Modeling observation error and its effects in a random walk/extinction model

This paper examines the consequences of observation errors for the "random walk with drift", a model that incorporates density independence and is frequently used in population viability analysis. Exact expressions are given for biases in estimates of the mean, variance and growth parameters under very general models for the observation errors. For other quantities, such as the finite rate of increase, and probabilities about population size in the future we provide and evaluate approximate expressions. These expressions explain the biases induced by observation error without relying exclusively on simulations, and also suggest ways to correct for observation error. A secondary contribution is a careful discussion of observation error models, presented in terms of either log-abundance or abundance. This discussion recognizes that the bias and variance in observation errors may change over time, the result of changing sampling effort or dependence on the underlying population being sampled.     

Linear age-dependent population growth with harvesting

This paper studies the effect of harvesting a fraction of a population where the population growth is modelled by a linear age-dependent model, the Von Foerster equation. Two harvesting strategies are considered: the first is where a fraction of the population greater than agec is removed, and the second is where a fraction of the population of age greater thanc but less thanc+n is removed. In the case where the death rate and fertility rate are time independent, the effect of harvesting on the stable age distribution is examined. Research done at the University of New Mexico and partially supported there by NIH Grant No. RR-08139.     

Population growth under the influence of random dispersal

A population is considered which grows according to the logistic equation while spreading out at random. An approximate method is used to obtain transient and steady-state values for various simple boundary conditions such as that of a population started in an infinite one- or two-dimensional region with or without reflecting or absorbing barriers. An approximate steady-state solution is given for the one-dimensional case of two neighboring regions having different growth rates, mobilities, and degrees of attractiveness.     

Note on the theory of mass behavior

A differential equation has been derived by A. Rapoport,Bull. Math. Biophysics,14, 159 (1952), giving the time course of the fraction of the population who have performed a given act. The general solution of this equation is obtained, some properties of the solution are deduced, and a special case presented in detail.     

Reversibility and the age of an allele II. Two-allele models,with selection and mutation

A Markov process with absorbing boundaries may be made recurrent by returning the process to the interior whenever a boundary is reached. The age of such a process may be defined as the length of time since the last return event. Examples drawn from two-allele genetic models are discussed, in which reversibility of the return process means that the age of an allele, whose present frequency in the population is known, has the same probability distribution as its future extinction time. Some discrete models are not reversible, yet if approximated by diffusion processes, the (approximate) age distribution is the same as the future extinction time distribution. Various results in the literature are unified by this viewpoint.     

The dynamics of hierarchical age-structured populations

An age-structured population is considered in which the birth and death rates of an individual of age a is a function of the density of individuals older and/or younger than a. An existence/uniqueness theorem is proved for the McKendrick equation that governs the dynamics of the age distribution function. This proof shows how a decoupled ordinary differential equation for the total population size can be derived. This result makes a study of the population's asymptotic dynamics (indeed, often its global asymptotic dynamics) mathematically tractable. Several applications to models for intra-specific competition and predation are given.     

On the asymptotic trajectory of the roots of Lotka's equation.

A mathematical formula is given for the asymptotic trajectory of the complex roots of Lotka's equation. This formula is obtained by use of a Taylor expansion of the net fertility function in the neighborhood of the age beyond which fertility is zero. The approximate trajectory is compared with an exact trajectory obtained by use of a computational algorithm for finding complex roots suggested by Turner. For two examples, the agreement is surprisingly good.     

Some stochastic features of bacterial constant growth apparatus

Stochastic models for bacterial constant growth apparatus such as the chemostat are posed and studied. Approximations are given for the mean and variance of the size of the bacterial population when the population is in steady state. Procedures for stimulating a chemostat are developed and the approximate moments are compared with simulated values. The distribution is derived for the waiting time until the occurrence of a population change-over to a faster growing strain. Research supported by National Institutes of Health Grant 5-R01-GM21214.     

The natural variability of vital rates and associated statistics

The first concern of this work is the development of approximations to the distributions of crude mortality rates, age-specific mortality rates, age-standardized rates, standardized mortality ratios, and the like for the case of a closed population or period study. It is found that assuming Poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate Poisson distribution. The Lexis diagram is seen to make direct use of the result. It is suggested that in a variety of cases, it will be satisfactory to approximate the distribution of the number of deaths given the population size, by a Poisson with mean proportional to the population size. It is further suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic Poisson distribution for the number of deaths, with mean proportional to the population size and an exponential function of a linear combination of the explanatories. Such a model is fit to mortality data for Canadian females classified by age and year. A dynamic variant of the model is further fit to the time series of total female deaths alone by year. The models with extra-Poisson variation are found to lead to substantially improved fits.     

