scholarly journals On a Maximum Sequence in a Critical Multitype Branching Process

1992 ◽  
Vol 20 (2) ◽  
pp. 746-752
Author(s):  
K. B. Athreya
Keyword(s):  
Branching Process ◽  
Multitype Branching Process ◽  
Maximum Sequence

A maximum sequence in a critical multitype branching process

1991 ◽  
Vol 28 (04) ◽  
pp. 893-897
Author(s):  
Aurel Spåtaru
Keyword(s):  
Spectral Radius ◽  
Branching Process ◽  
Multitype Branching Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
The Right ◽  
Maximum Sequence

Let (Zn ) be a p-type positively regular and non-singular critical Galton–Watson process with finite second moments. Associated with the spectral radius 1 of the mean matrix of (Zn ) consider the right eigenvector u = (u 1, · ··, up ) > 0, and set . It is shown that lim inf, lim sup whenever Z 0 = i, where .

A maximum sequence in a critical multitype branching process

1991 ◽  
Vol 28 (4) ◽  
pp. 893-897 ◽  
Author(s):  
Aurel Spåtaru
Keyword(s):  
Spectral Radius ◽  
Branching Process ◽  
Multitype Branching Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
The Right ◽  
Maximum Sequence

Let (Zn) be a p-type positively regular and non-singular critical Galton–Watson process with finite second moments. Associated with the spectral radius 1 of the mean matrix of (Zn) consider the right eigenvector u = (u1, · ··, up) > 0, and set . It is shown that lim inf, lim sup whenever Z0 = i, where .

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.

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.

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.

Conditions for extinction in certain bisexual Galton–Watson branching processes

1984 ◽  
Vol 21 (02) ◽  
pp. 414-418
Author(s):  
David M. Hull
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Multitype Branching Process ◽  
Extinction Probabilities ◽  
Multitype Branching Processes ◽  
Necessary And Sufficient

A multitype branching process, the n-family community mating process, is introduced for the purpose of comparing extinction probabilities with those of bisexual Galton–Watson branching processes. Consideration of known properties of standard multitype branching processes leads to conditions which are both necessary and sufficient for extinction in a bisexual Galton–Watson branching process. An application is then made to the counterexample of the author's earlier paper.

Estimation for multitype branching processes

1977 ◽  
Vol 14 (04) ◽  
pp. 829-835 ◽  
Author(s):  
M. P. Quine ◽  
P. Durham
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Natural Conditions ◽  
Multitype Branching Process ◽  
Large Samples ◽  
The Central Limit Theorem ◽  
The Matrix ◽  
Multitype Branching Processes ◽  
Branching Process With Immigration ◽  
Strongly Consistent

It is shown that certain estimators of the matrix of offspring means and the vector of stationary means in a subcritical multitype branching process with immigration are strongly consistent and obey the central limit theorem under fairly natural conditions. The results give some evidence of the robustness of the analogous autoregressive time series model in the case of large samples.

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