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.     

Limit theorems for the population size of a birth and death process allowing catastrophes

The linear birth and death process with catastrophes is formulated as a right continuous random walk on the non-negative integers which evolves in continuous time with an instantaneous jump rate proportional to the current value of the process. It is shown that distributions of the population size can be represented in terms of those of a certain Markov branching process. The ergodic theory of Markov branching process transition probabilities is then used to develop a fairly complete understanding of the behaviour of the population size of the birth-death-catastrophe process.Research done while on leave at Colorado State University from the University of Western Australia and partially supported by N.S.F. grant DMS-8501763     

Ewens' sampling formula and related formulae: combinatorial proofs, extensions to variable population size and applications to ages of alleles

Ewens' sampling formula, the probability distribution of a configuration of alleles in a sample of genes under the infinitely-many-alleles model of mutation, is proved by a direct combinatorial argument. The distribution is extended to a model where the population size may vary back in time. The distribution of age-ordered frequencies in the population is also derived in the model, extending the GEM distribution of age-ordered frequencies in a model with a constant-sized population. The genealogy of a rare allele is studied using a combinatorial approach. A connection is explored between the distribution of age-ordered frequencies and ladder indices and heights in a sequence of random variables. In a sample of n genes the connection is with ladder heights and indices in a sequence of draws from an urn containing balls labelled 1,2,...,n; and in the population the connection is with ladder heights and indices in a sequence of independent uniform random variables.     

Coalescent theory for a completely random mating monoecious population

Consider a large random mating monoecious diploid population that has N individuals in each generation. Let us assume that at time 0 a random sample of ninfinity. It is then possible to obtain a generalization of coalescent theory for haploid populations if the distribution of G1 has a finite second moment and E[G(1)(3)]/N-->0 as N-->infinity.     

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.     

Calculating the intrinsic growth rate: comparison of definition and model

It was shown that well known equation r = ln[N(t2)/N(t1)]/(t2 - t1) is the definition of the average value of intrinsic growth rate of population r within any given interval of time t2-t1 and changing arbitrarity its numbers N(t). The common opinion considering the equation as suitable only for exponentially growing population was found to be incorrect. The fundamentally different approach is based on the calculation of r within the framework of demographic model, realized as Euler - Lotka equation or population projection matrices. However this model requires simultaneous realization of several assumptions improbable for natural populations: exponential change in population size, stable age structure and maintaining constant age-dependent birth and death rates. The calculation of r by definition requires the data on the dynamics of population numbers, whereas calculation on the basis of the model requires the demographic tables of birth and death rate, but not the population numbers. With the example of American ginseng it was shown that evalution of r by definition and model approaches could produce opposite results.     

Inbreeding under a cyclical mating system

Summary General recursion formulae for the coefficient of inbreeding under a cyclical mating system were derived in which one male and one female are selected from each of the n families per generation (population size N = 2 n). Each male is given the family number of his sire in each generation, while his mate comes from another family, varying systematically in different generations. Males of the r-th family in generations 1, 2, 3,..., t = n–1 within each cycle mate with females from families r+1, r+2, r+3,..., r+t to produce generations 2, 3, 4,..., t+1=1, respectively. The change in heterozygosity shows a cyclical pattern of rises and falls, repeating in cycles of n–1 generations. The rate of inbreeding oscillates between <-3% to >6% in different generations within each cycle, irrespective of the population size. The average rate of inbreeding per generation is approximately 1/[4 N-(Log₂N+1)], which is the rate for the maximum avoidance of inbreeding. The average inbreeding effective population size is approximately 2 N–2.     

长白松自然同龄种群分布格局的研究

以长白松（Ｐｉｎｕｓｓｙｌｖｅｓｔｒｉｆｏｒｍｉｓ）种群为例,通过种群发展历史、种群大小结构研究,发现种群的胸径服从正态分布Ｎ（１６．８７,６．８０２）,确认该种群为自然同龄种群．种群的空间分布格局呈很高的均匀性,趋于随机分布甚至均匀分布,分布格局动态表现出由均匀分布向随机分布变化的趋势．     

Extinction times and moment closure in the stochastic logistic process

We investigate the statistics of extinction times for an isolated population, with an initially modest number M of individuals, whose dynamics are controlled by a stochastic logistic process (SLP). The coefficient of variation in the extinction time V is found to have a maximum value when the death and birth rates are close in value. For large habitat size K we find that Vmax is of order K1/4 / M1/2, which is much larger than unity so long as M is small compared to K1/2. We also present a study of the SLP using the moment closure approximation (MCA), and discuss the successes and failures of this method. Regarding the former, the MCA yields a steady-state distribution for the population when the death rate is low. Although not correct for the SLP model, the first three moments of this distribution coincide with those calculated exactly for an adjusted SLP in which extinction is forbidden. These exact calculations also pinpoint the breakdown of the MCA as the death rate is increased.     

