Peter Jagers

General branching processes as Markov fields

The natural Markov structure for population growth is that of genetics: newborns inherit types from their mothers, and given those they are independent of the history of their earlier ancestry. This leads to Markov fields on the space of sets of individuals, partially ordered by descent. The structure of such fields is investigated. It is proved that this Markov property implies branching, i.e. the conditional independence of disjoint daughter populations. The process also has the strong Markov property at certain optional sets of individuals. An intrinsic martingale (indexed by sets of individuals) is exhibited, that catches the stochastic element of population development. The deterministic part is analyzed by Markov renewal methods. Finally the strong Markov property found is used to divide the population into conditionally independent subpopulations. On those classical limit theory for sums of independent random variables can be used to catch the asymptotic population development, as real time passes.

GALTON-WATSON PROCESSES IN VARYING ENVIRONMENTS

Galton-Watson processes where the reproduction of individuals in different generations can have different distributions retain many characteristic features of classical processes.

A general stochastic model for population development

Abstract There are two main streams in the mathematical study of populations. The theory of branching processes, in its general formulations, describes the development in time of populations where each member has a random life length and at its death begets a random number of new individuals, or, in physical cases, splits into a random number of new particles. The basic assumptions of the so called stable population theory, on the other side, are often more implicit. But in all formulations of it an important feature is that individuals may give birth not only when they die. Whereas the theory of branching processes is rigorous but with a more limited scope of application (elementary particles, cells, bacteria), stable population theory is a vague but exciting blend of mathematics and intuitive reasoning. Whereas branching processes are stochastic, stable population theory is pseudo-deterministic; only conclusions concerning expectations are arrived at but intuitive probabilistic arguments are often relie...

Modelling the PCR amplification process by a size-dependent branching process and estimation of the efficiency

We propose a stochastic modelling of the PCR amplification process by a size-dependent branching process starting as a supercritical Bienaymé-Galton-Watson transient phase and then having a saturation near-critical size-dependent phase. This model allows us to estimate the probability of replication of a DNA molecule at each cycle of a single PCR trajectory with a very good accuracy.

Random variation and concentration effects in PCR

Even though the efficiency of the polymerase chain reaction (PCR) reaction decreases, analyses are made in terms of Galton-Watson processes, or simple deterministic models with constant replication probability (efficiency). Recently, Schnell and Mendoza have suggested that the form of the efficiency, can be derived from enzyme kinetics. This results in the sequence of molecules numbers forming a stochastic process with the properties of a branching process with population size dependence, which is supercritical, but has a mean reproduction number that approaches one. Such processes display ultimate linear growth, after an initial exponential phase, as is the case in PCR. It is also shown that the resulting stochastic process for a large Michaelis-Menten constant behaves like the deterministic sequence x(n) arising by iterations of the function f(x)=x+x/(1+x).

Population-size-dependent and age-dependent branching processes

Supercritical branching processes are considered which are Markovian in the age structure but where reproduction parameters may depend upon population size and even the age structure of the population. Such processes generalize Bellman-Harris processes as well as customary demographic processes where individuals give birth during their lives but in a purely age-determined manner. Although the total population size of such a process is not Markovian the age chart of all individuals certainly is. We give the generator of this process, and a stochastic equation from which the asymptotic behaviour of the process is obtained, provided individuals are measured in a suitable way (with weights according to Fishers reproductive value). The approach so far is that of stochastic calculus. General supercritical asymptotics then follows from a combination of L2 arguments and Tauberian theorems. It is shown that when the reproduction and life span parameters stabilise suitably during growth, then the process exhibits exponential growth as in the classical case. Application of the approach to, say, the classical Bellman-Harris process gives an alternative way of establishing its asymptotic theory and produces a number of martingales.

On the path to extinction

Populations can die out in many ways. We investigate one basic form of extinction, stable or intrinsic extinction, caused by individuals on the average not being able to replace themselves through reproduction. The archetypical such population is a subcritical branching process, i.e., a population of independent, asexually reproducing individuals, for which the expected number of progeny per individual is less than one. The main purpose is to uncover a fundamental pattern of nature. Mathematically, this emerges in large systems, in our case subcritical populations, starting from a large number, x, of individuals. First we describe the behavior of the time to extinction T: as x grows to infinity, it behaves like the logarithm of x, divided by r, where r is the absolute value of the Malthusian parameter. We give a more precise description in terms of extreme value distributions. Then we study population size partway (or u-way) to extinction, i.e., at times uT, for 0 < u < 1, e.g., u = 1/2 gives halfway to extinction. (Note that mathematically this is no stopping time.) If the population starts from x individuals, then for large x, the proper scaling for the population size at time uT is x into the power u − 1. Normed by this factor, the population u-way to extinction approaches a process, which involves constants that are determined by life span and reproduction distributions, and a random variable that follows the classical Gumbel distribution in the continuous time case. In the Markov case, where an explicit representation can be deduced, we also find a description of the behavior immediately before extinction.

Post-stratification against bias in sampling

Summary For a quite general sampling model, allowing bias (due to undercoverage, nonresponse, for example), the post-stratified estimator of the population mean is shown to be maximum likelihood and have a minimal variance property. Bounds are calculated for bias and variance. In an illustration it is shown how these bounds may be used to obtain approximate confidence intervals.

Convergence of general branching processes and functionals thereof

With each individual in a branching population associate a random function of the age. Count the population by the values of these functions. Different choices yield different processes. In the supercritical case a unified treatment of the asymptotics is possible for a wide class, including for example the number of individuals having some random age dependent property or integrals of branching processes. As an application, the demographic concept of average age at childbearing is given a rigorous interpretation. ALMOST SURE CONVERGENCE; GENERAL BRANCHING PROCESSES

Limit theorems for sums determined by branching and other exponentially growing processes

A branching process counted by a random characteristic has been defined as a process which at time t is the superposition of individual stochastic processes evaluated at the actual ages of the individuals of a branching population. Now characteristics which may depend not only on age but also on absolute time are considered. For supercritical processes a distributional limit theorem is proved, which implies that classical limit theorems for sums of characteristics evaluated at a fixed age point transfer into limit theorems for branching processes counted by these characteristics. A point is that, though characteristics of different individuals should be independent, the characteristics of an individual may well interplay with the reproduction of the latter. The result requires a sort of Lp-continuity for some 1 [less-than-or-equals, slant] p [less-than-or-equals, slant] 2. Its proof turns out to be valid for a wider class of processes than branching ones. For the case p = 1 a number of Poisson type limits follow and for p = 2 some normality approximations are concluded. For example results are obtained for processes for rare events, the age of the oldest individual, and the error of population predictions. This work has been supported by a grant from the Swedish Natural Science Research Council.