Model transformations to evaluate transient thermal responses at a tissue surface

For a spatially distributed model describing the transient temperature response of a thermistor-tissue system, Wei et al. [J. Biomech. Eng., 117:74-85, 1995] obtained an approximate transformation for fast analysis of the temperature response at the tissue surface. This approximate transformation reduces the model to a single ordinary differential equation. Here, we present an exact transformation that yields a single differential-integral equation. Numerical solutions from the approximate and exact transformations were compared to evaluate the differences with several sets of parameter values. The maximum difference between the exact and approximate solutions did not exceed 15 percent and occurred for only a short time interval. The root-mean-square error of the approximate solution was no more than 5 percent and within the level of experimental noise. Under the experimental conditions used by Wei et al., the approximate transformation is justified for estimating model parameters from transient thermal responses.     

The accuracy of Bartlett's small-fluctuation approximation for stochastic-difference-equation population models

A heuristic approximation procedure devised by Bartlett has often been used to estimate the stationary first- and second-order moments of difference-equation population models perturbed by “small” noise. Here, the approximation is proved to be valid under quite general assumptions: the exact and approximate moments differ by an amount of order σ³ as σ → 0, where σ² is the mean-square norm of the noise process. The existence of stationary solutions to the perturbed difference equation is also considered. If the noise is Markovian, stationary solutions satisfying the assumptions of the error analysis are proved to exist if the noise is “small” with probability 1. The results are applied to a population model with two age classes and variable recruitment.     

Deterministic epidemiological models at the individual level

In many fields of science including population dynamics, the vast state spaces inhabited by all but the very simplest of systems can preclude a deterministic analysis. Here, a class of approximate deterministic models is introduced into the field of epidemiology that reduces this state space to one that is numerically feasible. However, these reduced state space master equations do not in general form a closed set. To resolve this, the equations are approximated using closure approximations. This process results in a method for constructing deterministic differential equation models with a potentially large scope of application including dynamic directed contact networks and heterogeneous systems using time dependent parameters. The method is exemplified in the case of an SIR (susceptible-infectious-removed) epidemiological model and is numerically evaluated on a range of networks from spatially local to random. In the context of epidemics propagated on contact networks, this work assists in clarifying the link between stochastic simulation and traditional population level deterministic models.     

The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model

In this paper, we develop the theory of a state-reproduction number for a multistate class age structured epidemic system and apply it to examine the asymptomatic transmission model. We formulate a renewal integral equation system to describe the invasion of infectious diseases into a multistate class age structured host population. We define the state-reproduction number for a class age structured system, which is the net reproduction number of a specific host type and which plays an analogous role to the type-reproduction number [M.G. Roberts, J.A.P. Heesterbeek, A new method for estimating the effort required to control an infectious disease, Proc. R. Soc. Lond. B 270 (2003) 1359; J.A.P. Heesterbeek, M.G. Roberts, The type-reproduction number T in models for infectious disease control, Math. Biosci. 206 (2007) 3] in discussing the critical level of public health intervention. The renewal equation formulation permits computations not only of the state-reproduction number, but also of the generation time and the intrinsic growth rate of infectious diseases.Subsequently, the basic theory is applied to capture the dynamics of a directly transmitted disease within two types of infected populations, i.e., asymptomatic and symptomatic individuals, in which the symptomatic class is observable and hence a target host of the majority of interventions. The state-reproduction number of the symptomatic host is derived and expressed as a measurable quantity, leading to discussion on the critical level of case isolation. The serial interval and other epidemiologic indices are computed, clarifying the parameters on which these indices depend. As a practical example, we illustrate the eradication threshold for case isolation of smallpox. The generation time and serial interval are comparatively examined for pandemic influenza.     

A model of proliferating cell populations with inherited cycle length

A mathematical model of cell population growth introduced by J. L. Lebowitz and S. I. Rubinow is analyzed. Individual cells are distinguished by age and cell cycle length. The cell cycle length is viewed as an inherited property determined at birth. The density of the population satisfies a first order linear partial differential equation with initial and boundary conditions. The boundary condition models the process of cell division of mother cells and the inheritance of cycle length by daughter cells. The mathematical analysis of the model employs the theory of operator semigroups and the spectral theory of linear operators. It is proved that the solutions exhibit the property of asynchronous exponential growth.     

