Fixation in haploid populations exhibiting density dependence II: the quasi-neutral case

We determine fixation probabilities in a model of two competing types with density dependence. The model is defined as a two-dimensional birth-and-death process with density-independent death rates, and birth rates that are a linearly decreasing function of total population density. We treat the 'quasi-neutral case' where both types have the same equilibrium population densities. This condition results in birth rates that are proportional to death rates. This can be viewed as a life history trade-off. The deterministic dynamics possesses a stable manifold of mixtures of the two types. We show that the fixation probability is asymptotically equal to the fixation probability at the point where the deterministic flow intersects this manifold. The deterministic dynamics predicts an increase in the proportion of the type with higher birth rate in growing populations (and a decrease in shrinking populations). Growing (shrinking) populations therefore intersect the manifold at a higher (lower) than initial proportion of this type. On the center manifold, the fixation probability is a quadratic function of initial proportion, with a disadvantage to the type with higher birth rate. This disadvantage arises from the larger fluctuations in population density for this type. These results are asymptotically exact and have relevance for allele fixation, models of species abundance, and epidemiological models.     

Population processes under the influence of disasters occurring independently of population size

Markov branching processes and in particular birth-and-death processes are considered under the influence of disasters that arrive independently of the present population size. For these processes we derive an integral equation involving a shifted and rescaled argument. The main emphasis, however, is on the (random) probability of extinction. Its distribution density satisfies an equation which can be solved numerically at least up to a multiplicative constant. In an example it is also found by simulation.     

Distribution of branch lengths and phylogenetic diversity under homogeneous speciation models

The constant rate birth–death process is a popular null model for speciation and extinction. If one removes extinct and non-sampled lineages, this process induces ‘reconstructed trees’ which describe the relationship between extant lineages. We derive the probability density of the length of a randomly chosen pendant edge in a reconstructed tree. For the special case of a pure-birth process with complete sampling, we also provide the probability density of the length of an interior edge, of the length of an edge descending from the root, and of the diversity (which is the sum of all edge lengths). We show that the results depend on whether the reconstructed trees are conditioned on the number of leaves, the age, or both.     

Duality,ancestral and diffusion processes in models with selection

The ancestral selection graph in population genetics was introduced by Krone and Neuhauser [Krone, S.M., Neuhauser, C., 1997. Ancestral process with selection. Theor. Popul. Biol. 51, 210–237] as an analogue of the coalescent genealogy of a sample of genes from a neutrally evolving population. The number of particles in this graph, followed backwards in time, is a birth and death process with quadratic death and linear birth rates. In this paper an explicit form of the probability distribution of the number of particles is obtained by using the density of the allele frequency in the corresponding diffusion model obtained by Kimura [Kimura, M., 1955. Stochastic process and distribution of gene frequencies under natural selection. Cold Spring Harbor Symposia on Quantitative Biology 20, 33–53]. It is shown that the process of fixation of the allele in the diffusion model corresponds to convergence of the ancestral process to its stationary measure. The time to fixation of the allele conditional on fixation is studied in terms of the ancestral process.     

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.     

On the study of two interacting populations one of which acts independently of the other

Many ecological and biological systems can be studied in terms of a bivariate stochastic branching process, {X ₁ (t), X ₂ (t)}, each of whose components (or populations) varies in magnitude according to the laws of a generalized birth-death process. Of particular interest is such a model in which the birth and death rates of the first population,X ₁, are constant while those of the second population,X ₂, exhibit a functional dependence upon the magnitude of the first. It is shown, first, that the existence of the stochastic mean of a birth death process implies the existence of all higher moments. The values of all the factorial moments of such a process are then determined. The moments of the dependent population of the bivariate process are given in terms of its expectation and the joint probability density function of the process is determined. It is possible, therefore, to use Bayesian techniques to infer conclusions about the independent population, given information about the variation of the dependent one.     

