The point-process approach to age- and time-dependent branching processes

The multitype age-dependent branching process in varying environment is treated as a random stream (point process) of the birth and death events. We derive the point-process version of the Bellman–Harris equation, which provides us with a symbolic method of writing the equations for the arbitrary finite-dimensional distribution of the process. We also derive a new recurrence relation, the ‘principle of last generation'. This result is applied to obtain new moment equations for the process with immigration also. General considerations are illustrated by a simple cell kinetics model.

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Poisson law for Axiom A diffeomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700007513 ◽

1993 ◽

Vol 13 (3) ◽

pp. 533-556 ◽

Cited By ~ 79

Author(s):

Masaki Hirata

Keyword(s):

Point Process ◽

Gibbs Measure ◽

Poisson Point Process ◽

Axiom A ◽

Lipschitz Continuous ◽

Return Times ◽

Finite Dimensional ◽

Finite Dimensional Distribution ◽

Dimensional Distribution ◽

Continuous Potential

AbstractLet f be an Axiom A diffeomorphism, Ω its non wandering set, µ the Gibbs measure for the Lipschitz continuous potential. We consider the (suitably normalized) return times of the orbit to the ε-neighborhood of a point z ∈ Ω and prove that for µ-a.e. z the sequence of the normalized return times converges to the Poisson point process in finite dimensional distribution as ε → 0.

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On the Laplace transform of finite-dimensional distribution functions of semi-continuous random processes with reflecting and delaying screens

Exploring Stochastic Laws ◽

10.1515/9783112318768-019 ◽

1995 ◽

pp. 167-174 ◽

Cited By ~ 1

Keyword(s):

Laplace Transform ◽

Random Processes ◽

Distribution Functions ◽

Finite Dimensional ◽

Finite Dimensional Distribution ◽

The Laplace Transform ◽

Dimensional Distribution

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Stochastic models in cell kinetics

Journal of Applied Probability ◽

10.2307/3214149 ◽

1988 ◽

Vol 25 (A) ◽

pp. 91-111

Author(s):

Peter J. Brockwell

Keyword(s):

Life Cycle ◽

Stochastic Models ◽

Death Rate ◽

Cell Kinetics ◽

Branching Processes ◽

Experimental Studies ◽

Renewal Theory ◽

Biological Cell ◽

Age Dependent

We discuss the role of stochastic processes in modelling the life-cycle of a biological cell and the growth of cell populations. Results for multiphase age-dependent branching processes have proved invaluable for the interpretation of many of the basic experimental studies of the life-cycle. Moreover problems from cell kinetics, in particular those related to diurnal rhythm in cell-growth, have stimulated research into ‘periodic' renewal theory, and the asymptotic behaviour of populations of cells with periodic death rate.

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Sums of Lifetimes in Age Dependent Branching Processes

Journal of Applied Probability ◽

10.1017/s0021900200032630 ◽

1969 ◽

Vol 6 (01) ◽

pp. 195-200 ◽

Cited By ~ 2

Author(s):

J. Howard Weiner

Keyword(s):

Distribution Function ◽

Branching Process ◽

Past History ◽

Branching Processes ◽

Lifetime Distribution ◽

Parent Cell ◽

Absolutely Continuous ◽

Age Dependent ◽

A Cell ◽

Additional Assumptions

Consider a Bellman-Harris [1] age dependent branching process. At t = 0, a cell is born, has lifetime distribution function G(t), G(0) = 0, assumed to be absolutely continuous with density g(t). At the death of the cell, k new cells are born, each proceeding independently and identically as the parent cell, and independent of past history. Denote by h(s) = Σ k=0 ∞ pk s k and suppose h(1) ≡ m, and assume h”(1) < ∞. Additional assumptions will be added as required.

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Malthusian behaviour of the critical and subcritical age-dependent branching processes with arbitrary state space

Journal of Applied Probability ◽

10.2307/3212465 ◽

1976 ◽

Vol 13 (3) ◽

pp. 455-465

Author(s):

D. I. Saunders

Keyword(s):

State Space ◽

Branching Process ◽

Branching Processes ◽

Rates Of Convergence ◽

Expected Number ◽

Subcritical Case ◽

Arbitrary State ◽

Malthusian Parameter ◽

Age Dependent ◽

Number Of Individuals

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.

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A note on a functional equation arising in Galton-Watson branching processes

Journal of Applied Probability ◽

10.2307/3212181 ◽

1971 ◽

Vol 8 (3) ◽

pp. 589-598 ◽

Cited By ~ 18

Author(s):

Krishna B. Athreya

Keyword(s):

Functional Equation ◽

Continuous Function ◽

Generating Function ◽

Branching Process ◽

Branching Processes ◽

Probability Generating Function ◽

Age Dependent ◽

Watson Process ◽

Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

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Coalescence in Subcritical Bellman-Harris Age-Dependent Branching Processes

Journal of Applied Probability ◽

10.1239/jap/1371648962 ◽

2013 ◽

Vol 50 (2) ◽

pp. 576-591

Author(s):

Jyy-I Hong

Keyword(s):

Limit Distribution ◽

Branching Process ◽

Branching Processes ◽

Simple Random Sampling ◽

Single Type ◽

Last Common Ancestor ◽

Death Time ◽

Age Dependent ◽

Lines Of Descent ◽

First Time

We consider a continuous-time, single-type, age-dependent Bellman-Harris branching process. We investigate the limit distribution of the point process A(t)={at,i: 1≤ i≤ Z(t)}, where at,i is the age of the ith individual alive at time t, 1≤ i≤ Z(t), and Z(t) is the population size of individuals alive at time t. Also, if Z(t)≥ k, k≥2, is a positive integer, we pick k individuals from those who are alive at time t by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let Dk(t) be the coalescence time (the death time of the last common ancestor) of these k random chosen individuals. We study the distribution of Dk(t) and its limit distribution as t→∞.

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A note on a functional equation arising in Galton-Watson branching processes

Journal of Applied Probability ◽

10.1017/s002190020003566x ◽

1971 ◽

Vol 8 (03) ◽

pp. 589-598 ◽

Cited By ~ 6

Author(s):

Krishna B. Athreya

Keyword(s):

Functional Equation ◽

Continuous Function ◽

Generating Function ◽

Branching Process ◽

Branching Processes ◽

Probability Generating Function ◽

Age Dependent ◽

Watson Process ◽

Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

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Stochastic models in cell kinetics

Journal of Applied Probability ◽

10.1017/s0021900200040286 ◽

1988 ◽

Vol 25 (A) ◽

pp. 91-111

Author(s):

Peter J. Brockwell

Keyword(s):

Life Cycle ◽

Stochastic Models ◽

Death Rate ◽

Cell Kinetics ◽

Branching Processes ◽

Experimental Studies ◽

Renewal Theory ◽

Biological Cell ◽

Age Dependent

We discuss the role of stochastic processes in modelling the life-cycle of a biological cell and the growth of cell populations. Results for multiphase age-dependent branching processes have proved invaluable for the interpretation of many of the basic experimental studies of the life-cycle. Moreover problems from cell kinetics, in particular those related to diurnal rhythm in cell-growth, have stimulated research into ‘periodic' renewal theory, and the asymptotic behaviour of populations of cells with periodic death rate.

Download Full-text