Fixation in bisexual models with variable population sizes

1997 ◽  
Vol 34 (2) ◽  
pp. 436-448 ◽  
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.

Fixation in bisexual models with variable population sizes

1997 ◽  
Vol 34 (02) ◽  
pp. 436-448 ◽  
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.

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

1986 ◽  
Vol 23 (02) ◽  
pp. 283-296 ◽  
Author(s):  
Peter Donnelly
Keyword(s):  
Population Genetics ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Process Models ◽  
Moran Model ◽  
Fisher Model ◽  
Variable Population ◽  
Exchangeable Model ◽  
Necessary And Sufficient ◽  
Variable Population Size

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.

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

1986 ◽  
Vol 23 (2) ◽  
pp. 283-296 ◽  
Author(s):  
Peter Donnelly
Keyword(s):  
Population Genetics ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Process Models ◽  
Moran Model ◽  
Fisher Model ◽  
Variable Population ◽  
Exchangeable Model ◽  
Necessary And Sufficient ◽  
Variable Population Size

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.

The Wright–Fisher model with varying selection

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.

Conditions for fixation of an allele in the density-dependent wright–Fisher models

1988 ◽  
Vol 25 (2) ◽  
pp. 247-256 ◽  
Author(s):  
Fima C. Klebaner
Keyword(s):  
Density Dependence ◽  
Weighted Sums ◽  
Necessary Condition ◽  
Genetic Composition ◽  
Density Dependent ◽  
Fisher Model ◽  
Population Sizes ◽  
Necessary And Sufficient ◽  
Composition Process ◽  
Series Of Functions

A density-dependent Wright–Fisher model is a model where the population size changes randomly depending on the genetic composition process. If population sizes Mn vary without density dependence then the condition ΣMn–1 = ∞ is necessary and sufficient for fixation. It is shown that the above condition is no longer necessary for fixation in the density dependent models. Another necessary condition for fixation is given. Some known results on series of functions of sums of i.i.d. random variables are generalized to weighted sums.

Conditions for fixation of an allele in the density-dependent wright–Fisher models

1988 ◽  
Vol 25 (02) ◽  
pp. 247-256
Author(s):  
Fima C. Klebaner
Keyword(s):  
Density Dependence ◽  
Weighted Sums ◽  
Necessary Condition ◽  
Genetic Composition ◽  
Density Dependent ◽  
Fisher Model ◽  
Population Sizes ◽  
Necessary And Sufficient ◽  
Composition Process ◽  
Series Of Functions

A density-dependent Wright–Fisher model is a model where the population size changes randomly depending on the genetic composition process. If population sizes Mn vary without density dependence then the condition ΣMn –1 = ∞ is necessary and sufficient for fixation. It is shown that the above condition is no longer necessary for fixation in the density dependent models. Another necessary condition for fixation is given. Some known results on series of functions of sums of i.i.d. random variables are generalized to weighted sums.

The Wright–Fisher model with varying selection

1986 ◽  
Vol 23 (2) ◽  
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.

Optimal Distribution of City Sizes in a Region

1982 ◽  
Vol 14 (1) ◽  
pp. 21-32 ◽  
Author(s):  
T Tabuchi
Keyword(s):  
Population Distribution ◽  
Sufficient Conditions ◽  
Optimal Solution ◽  
Distribution Model ◽  
Tokyo Metropolitan Area ◽  
Congestion Cost ◽  
Spatial Distribution Model ◽  
Population Sizes ◽  
Optimal Population ◽  
Necessary And Sufficient

First, an optimal spatial distribution model is proposed of population sizes in a country. The objective function to be examined consists of the amount of interaction benefit which is formulated by means of accessibility, and the amount of intraaction congestion cost which is measured by means of population density. Second, the optimal population distribution is obtained by use of this optimization model, and the necessary and sufficient conditions for the optimal solution is given. Third, based upon the data analysis of population distribution in Japanese prefectures in 1975, it is shown that the Japanese population is undergoing suburbanization and that this leads to the optimal population distribution. Last, this model is used to obtain and analyze the optimal grid system population distribution of the Tokyo Metropolitan Area.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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