameters on numbers of territorial and nonterritorial fighters and waiters. Waiters predominate when injury rate, birth rate of nonterritorial individuals, and death rate of territory owners are high. Fighters predominate when rate of successful territory takeover, death rate of nonterritorial individuals, and birth rate of territory owners are high. We present supportive evidence for these preditions from the literature.     

有性生殖与食者—被食者体系中的极限环

1　引　言种群生物学的原始假设认为,在一定条件下,任一种群,不管是有性生殖种群还是无性生殖种群,其自然增长都遵守ＶｅｒｈｕｌｓｔＰｅａｒｌＬｏｇｉｓｔｉｃ方程。按照这一方程,种群有一个平衡点Ｓ,Ｓ=Ｋ,Ｋ是环境负荷容量;并且随着种群大小(或密度)Ｎ的增加,种群的个体(或相对)自然增长率ｄＮ/Ｎｄｔ单调下降。这是由于存在拥挤效应。Ｌｏｇｉｓｔｉｃ方程不含Ａｌｌｅｅ效应或过疏效应[1,2]。实际情况与Ｌｏｇｉｓｔｉｃ方程有所不同,由于存在“拥挤效应”和“过疏效应”,第一,有一个最适种群大小Ｎｍ:随着Ｎ增加,ｄＮ/Ｎｄｔ在Ｎ<Ｎ…     

Obtaining birth and mortality patterns from structured population trajectories

"A method is presented for unravelling the demographic equation for structured populations. A solution to the McKendrick-von Foerster equation is constructed using spline functions and this is fitted to stage-structured population data in such a way that the solution is smooth, positive, and does not imply negative death rates. The smoothness of the surface, and hence the complexity of the population model, is determined in a statistically optimum manner using cross validation. Time- and age-dependent death rates can be obtained as well as time-dependent birth rates. Confidence intervals are obtained for population size and death rates that give a 95% probability that the true population dynamics are within the intervals. Practical application of the method is demonstrated, and comparison made with three alternative methods."     

Demographic heterogeneity impacts density-dependent population dynamics

Among-individual variation in vital parameters such as birth and death rates that is unrelated to age, stage, sex, or environmental fluctuations is referred to as demographic heterogeneity. This kind of heterogeneity is prevalent in ecological populations, but is almost always left out of models. Demographic heterogeneity has been shown to affect demographic stochasticity in small populations and to increase growth rates for density-independent populations. The latter is due to ??cohort selection,?? where the most frail individuals die out first, lowering the cohort??s average mortality as it ages. The importance of cohort selection to population dynamics has only recently been recognized. We use a continuous-time model with density dependence, based on the logistic equation, to study the effects of demographic heterogeneity in mortality and reproduction. Reproductive heterogeneity is introduced in three ways: parent fertility, offspring viability, and parent?Coffspring correlation. We find that both the low-density growth rate and the equilibrium population size increase as the magnitude of mortality heterogeneity increases or as parent?Coffspring phenotypic correlation increases. Population dynamics are affected by complex interactions among the different types of heterogeneity, and trade-off scenarios are examined which can sometimes reverse the effect of increased heterogeneity. We show that there are a number of different homogeneous approximations to heterogeneous models, but all fail to capture important parts of the dynamics of the full model.     

Evolutionary tracking is determined by differential selection on demographic rates and density dependence

Recent ecological forecasts predict that ~25% of species worldwide will go extinct by 2050. However, these estimates are primarily based on environmental changes alone and fail to incorporate important biological mechanisms such as genetic adaptation via evolution. Thus, environmental change can affect population dynamics in ways that classical frameworks can neither describe nor predict. Furthermore, often due to a lack of data, forecasting models commonly describe changes in population demography by summarizing changes in fecundity and survival concurrently with the intrinsic growth rate (r). This has been shown to be an oversimplification as the environment may impose selective pressure on specific demographic rates (birth and death) rather than directly on r (the difference between the birth and death rates). This differential pressure may alter population response to density, in each demographic rate, further diluting the information combined to produce r. Thus, when we consider the potential for persistence via adaptive evolution, populations with the same r can have different abilities to persist amidst environmental change. Therefore, we cannot adequately forecast population response to climate change without accounting for demography and selection on density dependence. Using a continuous‐time Markov chain model to describe the stochastic dynamics of the logistic model of population growth and allow for trait evolution via mutations arising during birth events, we find persistence via evolutionary tracking more likely when environmental change alters birth rather than the death rate. Furthermore, species that evolve responses to changes in the strength of density dependence due to environmental change are less vulnerable to extinction than species that undergo selection independent of population density. By incorporating these key demographic considerations into our predictive models, we can better understand how species will respond to climate change.     

