Transition phenomena for age-dependent branching processes with discrete time. I

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

A renewal density theorem in the multi-dimensional case

Journal of Applied Probability ◽

10.2307/3212299 ◽

1967 ◽

Vol 4 (1) ◽

pp. 62-76 ◽

Cited By ~ 5

Author(s):

Charles J. Mode

Keyword(s):

Density Function ◽

Covariance Matrix ◽

Branching Processes ◽

Random Variable ◽

Density Theorem ◽

Positive Definite ◽

Dimensional Case ◽

Age Dependent ◽

Image Position ◽

Mean Vector

SummaryIn this note a renewal density theorem in the multi-dimensional case is formulated and proved. Let f(x) be the density function of a p-dimensional random variable with positive mean vector μ and positive-definite covariance matrix Σ, let hn(x) be the n-fold convolution of f(x) with itself, and set Then for arbitrary choice of integers k1, …, kp–1 distinct or not in the set (1, 2, …, p), it is shown that under certain conditions as all elements in the vector x = (x1, …, xp) become large. In the above expression μ‵ is interpreted as a row vector and μ a column vector. An application to the theory of a class of age-dependent branching processes is also presented.

Download Full-text

Two-Sex Branching Processes with Offspring and Mating in a Random Environment

Journal of Applied Probability ◽

10.1017/s0021900200006094 ◽

2009 ◽

Vol 46 (04) ◽

pp. 993-1004

Author(s):

S. Ma ◽

M. Molina

Keyword(s):

Mathematical Model ◽

Probability Distribution ◽

Discrete Time ◽

Random Environment ◽

Generating Functions ◽

Branching Processes ◽

Random Vectors ◽

Environmental Process ◽

Mating Function ◽

The Mathematical Model

We introduce a class of discrete-time two-sex branching processes where the offspring probability distribution and the mating function are governed by an environmental process. It is assumed that the environmental process is formed by independent but not necessarily identically distributed random vectors. For such a class, we determine some relationships among the probability generating functions involved in the mathematical model and derive expressions for the main moments. Also, by considering different probabilistic approaches we establish several results concerning the extinction probability. A simulated example is presented as an illustration.

Download Full-text

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.

Download Full-text

Diffusion limits of critical branching processes conditioned on extinction in the near future

Journal of Applied Probability ◽

10.1017/s0021900200094316 ◽

1976 ◽

Vol 13 (02) ◽

pp. 247-254

Author(s):

Warren W. Esty

Keyword(s):

Branching Processes ◽

Limit Laws ◽

Age Dependent ◽

Diffusion Limits ◽

Near Future

Limit laws and limiting diffusions are obtained for critical branching processes, either Galton-Watson or age-dependent, conditioned on extinction in the interval (T, cT], 1 < c, as T→∞, and also as T→∞ and c ↓ 1.

Download Full-text