The supercritical birth,death and catastrophe process: Limit theorems on the set of non-extinction

The skip-free property of the super-critical linear birth and death process with a linear catastrophe component is exploited to obtain an almost sure convergence theorem for the population size without any extra moment assumptions. This completes some earlier work of the author. A proof is sketched of a central limit theorem for the logarithm of the population size. Identification of the form of the norming constants for this result requires estimates of the extinction probabilities as functions of the initial population size. This is related to moment conditions on the decrement distribution.Most of this research was carried out at Colorado State University with the partial support of N.S.F. grant DMS-8501763.     

Evaluation of the evolution rate in Eigen's and Kuhn's models

The stage evolution of a "population" consisting of informational sequences having fixed length N is investigated. It is assumed that at each stage sequences of a "population" are selected, reproduced, mutated and diluted. Character of an evolution is revealed and the evolution rate is estimated for different values of N, "population" size n, and a number of different symbols lambda in sequences. According to the obtained estimations it is possible to find sequences in an evolution process which are close to the "optimum" one using far less number of sequences (approximately lambda N4) than for a random search (approximately lambda N).     

On the precision of the conditionally autoregressive prior in spatial models

Bayesian analyses of spatial data often use a conditionally autoregressive (CAR) prior, which can be written as the kernel of an improper density that depends on a precision parameter tau that is typically unknown. To include tau in the Bayesian analysis, the kernel must be multiplied by tau(k) for some k. This article rigorously derives k = (n - I)/2 for the L2 norm CAR prior (also called a Gaussian Markov random field model) and k = n - I for the L1 norm CAR prior, where n is the number of regions and I the number of "islands" (disconnected groups of regions) in the spatial map. Since I = 1 for a spatial structure defining a connected graph, this supports Knorr-Held's (2002, in Highly Structured Stochastic Systems, 260-264) suggestion that k = (n - 1)/2 in the L2 norm case, instead of the more common k = n/2. We illustrate the practical significance of our results using a periodontal example.     

