The extinction probability of descendants in bisexual models of fixed population size

In this paper exchangeable bisexual models with fixed population size and non-overlapping generations are introduced. Each generation consists of N pairs of individuals. The pairs of a generation have altogether 2N children. These individuals form randomly the N pairs of the next generation. The extinction probability of the descendants of a fixed number of pairs of generation 0 is discussed. Under suitable conditions it can be approximately described by the extinction probability of a Galton–Watson process, if the population size is large. Special examples are a bisexual Wright–Fisher model and models with a uniformly bounded number of children of a pair.

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Forward and backward processes in bisexual models with fixed population sizes

Journal of Applied Probability ◽

10.1017/s0021900200044855 ◽

1994 ◽

Vol 31 (02) ◽

pp. 309-332 ◽

Cited By ~ 5

Author(s):

M. Möhle

Keyword(s):

Overlapping Generations ◽

Weak Limit ◽

Fixed Number ◽

Uhlenbeck Process ◽

Fisher Model ◽

Extinction Probabilities ◽

Ornstein Uhlenbeck Process ◽

Population Sizes ◽

Watson Process ◽

Fixed Population

This paper introduces exchangeable bisexual models with fixed population sizes and non-overlapping generations. In each generation there are N pairs of individuals consisting of a female and a male. The N pairs of a generation produce N daughters and N sons altogether, and these 2N children form the N pairs of the next generation at random. First the extinction of the lines of descendants of a fixed number of pairs is studied, when the population size becomes large. Under suitable conditions this structure can be approximately treated in the framework of a Galton-Watson process. In particular it is shown for the Wright-Fisher model that the discrepancy between the extinction probabilities in the model and in the approximating Galton-Watson process is of order N. Next, the process of the number of ancestor-pairs of all pairs of a generation is analysed. Under suitable conditions this process, properly normed, has a weak limit as N becomes large. For the Wright-Fisher model this limit is an Ornstein–Uhlenbeck process (restricted to a discrete time-set). The corresponding stationary distributions of the backward processes converge to the normal distribution, as expected.

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Forward and backward processes in bisexual models with fixed population sizes

Journal of Applied Probability ◽

10.2307/3215026 ◽

1994 ◽

Vol 31 (2) ◽

pp. 309-332 ◽

Cited By ~ 10

Author(s):

M. Möhle

Keyword(s):

Overlapping Generations ◽

Weak Limit ◽

Fixed Number ◽

Uhlenbeck Process ◽

Fisher Model ◽

Extinction Probabilities ◽

Ornstein Uhlenbeck Process ◽

Population Sizes ◽

Watson Process ◽

Fixed Population

This paper introduces exchangeable bisexual models with fixed population sizes and non-overlapping generations. In each generation there are N pairs of individuals consisting of a female and a male. The N pairs of a generation produce N daughters and N sons altogether, and these 2N children form the N pairs of the next generation at random.First the extinction of the lines of descendants of a fixed number of pairs is studied, when the population size becomes large. Under suitable conditions this structure can be approximately treated in the framework of a Galton-Watson process. In particular it is shown for the Wright-Fisher model that the discrepancy between the extinction probabilities in the model and in the approximating Galton-Watson process is of order N.Next, the process of the number of ancestor-pairs of all pairs of a generation is analysed. Under suitable conditions this process, properly normed, has a weak limit as N becomes large. For the Wright-Fisher model this limit is an Ornstein–Uhlenbeck process (restricted to a discrete time-set). The corresponding stationary distributions of the backward processes converge to the normal distribution, as expected.

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Looking Forwards and Backwards in the Multi-Allelic Neutral Cannings Population Model

Journal of Applied Probability ◽

10.1017/s0021900200007026 ◽

2010 ◽

Vol 47 (03) ◽

pp. 713-731 ◽

Cited By ~ 1

Author(s):

M. Möhle

Keyword(s):

Markov Chain ◽

Population Size ◽

Side Effect ◽

Population Model ◽

Inversion Formula ◽

Transition Matrix ◽

Duality Relation ◽

Moran Model ◽

Fisher Model ◽

Fixed Population

We look forwards and backwards in the multi-allelic neutral exchangeable Cannings model with fixed population size and nonoverlapping generations. The Markov chain X is studied which describes the allelic composition of the population forward in time. A duality relation (inversion formula) between the transition matrix of X and an appropriate backward matrix is discussed. The probabilities of the backward matrix are explicitly expressed in terms of the offspring distribution, complementing the work of Gladstien (1978). The results are applied to fundamental multi-allelic Cannings models, among them the Moran model, the Wright-Fisher model, the Kimura model, and the Karlin and McGregor model. As a side effect, number theoretical sieve formulae occur in these examples.

