A genealogical approach to variable-population-size models in population genetics

A general exchangeable model is introduced to study gene survival in populations whose size changes without density dependence. Necessary and sufficient conditions for the occurrence of fixation (that is the proportion of one of the types tending to 1 with probability 1) are obtained. These are then applied to the Wright–Fisher model, the Moran model, and conditioned branching-process models. For the Wright–Fisher model it is shown that certain fixation is equivalent to certain extinction of one of the types, but that this is not the case for the Moran model.

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Fixation in bisexual models with variable population sizes

Journal of Applied Probability ◽

10.1017/s002190020010107x ◽

1997 ◽

Vol 34 (02) ◽

pp. 436-448 ◽

Cited By ~ 3

Author(s):

M. Möhle

Keyword(s):

Main Part ◽

Sufficient Conditions ◽

Moran Model ◽

Fisher Model ◽

Population Sizes ◽

Variable Population ◽

Allelic Type ◽

Ultimate Homozygosity ◽

Necessary And Sufficient ◽

Forward Process

A general exchangeable bisexual model with variable population sizes is introduced. First the forward process, i.e. the number of certain descending pairs, is studied. For the bisexual Wright-Fisher model fixation of the descendants occurs, i.e. their proportion tends to 0 or 1 almost surely. The main part of this article deals with necessary and sufficient conditions for ultimate homozygosity, i.e. the proportion of an arbitrarily chosen allelic type tends to 0 or 1 almost surely. The results are applied to a bisexual Wright-Fisher model and to a bisexual Moran model.

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Fixation in bisexual models with variable population sizes

Journal of Applied Probability ◽

10.2307/3215383 ◽

1997 ◽

Vol 34 (2) ◽

pp. 436-448 ◽

Cited By ~ 3

Author(s):

M. Möhle

Keyword(s):

Main Part ◽

Sufficient Conditions ◽

Moran Model ◽

Fisher Model ◽

Population Sizes ◽

Variable Population ◽

Allelic Type ◽

Ultimate Homozygosity ◽

Necessary And Sufficient ◽

Forward Process

A general exchangeable bisexual model with variable population sizes is introduced. First the forward process, i.e. the number of certain descending pairs, is studied. For the bisexual Wright-Fisher model fixation of the descendants occurs, i.e. their proportion tends to 0 or 1 almost surely.The main part of this article deals with necessary and sufficient conditions for ultimate homozygosity, i.e. the proportion of an arbitrarily chosen allelic type tends to 0 or 1 almost surely. The results are applied to a bisexual Wright-Fisher model and to a bisexual Moran model.

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Fixation in Conditional Branching Process Models in Population Genetics

Journal of Applied Probability ◽

10.1239/jap/1197908828 ◽

2007 ◽

Vol 44 (4) ◽

pp. 1103-1110 ◽

Cited By ~ 1

Author(s):

Thomas Prince ◽

Neville Weber

Keyword(s):

Population Genetics ◽

Process Model ◽

Branching Process ◽

Process Models ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Alternative Version ◽

Necessary And Sufficient ◽

Insight Into

An alternative version of the necessary and sufficient condition for almost sure fixation in the conditional branching process model is derived. This formulation provides an insight into why the examples considered in Buckley and Seneta (1983) all have the same condition for fixation.

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Fixation in Conditional Branching Process Models in Population Genetics

Journal of Applied Probability ◽

10.1017/s0021900200003776 ◽

2007 ◽

Vol 44 (04) ◽

pp. 1103-1110

Author(s):

Thomas Prince ◽

Neville Weber

Keyword(s):

Population Genetics ◽

Process Model ◽

Branching Process ◽

Process Models ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Alternative Version ◽

Necessary And Sufficient ◽

Insight Into

An alternative version of the necessary and sufficient condition for almost sure fixation in the conditional branching process model is derived. This formulation provides an insight into why the examples considered in Buckley and Seneta (1983) all have the same condition for fixation.

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On the L 2-convergence of a superadditive bisexual Galton-Watson branching process

Journal of Applied Probability ◽

10.1017/s0021900200101251 ◽

1997 ◽

Vol 34 (03) ◽

pp. 575-582 ◽

Cited By ~ 1

Author(s):

M. González ◽

M. Molina

Keyword(s):

Branching Process ◽

Final Section ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Necessary And Sufficient

In this paper the L 2-convergence of a superadditive bisexual Galton–Watson branching process is studied. Necessary and sufficient conditions for the convergence of the suitably normed process are given. In the final section, a result about one of the most important bisexual models is proved.