The parasite capacity of the host population

The estimation of parasitic pressure on the host populations is frequently required in parasitological investigations. The empirical values of prevalence of infection are used for this, however the latter one as an estimation of parasitic pressure on the host population is insufficient. For example, the same prevalence of infection can be insignificant for the population with high reproductive potential and excessive for the population with the low reproductive potential. Therefore the development of methods of an estimation of the parasitic pressure on the population, which take into account the features the host population, is necessary. Appropriate parameters are to be independent on view of the researcher, have a clear biological sense and be based on easily available characteristics. The methods of estimation of parasitic pressure on the host at the organism level are based on various individual viability parameters: longevity, resistance to difficult environment etc. The natural development of this approach for population level is the analysis of viability parameters of groups, namely, the changing of extinction probability of host population under the influence of parasites. Obviously, some critical values of prevalence of infection should exist; above theme the host population dies out. Therefore the heaviest prevalence of infection, at which the probability of host population size decreases during the some period is less than probability of that increases or preserves, can serve as an indicator of permissible parasitic pressure on the host population. For its designation the term "parasite capacity of the host population" is proposed. The real parasitic pressure on the host population should be estimated on the comparison with its parasite capacity. Parasite capacity of the host population is the heaviest possible prevalence of infection, at which, with the generation number T approaching infinity, there exists at least one initial population size ni(0) for which the probability of size decrease through T generations is less than the probability of its increase. [formula: see text] The estimation of the probabilities of host population size changes is necessary for the parasite capacity determination. The classical methods for the estimation of extinction probability of population are unsuitable in this case, as these methods require the knowledge of population growth rates and their variances for all possible population sizes. Thus, the development methods of estimate of extinction probability of population, based on the using of available parameters (sex ratio, fecundity, mortality, prevalence of infection PI) is necessary. The population size change can be considered as the Markov process. The probabilities of all changes of population size for a generation in this case are described by a matrix of transition probabilities of Markov process (pi) with dimensions Nmax x Nmax (maximum population size). The probabilities of all possible size changes for T generations can be calculated as pi T. Analyzing the behaviour matrix of transition at various prevalence of infection, it is possible to determine the parasite capacity of the host population. In constructing of the matrix of transition probabilities, should to be taken into account the features the host population and the influence of parasites on its reproductive potential. The set of the possible population size at a generation corresponds to each initial population size. The transition probabilities for the possible population sizes at a generation can be approximated to the binomial distribution. The possible population sizes at a generation nj(t + 1) can be calculated as sums of the number of survived parents N1 and posterities N2; their probabilities--as P(N1) x P(N2). The probabilities of equal sums N1 + N2 and nj(t + 1) > or = Nmax are added. The number of survived parents N1 may range from 0 to (1-PI) x ni(t). The survival probabilities can be estimated for each N1 as [formula: see text] The number of survived posterities N2 may range from 0 to N2max (the maximum number of posterities). N2max is [formula: see text] and the survival probabilities for each N2, is defined as [formula: see text] where [formula: see text], ni(t) is the initial population size (including of males and infected specimens of host), PI is the prevalence of infection, Q1 is the survival probabilities of parents, Pfemales is the frequency of females in the host population, K is the number of posterities per a female, and Q2 is the survival probabilities of posterities. When constructing matrix of transition probabilities of Markov process (pi), the procedure outlined above should be repeated for all possible initial population size. Matrix of transition probabilities for T generations is defined as pi T. This matrix (pi T) embodies all possible transition probabilities from the initial population sizes to the final population sizes and contains a wealth of information by itself. From the practical point of view, however, the plots of the probability of population size decrease are more suitable for analysis. They can be received by summing the probabilities within of lines of matrix from 0 to ni--1 (ni--the population size, which corresponds to the line of the matrix). Offered parameter has the number of advantages. Firstly, it is independent on a view of researcher. Secondly, it has a clear biological sense--this is a limit of prevalence, which is safe for host population. Thirdly, only available parameters are used in the calculation of parasite capacity: population size, sex ratio, fecundity, mortality. Lastly, with the availability of modern computers calculations do not make large labour. Drawbacks of this parameter: 1. The assumption that prevalence of infection, mortality, fecundity and sex ratio are constant in time (the situations are possible when the variability of this parameters can not be neglected); 2. The term "maximum population size" has no clear biological sense; 3. Objective restrictions exist for applications of this mathematical approach for populations with size, which exceeds 1000 specimens (huge quantity of computing operations--order Nmax 3*(T-1), work with very low probabilities). The further evolution of the proposed approach will allow to transfer from the probabilities of size changes of individual populations to be probabilities of size changes of population systems under the influence of parasites. This approach can be used at the epidemiology and in the conservation biology.     

Novel bivariate moment-closure approximations

Nonlinear stochastic models are typically intractable to analytic solutions and hence, moment-closure schemes are used to provide approximations to these models. Existing closure approximations are often unable to describe transient aspects caused by extinction behaviour in a stochastic process. Recent work has tackled this problem in the univariate case. In this study, we address this problem by introducing novel bivariate moment-closure methods based on mixture distributions. Novel closure approximations are developed, based on the beta-binomial, zero-modified distributions and the log-Normal, designed to capture the behaviour of the stochastic SIS model with varying population size, around the threshold between persistence and extinction of disease. The idea of conditional dependence between variables of interest underlies these mixture approximations. In the first approximation, we assume that the distribution of infectives (I) conditional on population size (N) is governed by the beta-binomial and for the second form, we assume that I is governed by zero-modified beta-binomial distribution where in either case N follows a log-Normal distribution. We analyse the impact of coupling and inter-dependency between population variables on the behaviour of the approximations developed. Thus, the approximations are applied in two situations in the case of the SIS model where: (1) the death rate is independent of disease status; and (2) the death rate is disease-dependent. Comparison with simulation shows that these mixture approximations are able to predict disease extinction behaviour and describe transient aspects of the process.     

Kinetic evidence for inefficient and error-prone bypass across bulky N2-guanine DNA adducts by human DNA polymerase iota