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Looking Forwards and Backwards in the Multi-Allelic Neutral Cannings Population Model

Journal of Applied Probability ◽

10.1239/jap/1285335405 ◽

2010 ◽

Vol 47 (3) ◽

pp. 713-731 ◽

Cited By ~ 4

Author(s):

M. Möhle

Keyword(s):

Markov Chain ◽

Population Size ◽

Side Effect ◽

Population Model ◽

Inversion Formula ◽

Transition Matrix ◽

Duality Relation ◽

Moran Model ◽

Fisher Model ◽

Fixed Population

We look forwards and backwards in the multi-allelic neutral exchangeable Cannings model with fixed population size and nonoverlapping generations. The Markov chain X is studied which describes the allelic composition of the population forward in time. A duality relation (inversion formula) between the transition matrix of X and an appropriate backward matrix is discussed. The probabilities of the backward matrix are explicitly expressed in terms of the offspring distribution, complementing the work of Gladstien (1978). The results are applied to fundamental multi-allelic Cannings models, among them the Moran model, the Wright-Fisher model, the Kimura model, and the Karlin and McGregor model. As a side effect, number theoretical sieve formulae occur in these examples.

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A phylogenetic estimator of effective population size or mutation rate.

Genetics ◽

10.1093/genetics/136.2.685 ◽

1994 ◽

Vol 136 (2) ◽

pp. 685-692 ◽

Cited By ~ 1

Author(s):

Y X Fu

Keyword(s):

Population Size ◽

Effective Population Size ◽

Mutation Rate ◽

Dna Sequences ◽

High Efficiency ◽

Minimum Variance ◽

Population Subdivision ◽

Effective Population ◽

Fisher Model ◽

Mitochondrial Sequences

Abstract A new estimator of the essential parameter theta = 4Ne mu from DNA polymorphism data is developed under the neutral Wright-Fisher model without recombination and population subdivision, where Ne is the effective population size and mu is the mutation rate per locus per generation. The new estimator has a variance only slightly larger than the minimum variance of all possible unbiased estimators of the parameter and is substantially smaller than that of any existing estimator. The high efficiency of the new estimator is achieved by making full use of phylogenetic information in a sample of DNA sequences from a population. An example of estimating theta by the new method is presented using the mitochondrial sequences from an American Indian population.

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Stochastic Models for a Chemostat and Long-Time Behavior

Advances in Applied Probability ◽

10.1017/s0001867800006595 ◽

2013 ◽

Vol 45 (03) ◽

pp. 822-836 ◽

Cited By ~ 2

Author(s):

Pierre Collet ◽

Servet Martínez ◽

Sylvie Méléard ◽

Jaime San Martín

Keyword(s):

Population Size ◽

Nutrient Concentration ◽

Bacterial Population ◽

Fixed Number ◽

Death Process ◽

Time Behavior ◽

Long Time Behavior ◽

Nutrient Distribution ◽

Long Time ◽

Number Of Individuals

We introduce two stochastic chemostat models consisting of a coupled population-nutrient process reflecting the interaction between the nutrient and the bacteria in the chemostat with finite volume. The nutrient concentration evolves continuously but depends on the population size, while the population size is a birth-and-death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long-time behavior of the bacterial population conditioned to nonextinction. We prove the global existence of the process and its almost-sure extinction. The existence of quasistationary distributions is obtained based on a general fixed-point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities.

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The Wright–Fisher model with varying selection

Journal of Applied Probability ◽

10.1017/s0021900200029788 ◽

1986 ◽

Vol 23 (02) ◽

pp. 504-508

Author(s):

N. C. Weber

Keyword(s):

Population Growth ◽

Population Size ◽

Sufficient Conditions ◽

Selective Advantage ◽

Fisher Model ◽

Ultimate Homozygosity ◽

Varying Population

The Wright–Fisher model with varying population size is examined in the case where the selective advantage varies from generation to generation. Models are considered where the selective advantage is not always in favour of a particular genotype. Sufficient conditions in terms of the selection coefficients and the population growth are given to ensure ultimate homozygosity.

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A note on the probability of extinction in a class of population-size-dependent Galton-Watson processes

Journal of Applied Probability ◽

10.1017/s0021900200108150 ◽

1985 ◽

Vol 22 (04) ◽

pp. 920-925 ◽

Cited By ~ 12

Author(s):

R. Höpfner

Keyword(s):

Asymptotic Behavior ◽

Green’S Function ◽

Population Size ◽

Green's Function ◽

Size Dependent ◽

Rate Of Decay ◽

Probability Of Extinction ◽

Watson Process

In a class of population-size-dependent Galton-Watson processes where extinction does not occur with probability 1 we describe the rate of decay of qi (the probability that the process starting from i ancestors will become extinct) as the number i of ancestors increases. The results are related to the asymptotic behavior of the Green's function of the critical Galton-Watson process with immigration.

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The asymptotic behaviour of extinction probability in the Smith–Wilkinson branching process

Advances in Applied Probability ◽

10.1017/s0001867800025349 ◽

1993 ◽

Vol 25 (02) ◽

pp. 263-289

Author(s):

D. R. Grey ◽

Lu Zhunwei

Keyword(s):

Asymptotic Behaviour ◽

Rate Of Convergence ◽

Population Size ◽

Random Environment ◽

Critical State ◽

Branching Process ◽

Extinction Probability ◽

Subcritical State ◽

Regularity Conditions

Under some regularity conditions, in the supercritical Smith–Wilkinson branching process it is shown that as k, the starting population size, tends to infinity, the rate of convergence of qk, the corresponding extinction probability, to zero is similar to that of: k–θ, if there exists at least one subcritical state in the random environment space; xkk–α , if there exist only supercritical states in the random environment space; exp , if there exists at least one critical state and the others are supercritical in the random environment space. Here θ, x, α and c are positive constants determined by the process.

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