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Recurrence Equations for the Probability Distribution of Sample Configurations in Exact Population Genetics Models

Journal of Applied Probability ◽

10.1017/s0021900200007038 ◽

2010 ◽

Vol 47 (03) ◽

pp. 732-751 ◽

Cited By ~ 2

Author(s):

Sabin Lessard

Keyword(s):

Mutation Rate ◽

Overlapping Generations ◽

Finite Population ◽

The Other ◽

Moran Model ◽

Recurrence Equations ◽

Sampling Formula ◽

Fisher Model ◽

Variation Distance ◽

Exchangeable Model

Recurrence equations for the number of types and the frequency of each type in a random sample drawn from a finite population undergoing discrete, nonoverlapping generations and reproducing according to the Cannings exchangeable model are deduced under the assumption of a mutation scheme with infinitely many types. The case of overlapping generations in discrete time is also considered. The equations are developed for the Wright-Fisher model and the Moran model, and extended to the case of the limit coalescent with nonrecurrent mutation as the population size goes to ∞ and the mutation rate to 0. Computations of the total variation distance for the distribution of the number of types in the sample suggest that the exact Moran model provides a better approximation for the sampling formula under the exact Wright-Fisher model than the Ewens sampling formula in the limit of the Kingman coalescent with nonrecurrent mutation. On the other hand, this model seems to provide a good approximation for a Λ-coalescent with nonrecurrent mutation as long as the probability of multiple mergers and the mutation rate are small enough.

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Extinction Probability of Interacting Branching Collision Processes

Advances in Applied Probability ◽

10.1239/aap/1331216651 ◽

2012 ◽

Vol 44 (1) ◽

pp. 226-259 ◽

Cited By ~ 6

Author(s):

Anyue Chen ◽

Junping Li ◽

Yiqing Chen ◽

Dingxuan Zhou

Keyword(s):

Branching Process ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Extinction Probability ◽

Hitting Times ◽

Collision Process ◽

Markov Branching Process ◽

Extinction Probabilities ◽

Collision Processes ◽

Necessary And Sufficient

We consider the uniqueness and extinction properties of the interacting branching collision process (IBCP), which consists of two strongly interacting components: an ordinary Markov branching process and a collision branching process. We establish that there is a unique IBCP, and derive necessary and sufficient conditions for it to be nonexplosive that are easily checked. Explicit expressions are obtained for the extinction probabilities for both regular and irregular cases. The associated expected hitting times are also considered. Examples are provided to illustrate our results.

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Event and time averages: a review

Advances in Applied Probability ◽

10.2307/1427697 ◽

1992 ◽

Vol 24 (2) ◽

pp. 377-411 ◽

Cited By ~ 25

Author(s):

Pierre Brémaud ◽

Raghavan Kannurpatti ◽

Ravi Mazumdar

Keyword(s):

Continuous Time ◽

Short Proof ◽

Sufficient Conditions ◽

Process Models ◽

Stationary Case ◽

Unified Framework ◽

Martingale Approach ◽

Point Process Models ◽

Time Averages ◽

Necessary And Sufficient

This article reviews results related to event and time averages (EATA) for point process models, including PASTA, ASTA and ANTIPASTA under general hypotheses. In particular, the results for the stationary case relating the Palm and martingale approach are reviewed. The non-stationary case is discussed in the martingale framework where minimal conditions for ASTA generalizing earlier work are presented in a unified framework for the discrete- and continuous-time cases. In addition, necessary and sufficient conditions for ASTA to hold in the stationary case are discussed in the case even when stochastic intensities may not exist and a short proof of the ANTIPASTA results known to date are given.

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A continuous time Markov branching model with random environments

Advances in Applied Probability ◽

10.2307/1425963 ◽

1973 ◽

Vol 5 (1) ◽

pp. 37-54 ◽

Cited By ~ 16

Author(s):

Norman Kaplan

Keyword(s):

Discrete Time ◽

Continuous Time ◽

Population Model ◽

Limit Theorems ◽

Branching Process ◽

Sufficient Conditions ◽

Random Environments ◽

Markov Branching Process ◽

Branching Model ◽

Necessary And Sufficient

A population model is constructed which combines the ideas of a discrete time branching process with random environments and a continuous time non-homogeneous Markov branching process. The extinction problem is considered and necessary and sufficient conditions for extinction are determined. Also discussed are limit theorems for what corresponds to the supercritical case.

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