On the evaluation of firing densities for periodically driven neuron models

The leaky integrate-and-fire model for neuronal spiking events driven by a periodic stimulus is studied by using the Fokker-Planck formulation. To this purpose, an essential use is made of the asymptotic behavior of the first-passage-time probability density function of a time homogeneous diffusion process through an asymptotically periodic threshold. Numerical comparisons with some recently published results derived by a different approach are performed. Use of a new asymptotic approximation is then made in order to design a numerical algorithm of predictor-corrector type to solve the integral equation in the unknown first-passage-time probability density function. Such algorithm, characterized by a reduced (linear) computation time, is seen to provide a high computation accuracy. Finally, it is shown that such an approach yields excellent approximations to the firing probability density function for a wide range of parameters, including the case of high stimulus frequencies.     

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.     

The frequency spectrum of a mutation,and its age,in a general diffusion model

General formulae are derived for the probability density and expected age of a mutation of frequency x in a population, and similarly for a mutation with b copies in a sample of n genes. A general formula is derived for the frequency spectrum of a mutation in a sample. Variable population size models are included. Results are derived in two frameworks: diffusion process models for the frequency of the mutation; and birth and death process models. The coalescent structure within the mutant gene group and the non-mutant group is considered.     

On the classical trapping problem

We consider the problem of finding the probability density for the location of an untrapped pest satisfying a diffusion equation with the scaled Laplacian. Then by taking the initial data to the diffusion process to be in some Lp spaces, we establish the existence and uniqueness of local solutions in these spaces and the existence and uniqueness of weak global solutions in Lp,q; p,q greater than 3. The interest of this method relies on the fact that it is by successive approximations and hence amenable to numerical treatment.     

A Note on the Application of Martingales in a Problem of Non-Communicable Epidemics

The present paper is a Martingale approach to some non-communicable epidemic problem (e.g. cervical cancer). It is assumed the progress of the disease from pre-cancerous lesions to several grades of dysplasia and ultimately leading to carcinomia in situ and invasive cancer follows by consecutive hittings; and the regression (or the backward movement) from these states to ultimately non cancerous state; may be analogous to consecutive healings. Each hitting and healing thus considered to be a birth and death respectively in the density dependent linear birth and death process. Given that a patient is in some states of dysplasia the problem lies in finding the proportion of patients coming back to noncancerous state and the expected time for the same. Martingales constructed on a linear birth and death process have been employed to answer the problems.     

Fixation in haploid populations exhibiting density dependence I: The non-neutral case

We extend the one-locus two allele Moran model of fixation in a haploid population to the case where the total size of the population is not fixed. The model is defined as a two-dimensional birth-and-death process for allele number. Changes in allele number occur through density-independent death events and birth events whose per capita rate decreases linearly with the total population density. Uniquely for models of this type, the latter is determined by these same birth-and-death events. This provides a framework for investigating both the effects of fluctuation in total population number through demographic stochasticity, and deterministic density-dependent changes in mean density, on allele fixation. We analyze this model using a combination of asymptotic analytic approximations supported by numerics. We find that for advantageous mutants demographic stochasticity of the resident population does not affect the fixation probability, but that deterministic changes in total density do. In contrast, for deleterious mutants, the fixation probability increases with increasing resident population fluctuation size, but is relatively insensitive to initial density. These phenomena cannot be described by simply using a harmonic mean effective population size.     

Resolution of steroid binding heterogeneity by Fourier-derived affinity spectrum analysis (FASA)

A new mathematical method of analyzing radioreceptor assay data is presented. When there are many binding classes with different affinities, the probability-density function B(p) is described by the equation B(p) = (integral negative infinity to infinity) q(k)f(p-k)dk, where q(k) is the affinity spectrum (density of a particular binding class as a function of affinity) and f(p-k) is a probability function (probability that dissociation constants will fall between k and p-k, where p is the free ligand concentration). This equation is solved for q(k) and evaluated explicitly by Fourier transformation, namely, q(w) = b(w)/f(w), where w is frequency. Since division by f(w) can amplify and high frequency noise present in the experimental data, a Gaussian smoothing function is introduced thus: qs(w) = q(w)e(-w/W0)2, where W0 is a constant. This produces an affinity spectrum defined as a plot of the number of binding sites, qs(k), versus their respective dissociation constants, k. Using a FORTRAN computer program, we verify this algorithm using simulated data. We also apply the procedure to resolve heterogeneous populations of estrogen binders in human endometrium using [3H]estradiol as ligand. Two estrogen binder classes are revealed with dissociation constants approximately 2.5 natural logarithmic units apart. We identify one high-affinity (Kd = 0.18 nM)-low density (70 pM [or 72 fmol/mg protein]) subpopulation and one low affinity (Kd = 2.5 nM)-high density (101 pM [or 102 fmol/mg protein]) subpopulation of estradiol binders. The management of experimental error, sampling limitations, and nonspecific binding are discussed. This method directly transforms experimental data into an easily interpretable representation without mathematical modeling or statistical procedures.     

