Galton-Watson processes where the reproduction of individuals in different generations can have d... more Galton-Watson processes where the reproduction of individuals in different generations can have different distributions retain many characteristic features of classical processes.
International Statistical Review / Revue Internationale de Statistique, 1985
Summary Post-stratification is studied in sampling models which result in independently and ident... more Summary Post-stratification is studied in sampling models which result in independently and identically distributed observations. Examples of such models are simple random sampling with replacement in the classical fixed unknown population vector setting, or simple random sampling without replacement in superpopulation models of independently and identically distributed individuals. It is shown that post-stratified estimators are maximum likelihood and have a conditional Cramer-Rao property. They are compared to certain ratio estimators and studied asymptotically as sample size increases. Conditional inference is discussed and applications are made to opinion polls.
Kingman's classical theory of the coalescent uncovered the basic pattern of genealogical tree... more Kingman's classical theory of the coalescent uncovered the basic pattern of genealogical trees of random samples of individuals in large but time-constant populations. Time is viewed as being discrete and is identified with non-overlapping generations. Reproduction can be very generally taken as exchangeable (meaning that the labelling of individuals in each generation carries no significance). Recent generalisations have dealt with population sizes exhibiting given deterministic or (minor) random fluctuations. We consider population sizes which constitute a stationary Markov chain, explicitly allowing large fluctuations in short times. Convergence of the genealogical tree, as population size tends to infinity, towards the (time-scaled) coalescent is proved under minimal conditions. As a result, we obtain a formula for effective population size, generalising the well-known harmonic mean expression for effective size.
It is well known that a simple, supercritical Bienaym\'e-Galton-Watson process turns into a s... more It is well known that a simple, supercritical Bienaym\'e-Galton-Watson process turns into a subcritical such process, if conditioned to die out. We prove that the corresponding holds true for general, multi-type branching, where child-bearing may occur at different ages, life span may depend upon reproduction, and the whole course of events is thus affected by conditioning upon extinction.
Comprehensive models of stochastic, clonally reproducing populations are defined in terms of gene... more Comprehensive models of stochastic, clonally reproducing populations are defined in terms of general branching processes, allowing birth during maternal life, as for higher organisms, or by splitting, as in cell division. The populations are assumed to start small, by mutation or immigration, reproduce supercritically while smaller than the habitat carrying capacity but subcritically above it. Such populations establish themselves with a probability wellknown from branching process theory. Once established, they grow up to a band around the carrying capacity in a time that is logarithmic in the latter, assumed large. There they prevail during a time period whose duration is exponential in the carrying capacity. Even populations whose life style is sustainble in the sense that the habitat carrying capacity is not eroded but remains the same, ultimately enter an extinction phase, which again lasts for a time logarithmic in the carrying capacity. However, if the habitat can carry a pop...
We establish convergence to the Kingman coalescent for a class of age-structured population model... more We establish convergence to the Kingman coalescent for a class of age-structured population models with time-constant population size. Time is discrete with unit called a year. Offspring numbers in a year may depend on mother’s age. 1. Introduction. The
Biology takes a special place among the other natural sciences because biological units, be they ... more Biology takes a special place among the other natural sciences because biological units, be they pieces of DNA, cells, or organisms, reproduce more or less faithfully. Like any other biological process, reproduction has a large random component. The theory of branching processes was developed especially as a mathematical counterpart to this most fundamental of biological processes. This active and rich research area allows us to determine extinction risks and predict the development of population composition, and also uncover aspects of a population's history from its current genetic composition. Branching processes play an increasingly important role in models of genetics, molecular biology, microbiology, ecology, and evolutionary theory. This book presents this body of mathematical ideas for a biological audience, but should also be enjoyable to mathematicians, if only for its rich stock of rich biological examples. It can be read by anyone with a basic command of calculus, ma...
Few fields within the mathematical sciences cherish their past like branching processes. Ted Harr... more Few fields within the mathematical sciences cherish their past like branching processes. Ted Harris’s classical treatise from 1963 opens by a terse but appetising two-page flashback. Three years later, David Kendall’s elegant overview was published, and like Charles Mode in his monograph (1971), I could borrow from that for the historical sketch opening in my 1975 book, but also add some observations of my own.
Classical branching processes, even the most general, share the property that individuals are sup... more Classical branching processes, even the most general, share the property that individuals are supposed to multiply independently of one another, at least given some environment that in its turn is supposed to be unaffected by the population. Only more recently have birth-and-death and branching processes been considered which allow individual reproduction to be influenced by population size. The first results, due to Klebaner [14], deal with Galton-Watson processes. Work on general, age-structured processes and habitats with a threshold, a so called carrying capacity, came only decades later, cf. [5, 11, 13, 15, 16, 21], inspired by deterministic population dynamics [6, 7, 8, 22].
If a general branching process evolves in a habitat with a finite carrying capacity, i. e. a numb... more If a general branching process evolves in a habitat with a finite carrying capacity, i. e. a number such that reproduction turns subcritical as soon as population size exceeds that number, then the population may either die out quickly, or else grow up to arround the carrying capacity, where it will linger for a long time, until it starts decaying exponentially to extinction.
A general multi-type population model is considered, where individuals live and reproduce accordi... more A general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the population as a measure-valued process and obtain its asymptotics as the population grows with the environmental carrying capacity. Thus, a deterministic approximation is given, in the form of a law of large numbers, as well as a central limit theorem. This general framework is then adapted to model sexual reproduction, with a special section on serial monogamic mating systems.