Dissipative structures associated to certain reaction-diffusion equations in mathematical ecology

Sufficient conditions are determined for the existence of stable spatially heterogeneous solutions of the Lotka-Volterra many species equations extended to include spatial diffusion. Some properties of such solutions are obtained. A general definition is proposed for the concept of a dissipative structure associated with a given reaction-diffusion equation. Finally, an approximate solution is presented for two interacting species in one spatial dimension, although the question of stability for this example is left open. Supported in part by NSERC A-7667 to P. L. Antonelli and by a University of Alberta President's Fund grant to J. R. Royce.     

A quantitative method for the analysis of mammalian cell proliferation in culture in terms of dividing and non-dividing cells

The application of the exponential growth equation is the standard method employed in the quantitative analyses of mammalian cell proliferation in culture. This method is based on the implicit assumption that, within a cell population under study, all division events give rise to daughter cells that always divide. When a cell population does not adhere to this assumption, use of the exponential growth equation leads to errors in the determination of both population doubling time and cell generation time. We have derived a more general growth equation that defines cell growth in terms of the dividing fraction of daughter cells. This equation can account for population growth kinetics that derive from the generation of both dividing and non-dividing cells. As such, it provides a sensitive method for detecting non-exponential division dynamics. In addition, this equation can be used to determine when it is appropriate to use the standard exponential growth equation for the estimation of doubling time and generation time.     

Confidence intervals for population growth rate of organisms with two-stage life histories

Although life histories can be modelled with great generality using projection matrices, for organisms with life histories that can be accurately described by a simplified set of parameters, e.g. when adult fecundity and mortality are independent of age, more accurate estimates of life table parameters and of population growth rate and its standard error can be readily obtained. Here an analytic method for calculating approximate confidence intervals for population growth rate is given for two-stage life histories that can be described by four variables representing age at first breeding, fecundity per unit time, and juvenile and adult survivorships per unit time. The method is applied to experimental data on Capitella sp. I obtained by Hansen et al., and quite good agreement is found between the analytic and bootstrap estimates of the standard error of Λ. The analytic estimates were a little conservative, probably because of the way the action of mortality was modelled. Alternative life-history models are briefly discussed, and the desirability of formulating life-history models so that the variables involved are independent of each other is stressed. Analytic estimates of Λ may be biassed if an inappropriate model is chosen or if variables are not independent and the correlations between them are not measured. To allow for these possibilities, where necessary a conservative approach should be taken to significance testing using the analytic method.     

Extensions of models for the estimation of mating systems using n independent loci

Inferences about plant mating systems increasingly use highly informative genetic markers, and investigate finer facets of the mating system. Here, four extensions of models for the estimation mating systems are described. (1) Multiallelic probabilities for the mixed selfing-random mating model are given; these are especially suitable for microsatellites; a generalized Kronecker operator is basis of this formula. (2) Multilocus probabilities for the "correlated-matings model" are given; interestingly, comparisons between single- vs multilocus estimates of correlated-paternity can provide a new measure of population substructure. (3) A measure of biparental inbreeding, the "correlation of selfing among loci", is shown to approximate the fraction of selfing due to uniparental (as opposed to biparental) inbreeding; also joint estimation of 1- 2- and 3-locus selfing rates allow separation, under a simple model, of the frequency vs the magnitude of biparental inbreeding. (4) Method-of-moments estimators for individual outcrossing rates are given. Formulae are given for both gymnosperms and angiosperms, and the computer program "MLTR" implements these methods.     

Random effects in population models with hereditary effects

This paper discusses the influence of environmental noise on the dynamics of single species population models with hereditary effects. A detailed analysis is carried out for the logistic equation with discrete delay in the resource limitation term (Hutchinson's equation). When the system undergoes Hopf bifurcation, we find the stationary probability density distribution for the amplitude of the periodic solution by means of an averaged Fokker-Planck equation. Finally, we estimate the persistence time of the species when the population density has a lower bound beyond which it goes extinct.     

HYDROXYUREA: USE IN DETERMINING THE DURATION OF G2 IN TETRAHYMENA

Observation of division of individual cells in microdrops, plus autoradiographic studies using tritiated thymidine and standard cell cycle analysis techniques, reveal that hydroxyurea (10 DIM) reversibly arrests the normal progression of exponentially growing Tetrahymena pyriformis through the initial 92 % of S-phase while not affecting cells in the terminal 8 % and in G₂ and division. Thus the fraction of the population of cells that is in G₂ can be approximately determined by the fraction of the population able to divide in the presence of hydroxyurea. This fraction can be related to the approximate duration of G₂ by calculations which compensate for the age gradient.