Complexity and demographic stability in population models

This article is concerned with relating the stability of a population, as defined by the rate of decay of fluctuations induced by demographic stochasticity, with its heterogeneity in age-specific birth and death rates. We invoke the theory of large deviations to establish a fluctuation theorem: The demographic stability of a population is positively correlated with evolutionary entropy, a measure of the variability in the age of reproducing individuals in the population. This theorem is exploited to predict certain correlations between ecological constraints and evolutionary trends in demographic stability, namely, (i) bounded growth constraints--a uni-directional increase in stability, (ii) unbounded growth constraints (large population size)--a uni-directional decrease in stability, (iii) unbounded growth constraints (small population size)--random, non-directional change in stability. These principles relating ecological constraints with trends in demographic stability are shown to be far reaching generalizations of the tenets derived from classical studies of stability in an evolutionary context. These results thus provide a new conceptual framework for explaining patterns of variation in population numbers observed in natural populations.     

Genetic effective size of a wild primate population: influence of current and historical demography

A comprehensive assessment of the determinants of effective population size (N(e)) requires estimates of variance in lifetime reproductive success and past changes in census numbers. For natural populations, such information can be best obtained by combining longitudinal data on individual life histories and genetic marker-based inferences of demographic history. Independent estimates of the variance effective size (N(ev), obtained from life-history data) and the inbreeding effective size (N((eI), obtained from genetic data) provide a means of disentangling the effects of current and historical demography. The purpose of this study was to assess the demographic determinants of N(e) in one of the most intensively studied natural populations of a vertebrate species: the population of savannah baboons (Papio cynocephalus) in the Amboseli Basin, southern Kenya. We tested the hypotheses that N(eV) < N < N(eI) (where N = population census number) due to a recent demographic bottleneck. N(eV) was estimated using a stochastic demographic model based on detailed life-history data spanning a 28-year period. Using empirical estimates of age-specific rates of survival and fertility for both sexes, individual-based simulations were used to estimate the variance in lifetime reproductive success. The resultant values translated into an N(eV)/N estimate of 0.329 (SD = 0.116, 95% CI = 0.172-0.537). Historical N(eI), was estimated from 14-locus microsatellite genotypes using a coalescent-based simulation model. Estimates of N(eI) were 2.2 to 7.2 times higher than the contemporary census number of the Amboseli baboon population. In addition to the effects of immigration, the disparity between historical N(eI) and contemporary N is likely attributable to the time lag between the recent drop in census numbers and the rate of increase in the average probability of allelic identity-by-descent. Thus, observed levels of genetic diversity may primarily reflect the population's prebottleneck history rather than its current demography.     

Population dynamics and production of Daphnia hyalina in a eutrophic reservoir

The population dynamics and production of Daphnia hyaiina^ the dominant cladoceran i n Eglwys Nynydd, a shallow eutrophic reservoir in South Wales, were studied for 2 years against a background of limnological measurements. The appearance and development of successive generations from egg to adult could be followed from changing numbers in arbitrarily defined size classes. Seasonal variations in mean length, mean brood-size and proportion of gravid adults were recorded and mean brood-size was related to changing food and temperature conditions. Egg-development times for D. hyaiina were determined in culture and the population parameters finite birth (S), instantaneous birth (b′), instantaneous population change (r′), instantaneous death (d′) and finite death rates (D) were estimated from field data. Turnover and production estimates were calculated from finite death rates and biomass. The calculated potential rate of increase (b′) was nearly always greater than the observed rate of increase (r′): seasonal changes in death rate (d′) generally parallel changes in birth rate (b′) but remain somewhat out of phase. Population oscillations are probably due t o a delay in the expression of the effects of population density upon birth and death rates. The mean biomass of Daphnia in 1970 was 0-57 mg dry wt/l (0-88 g C/m²) and in 1971 0-32 mg dry wt/l (0.49 g C/m²). Annual production for Daphnia was 11-8 mg dry wt/l (18-2 g C/m²) in 1970 and 8-30 mg dry wt/l (12 8 g C/m²) in 1971. Information available on primary production in the reservoir suggests that the production of Daphnia accounts for less than 2% of gross primary production. However, the pattern of population growth of Daphnia in Eglwys Nynydd almost certainly reflects a food limited system. In summer, blue-green algae may be abundant but serve as a poor food source: throughout the blue-green bloom egg production remains low, at times remaining below 0-5 eggs/adult.     

