A bisexual multitype branching process with applications in population genetics

1972 ◽  
Vol 34 (1) ◽  
pp. 13-31 ◽  
Author(s):  
Charles J. Mode
Keyword(s):  
Population Genetics ◽  
Branching Process ◽  
Multitype Branching Process

scholarly journals Simulating the Emergence of Mutations and Their Subsequent Evolution in an Age-Structured Stochastic Self-Regulating Process with Two Sexes

2013 ◽  
Vol 2013 ◽  
pp. 1-23 ◽  
Author(s):  
Charles J. Mode ◽  
Candace K. Sleeman ◽  
Towfique Raj
Keyword(s):  
Population Genetics ◽  
Natural Selection ◽  
Branching Process ◽  
Initial Population ◽  
Expected Number ◽  
Parametric Function ◽  
Simulation Experiments ◽  
Multitype Branching Process ◽  
Male Sexual Partner ◽  
Age Structured

The stochastic process under consideration is intended to be not only part of the working paradigm of evolutionary and population genetics but also that of applied probability and stochastic processes with an emphasis on computer intensive methods. In particular, the process is an age-structured self-regulating multitype branching process with a genetic component consisting of an autosomal locus with two alleles for females and males. It is within this simple context that mutation will be quantified in terms of probabilities that a given allele mutates to the other per meiosis. But, unlike many models that are currently being used in mathematical population genetics, in which natural selection is often characterized in terms of parameters called fitness by genotype or phenotype, in this paper the parameterization of submodules of the model provides a framework for characterizing natural selection in terms of some of its components. One of these modules consists of reproductive success that is quantified in terms of the total expected number of offspring a female contributes to the population throughout her fertile years. Another component consists of survival probabilities that characterize an individual’s ability to compete for limited environmental resources. A third module consists of a parametric function that expresses the probabilities of survival in a birth cohort of individuals by age for both females and males. A forth module of the model as an acceptance matrix of conditional probabilities such female may show a preference for the genotype or phenotype as her male sexual partner. It is assumed that any force of natural selection acts at the level of the three genotypes under consideration for each sex. By assigning values of the parameters in each of the modules under consideration, it is possible to conduct Monte Carlo simulation experiments designed to study the effects of each component of selection separately or in any combination on a population evolving from a given initial population over some specified period of time.

scholarly journals Demographic Dynamics in Multitype Populations with Migrations

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 246
Author(s):  
Manuel Molina-Fernández ◽  
Manuel Mota-Medina
Keyword(s):  
Mathematical Modeling ◽  
Population Dynamics ◽  
Branching Process ◽  
Probability Model ◽  
Biological Systems ◽  
Research Work ◽  
General Setting ◽  
Process Theory ◽  
Multitype Branching Process ◽  
Demographic Dynamics

This research work deals with mathematical modeling in complex biological systems in which several types of individuals coexist in various populations. Migratory phenomena among the populations are allowed. We propose a class of mathematical models to describe the demographic dynamics of these type of complex systems. The probability model is defined through a sequence of random matrices in which rows and columns represent the various populations and the several types of individuals, respectively. We prove that this stochastic sequence can be studied under the general setting provided by the multitype branching process theory. Probabilistic properties and limiting results are then established. As application, we present an illustrative example about the population dynamics of biological systems formed by long-lived raptor colonies.

Branching processes and functional differential equations determining steady-size distributions in cell populations

1995 ◽  
Vol 32 (01) ◽  
pp. 1-10
Author(s):  
Ziad Taib
Keyword(s):  
Differential Equation ◽  
Branching Process ◽  
Branching Processes ◽  
Functional Differential Equations ◽  
Cell Populations ◽  
Size Distributions ◽  
Multitype Branching Process ◽  
Functional Differential ◽  
Intuitive Interpretation ◽  
Infinite Sum

The functional differential equation y′(x) = ay(λx) + by(x) arises in many different situations. The purpose of this note is to show how it arises in some multitype branching process cell population models. We also show how its solution can be given an intuitive interpretation as the probability density function of an infinite sum of independent but not identically distributed random variables.

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.

scholarly journals Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector

2019 ◽  
Vol 23 ◽  
pp. 797-802
Author(s):  
Raphaël Cerf ◽  
Joseba Dalmau
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Branching Process ◽  
Invariant Probability Measure ◽  
Classical Formula ◽  
Multitype Branching Process ◽  
Primitive Matrix ◽  
Watson Process

Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A, as well as of a multitype branching process with mean matrix e(A−I)t. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

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.

scholarly journals The extinction time of a subcritical branching process related to the SIR epidemic on a random graph

2015 ◽  
Vol 52 (04) ◽  
pp. 1195-1201 ◽  
Author(s):  
Peter Windridge
Keyword(s):  
Random Graph ◽  
Epidemic Model ◽  
Branching Process ◽  
Large Population ◽  
Extinction Time ◽  
Exponential Tail ◽  
Sir Epidemic Model ◽  
Multitype Branching Process ◽  
Type Distribution ◽  
Sir Epidemic

We give an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to∞). We only require a second moment for the offspring-type distribution featuring in our model.

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