Galton-Watson processes where the reproduction of individuals in different generations can have d... more Galton-Watson processes where the reproduction of individuals in different generations can have different distributions retain many characteristic features of classical processes.
International Statistical Review / Revue Internationale de Statistique, 1985
Summary Post-stratification is studied in sampling models which result in independently and ident... more Summary Post-stratification is studied in sampling models which result in independently and identically distributed observations. Examples of such models are simple random sampling with replacement in the classical fixed unknown population vector setting, or simple random sampling without replacement in superpopulation models of independently and identically distributed individuals. It is shown that post-stratified estimators are maximum likelihood and have a conditional Cramer-Rao property. They are compared to certain ratio estimators and studied asymptotically as sample size increases. Conditional inference is discussed and applications are made to opinion polls.
Kingman's classical theory of the coalescent uncovered the basic pattern of genealogical tree... more Kingman's classical theory of the coalescent uncovered the basic pattern of genealogical trees of random samples of individuals in large but time-constant populations. Time is viewed as being discrete and is identified with non-overlapping generations. Reproduction can be very generally taken as exchangeable (meaning that the labelling of individuals in each generation carries no significance). Recent generalisations have dealt with population sizes exhibiting given deterministic or (minor) random fluctuations. We consider population sizes which constitute a stationary Markov chain, explicitly allowing large fluctuations in short times. Convergence of the genealogical tree, as population size tends to infinity, towards the (time-scaled) coalescent is proved under minimal conditions. As a result, we obtain a formula for effective population size, generalising the well-known harmonic mean expression for effective size.
It is well known that a simple, supercritical Bienaym\'e-Galton-Watson process turns into a s... more It is well known that a simple, supercritical Bienaym\'e-Galton-Watson process turns into a subcritical such process, if conditioned to die out. We prove that the corresponding holds true for general, multi-type branching, where child-bearing may occur at different ages, life span may depend upon reproduction, and the whole course of events is thus affected by conditioning upon extinction.
Comprehensive models of stochastic, clonally reproducing populations are defined in terms of gene... more Comprehensive models of stochastic, clonally reproducing populations are defined in terms of general branching processes, allowing birth during maternal life, as for higher organisms, or by splitting, as in cell division. The populations are assumed to start small, by mutation or immigration, reproduce supercritically while smaller than the habitat carrying capacity but subcritically above it. Such populations establish themselves with a probability wellknown from branching process theory. Once established, they grow up to a band around the carrying capacity in a time that is logarithmic in the latter, assumed large. There they prevail during a time period whose duration is exponential in the carrying capacity. Even populations whose life style is sustainble in the sense that the habitat carrying capacity is not eroded but remains the same, ultimately enter an extinction phase, which again lasts for a time logarithmic in the carrying capacity. However, if the habitat can carry a pop...
We establish convergence to the Kingman coalescent for a class of age-structured population model... more We establish convergence to the Kingman coalescent for a class of age-structured population models with time-constant population size. Time is discrete with unit called a year. Offspring numbers in a year may depend on mother’s age. 1. Introduction. The
Biology takes a special place among the other natural sciences because biological units, be they ... more Biology takes a special place among the other natural sciences because biological units, be they pieces of DNA, cells, or organisms, reproduce more or less faithfully. Like any other biological process, reproduction has a large random component. The theory of branching processes was developed especially as a mathematical counterpart to this most fundamental of biological processes. This active and rich research area allows us to determine extinction risks and predict the development of population composition, and also uncover aspects of a population's history from its current genetic composition. Branching processes play an increasingly important role in models of genetics, molecular biology, microbiology, ecology, and evolutionary theory. This book presents this body of mathematical ideas for a biological audience, but should also be enjoyable to mathematicians, if only for its rich stock of rich biological examples. It can be read by anyone with a basic command of calculus, ma...
Few fields within the mathematical sciences cherish their past like branching processes. Ted Harr... more Few fields within the mathematical sciences cherish their past like branching processes. Ted Harris’s classical treatise from 1963 opens by a terse but appetising two-page flashback. Three years later, David Kendall’s elegant overview was published, and like Charles Mode in his monograph (1971), I could borrow from that for the historical sketch opening in my 1975 book, but also add some observations of my own.
Classical branching processes, even the most general, share the property that individuals are sup... more Classical branching processes, even the most general, share the property that individuals are supposed to multiply independently of one another, at least given some environment that in its turn is supposed to be unaffected by the population. Only more recently have birth-and-death and branching processes been considered which allow individual reproduction to be influenced by population size. The first results, due to Klebaner [14], deal with Galton-Watson processes. Work on general, age-structured processes and habitats with a threshold, a so called carrying capacity, came only decades later, cf. [5, 11, 13, 15, 16, 21], inspired by deterministic population dynamics [6, 7, 8, 22].
If a general branching process evolves in a habitat with a finite carrying capacity, i. e. a numb... more If a general branching process evolves in a habitat with a finite carrying capacity, i. e. a number such that reproduction turns subcritical as soon as population size exceeds that number, then the population may either die out quickly, or else grow up to arround the carrying capacity, where it will linger for a long time, until it starts decaying exponentially to extinction.
A general multi-type population model is considered, where individuals live and reproduce accordi... more A general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the population as a measure-valued process and obtain its asymptotics as the population grows with the environmental carrying capacity. Thus, a deterministic approximation is given, in the form of a law of large numbers, as well as a central limit theorem. This general framework is then adapted to model sexual reproduction, with a special section on serial monogamic mating systems.
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