Periodic difference equations, population biology and the Cushing-Henson conjectures

We show that for a k-periodic difference equation, if a periodic orbit of period r is globally asymptotically stable (GAS), then r must be a divisor of k. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and which has a GAS periodic orbit with minimum period r. Our method uses the technique of skew-product dynamical systems. Our methods are then applied to prove two conjectures of Cushing and Henson concerning a non-autonomous Beverton-Holt equation which arises in the study of the response of a population to a periodically fluctuating environmental force such as seasonal fluctuations in carrying capacity or demographic parameters like birth or death rates. We give an equality linking the average population with the growth rates and carrying capacities (in the 2-periodic case) which shows that out-of-phase oscillations in these quantities always have a deleterious effect on the average population. We give an example where in-phase oscillations cause the opposite to occur.     

Comparison of logistic equations for population growth.

Two different forms of the logistic equation for population growth appear in the ecological literature. In the form of the logistic equation that appears in recent ecology textbooks the parameters are the instantaneous rate of natural increase per individual and the carrying capacity of the environment. In the form of the logistic equation that appears in some older literature the parameters are the instantaneous birth rate per individual and the carrying capacity. The decision whether to use one form or the other depends on which form of the equation is biologically more realistic. In this study the form of the logistic equation in which the instantaneous birth rate per individual is a parameter is shown to be more realistic in terms of the birth and death processes of population growth. Application of the logistic equation to calculate yield from an exploited fish population also shows that the parameters must be the instantaneous birth rate per individual and the carrying capacity.     

黑线仓鼠种群数量动态预测研究

１９８４－１９９１年３－１０月,每月中旬在中国农科院草原研究所试验场的不同牧草和作物地进行调查,１９８４－１９９１年共捕获鼠４０９３只,其中黑线仓鼠２９２０只占７１．３４％。黑线仓鼠种群数量季节和年度变化明显,利用电子计算机对黑线仓鼠种数量进行分析,提出种群数量繁殖指数和动态模型以及短、中、长期预测公式、预测准确率为９０．０％。并对影响种群数量的因素进行了初步分析。     

A comparison of persistence-time estimation for discrete and continuous stochastic population models that include demographic and environmental variability

A discrete-time Markov chain model, a continuous-time Markov chain model, and a stochastic differential equation model are compared for a population experiencing demographic and environmental variability. It is assumed that the environment produces random changes in the per capita birth and death rates, which are independent from the inherent random (demographic) variations in the number of births and deaths for any time interval. An existence and uniqueness result is proved for the stochastic differential equation system. Similarities between the models are demonstrated analytically and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models satisfy certain consistency conditions.     

Existence of a non-constant periodic solution of a non-linear autonomous functional differential equation representing the growth of a single species population

Summary We consider an integro-differential equation for the densityn of a single species population where the birth rate is constant and the death rate depends on the values ofn in an interval of length — 1 > 0. We prove the existence of a non-constant periodic solution under the conditions birth rate b > /2 and - 1 small enough. The basic idea of proof (due to R. D. Nussbaum) is to employ a theorem about non-ejective fixed points for a translation operator associated with the solutions of the equation.A proof of existence was also announced by G. Dunkel in [1].     

Morphometric analysis of primordial follicle number in pigtailed monkey ovaries: symmetry and relationship with age.