DNA polymerase (pol) iota has been proposed to be involved in translesion synthesis past minor groove DNA adducts via Hoogsteen base pairing. The N2 position of G, located in minor groove side of duplex DNA, is a major site for DNA modification by various carcinogens. Oligonucleotides with varying adduct size at G N2 were analyzed for bypass ability and fidelity with human pol iota. Pol iota effectively bypassed N2-methyl (Me)G and N2-ethyl(Et)G, partially bypassed N2-isobutyl(Ib)G and N2-benzylG, and was blocked at N2-CH2(2-naphthyl)G (N2-NaphG), N2-CH2(9-anthracenyl)G (N2-AnthG), and N2-CH2(6-benzo[a]pyrenyl)G. Steady-state kinetic analysis showed decreases of kcat/Km for dCTP insertion opposite N2-G adducts according to size, with a maximal decrease opposite N2-AnthG (61-fold). dTTP misinsertion frequency opposite template G was increased 3-11-fold opposite adducts (highest with N2-NaphG), indicating the additive effect of bulk (or possibly hydrophobicity) on T misincorporation. N2-IbG, N2-NaphG, and N2-AnthG also decreased the pre-steady-state kinetic burst rate compared with unmodified G. High kinetic thio effects (S(p)-2'-deoxycytidine 5'-O-(1-thiotriphosphate)) opposite N2-EtG and N2-AnthG (but not G) suggest that the chemistry step is largely interfered with by adducts. Severe inhibition of polymerization opposite N2,N2-diMeG compared with N2-EtG by pol eta but not by pol iota is consistent with Hoogsteen base pairing by pol iota. Thus, polymerization by pol iota is severely inhibited by a bulky group at G N2 despite an advantageous mode of Hoogsteen base pairing; pol iota may play a limited role in translesion synthesis on bulky N2-G adducts in cells.     

Dynamic behaviors of the Ricker population model under a set of randomized perturbations

We studied the dynamics of the Ricker population model under perturbations by the discrete random variable epsilon which follows distribution P?epsilon=a(i)?=p(i),i=1,ellipsis,n,0/=1. Under the perturbations, n+1 blurred orbits appeared in the bifurcation diagram. Each of the n+1 blurred orbits consisted of n sub-orbits. The asymptotes of the n sub-orbits in one of the n+1 blurred orbits were N(t)=a(i) for i=1,ellipsis,n. For other n blurred orbits, the asymptotes of the n sub-orbits were N(t)=a(i)exp[r(1-a(i))]+a(j),j=1,2,ellipsis,n, for i=1,ellipsis,n, respectively. The effects of variances of the random variable epsilon on the bifurcation diagrams were examined. As the variance value increased, the bifurcation diagram became more blurred. Perturbation effects of the approximate continuous uniform random variable and random error were compared. The effects of the two perturbations on dynamics of the Ricker model were similar, but with differences. Under different perturbations, the attracting equilibrium points and two-cycle periods in the Ricker model were relatively stable. However, some dynamic properties, such as the periodic windows and the n-cycle periods (4), could not be observed even when the variance of a perturbation variable was very small. The process of reversal of the period-doubling, an important feature of the Ricker and other population models observed under constant perturbations, was relatively unstable under random perturbations.     

Extinction of populations by random influences

A unifying approach described by a random birth and death process which includes both environmental and demographic noise is introduced. It is shown that both of these noise sources play an essential role in extinction processes in general. The probability distribution of the lifetime of a population is determined and its dependence on the parameters of the model is discussed. Finally a population divided into subpopulations is modeled. The lifetime of this ensemble of subpopulations is compared to the lifetime of one large population.     

A model of ion channel kinetics using deterministic chaotic rather than stochastic processes.

Models of ion channel kinetics have previously assumed that the switching between the open and closed states is an intrinsically random process. Here, we present an alternative model based on a deterministic process. This model is a piecewise linear iterated map. We calculate the dwell time distributions, autocorrelation function, and power spectrum of this map. We also explore non-linear generalizations of this map. The chaotic nature of our model implies that its long-term behavior mimics the stochastic properties of a random process. In particular, the linear map produces an exponential probability distribution of dwell times in the open and closed states, the same as that produced by the two-state, closed in equilibrium open, Markov model. We show how deterministic and random models can be distinguished by their different phase space portraits. A test of some experimental data seems to favor the deterministic model, but further experimental evidence is needed for an unequivocal decision.     

Constitution and By-Laws of the American Association of Physical Anthropologists

From parent populations (N = 50,000) statistically generated, representing different levels of correlation (r) between the age at death and a hypothetical biological indicator (r = 0.8-0.98), reference samples and target demographic samples are randomly drawn. Two iterative techniques, proportional fitting procedure and Bayesian, are used to estimate from the reference samples the age distribution of the targets. Due to the random fluctuations of the pattern of aging, both in the reference and target samples, these techniques converge only in expectation toward the true value of a distribution, but not in practice for any particular realization. Nevertheless, these techniques allow the estimation of the average of an age distribution, even if its shape is unknown. Under the hypothesis that the target sample is drawn from a stationary population, this average represents the life expectancy at 20 years (plus 20 years). Using this mean age at death for the adults and the juvenility index at death (D5-14/D20-ω), a new set of paleodemographic estimators were derived from 40 archaic life tables. For a hypothesized stable population, they give the life expectancy at birth and at 20 years, and the probability of death at 1 and 5 years. © 1996 Wiley-Liss, Inc.     

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…