On the meaning of density dependence

Summary Despite a long history, the term density dependence lacks a generally accepted definition. A definition is offered that seems consistent with most other definitions and general usage, that is, a density-dependent factor is any component of the environment whose intensity is correlated with population density and whose action affects survival and reproduction. This definition is used in evaluating the role of territorial behavior, the availability of nest sites, and competition in determining the size of a population. Because neither territory size nor the number of nest sites is correlated with either density or with changes in the birth and death rates of these populations, these cannot be considered density-dependent factors. Competition determines who does breed and who does not rather than the number of breeders, and thus it is not a density-dependent factor determining a population's size.     

Monitoring Chinese hamster ovary cell culture by the analysis of glucose and lactate metabolism

Monitoring cell growth is crucial to the success of an animal cell culture process that can be accomplished by a variety of direct or indirect methodologies. Glucose is a major carbon and energy source for cultured mammalian cells in most cases, but glycolytic metabolism often results in the accumulation of lactate. Glucose and lactate levels are therefore routinely measured to determine metabolic activities of a culture. Typically, neither glucose consumption rate nor lactate accumulation rate has a direct correlation with cell density due to the changes in culture environment and cell physiology. We discovered that although the metabolic rate of glucose or lactate varies depending on the stages of a culture, the cumulative consumption of glucose and lactate combined (Q(GL)) exhibits a linear relationship relative to the integral of viable cells (IVC), with the slope indicating the specific consumption rate of glucose and lactate combined (q(GL)). Additional studies also showed that the q(GL) remains relatively constant under different culture conditions. The insensitivity of the q(GL) to process variations allows a potentially easy and accurate determination of viable cell density by the measurement of glucose and lactate. In addition, the more predictable nature of a linear relationship will aid the design of better forward control strategies to improve cell culture processes.     

Time Inhomogeneous Mutation Models with Birth Date Dependence

The classic Luria–Delbrück model for fluctuation analysis is extended to the case where the split instant distributions of cells are not i.i.d.: the lifetime of each cell is assumed to depend on its birth date. This model takes also into account cell deaths and non-exponentially distributed lifetimes. In particular, it is possible to consider subprobability distributions and to model non-exponential growth. The extended model leads to a family of probability distributions which depend on the expected number of mutations, the death probability of mutant cells, and the split instant distributions of normal and mutant cells. This is deduced from the Bellman–Harris integral equation, written for the birth date inhomogeneous case. A new theorem of convergence for the final mutant counts is proved, using an analytic method. Particular examples like the Haldane model or the case where hazard functions of the split-instant distributions are proportional are studied. The Luria–Delbrück distribution with cell deaths is recovered. A computation algorithm for the probabilities is provided.     

Measures of success in a class of evolutionary models with fixed population size and structure

We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth–death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.     

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.     

On a Continuous Branching Process and Birth-Death and Immigration Models for Testing the Efficacy of Some Control Programmes for AIDS

The present communication is an attempt to describe the mode of propagation of AIDS epidemic and its control programme using a branching process as well as a birth-death and immigration model. A comparison of the project of AIDS control programme on the basis of its propagation by a continuous branching process model with that of a linear birth and death process with immigration shows a remarkable contrast. Branching process model shows that it is possible to control the propagation of the disease by suitably increasing the detection rate and lowering the infection rate. However, the propagation of AIDS models by birth and death Process with or without immigration shows that it is increasingly difficult to control the invasion of AIDS merely by controlling the birth, death and immigration parameters.