We previously described a modern, three-dimensional counting method for determining primordial follicle (PF) numbers in primate ovaries using a combination of fractionator and physical dissector techniques. The purposes of our current study were 1) to apply our method to describe intraindividual differences in PF numbers between ovaries and 2) perform a linear regression analysis of age versus mean PF number per ovary. Ovaries from 16 pigtailed monkeys (Macaca nemestrina) age 0.85-12.5 yr were examined. Both ovaries were available from 11 subjects. The difference between ovaries ranged from 2% to 22% (mean +/- SD, 10 +/- 7%) and was not statistically significant. Regression analysis of data from all 16 subjects displayed a log-linear relationship according to the equation log N(a) = 4.8542 - 0.0714(age) where N(a) is the number of PF at a given chronological age. The fit for this model was highly significant (r(2) = 0.73, p 相似文献   

Stochastic analysis of predator–prey type ecosystems

A stochastic model for the predator–prey type ecosystems in a random environment is proposed and investigated. The model is a variation of the Lotka–Volterra type with an additional self-competition mechanism within the prey population. Two different situations are considered: (1) saturation of predators, and (2) competition among predators. Changes in the birth rate of the preys and the death rate of the predators are modeled as random processes. The stochastic averaging procedure of Stratonovich and Khasminskii is applied to obtain the probability distribution of the system state variables at the state of statistical stationarity. Asymptotic behaviors of the system are also investigated. Effects on the ecosystem behaviors are evaluated of (1) prey self-competition, (2) predator saturation and predator competition, (3) random variation in the prey birth rate, and (4) random variation in the predator death rate.     

种群统计随机性和环境随机性对种群绝灭的影响

影响种群绝灭的随机干扰可分为种群统计随机性、环境随机性和随机灾害三大类。在相对稳定的环境条件下和相对较短的时间内,以前两类随机干扰对种群绝灭的影响为生态学家关注的焦点。但是,由于自然种群动态及其影响因子的复杂特征,进一步深入研究随机干扰对种群绝灭的作用在理论上和实践上都必须发展新的技术手段。本文回顾了种群统计随机性与环境随机性的概念起源与发展,系统阐述了其分析方法。归纳了两类随机性在种群绝灭研究中的应用范围、作用方式和特点的异同和区别方法。各类随机作用与种群动态之间关系的理论研究与对种群绝灭机理的实践研究紧密相关。根据理论模型模拟和自然种群实际分析两方面的研究现状,作者提出了进一步深入研究随机作用与种群非线性动态方法的策略。指出了随机干扰影响种群绝灭过程的研究的方向：更多的研究将从单纯的定性分析随机干扰对种群动力学简单性质的作用,转向结合特定的种群非线性动态特征和各类随机力作用特点具体分析绝灭极端动态的成因,以期做出精确的预测。     

Bayesian inference on age-specific survival for censored and truncated data

1. Traditional estimation of age-specific survival and mortality rates in vertebrates is limited to individuals with known age. Although this subject has been studied extensively using effective capture-recapture and capture-recovery models, inference remains challenging because of large numbers of incomplete records (i.e. unknown age of many individuals) and because of the inadequate duration of the studies. 2. Here, we present a hierarchical model for capture-recapture/recovery (CRR) data sets with large proportions of unknown times of birth and death. The model uses a Bayesian framework to draw inference on population-level age-specific demographic rates using parametric survival functions and applies this information to reconstruct times of birth and death for individuals with unknown age. 3. We simulated a set of CRR data sets with varying study span and proportions of individuals with known age, and varying recapture and recovery probabilities. We used these data sets to compare our method to a traditional CRR model, which requires knowledge of individual ages. Subsequently, we applied our method to a subset of a long-term CRR data set on Soay sheep. 4. Our results show that this method performs better than the common CRR model when sample sizes are low. Still, our model is sensitive to the choice of priors with low recapture probability and short studies. In such cases, priors that overestimate survival perform better than those that underestimate it. Also, the model was able to estimate accurately ages at death for Soay sheep, with an average error of 0.94 years and to identify differences in mortality rate between sexes. 5. Although many of the problems in the estimation of age-specific survival can be reduced through more efficient sampling schemes, most ecological data sets are still sparse and with a large proportion of missing records. Thus, improved sampling needs still to be combined with statistical models capable of overcoming the unavoidable limitations of any fieldwork. We show that our approach provides reliable estimates of parameters and unknown times of birth and death even with the most incomplete data sets while being flexible enough to accommodate multiple recapture probabilities and covariates.