The linear cell-size-dependent branching process

1972 ◽  
Vol 9 (04) ◽  
pp. 687-696 ◽  
Author(s):  
Peter Clifford ◽  
Aidan Sudbury
Keyword(s):  
Cell Size ◽  
Correlation Coefficient ◽  
Branching Process ◽  
Branching Processes ◽  
Linear Case ◽  
Size Dependent ◽  
Rate Of Growth ◽  
Linear Cell

In this paper is developed a theory of branching processes in which the division probability and rate of growth of cells depend only on their ‘size’ and in which ‘size’ is shared between daughters. Specific results are obtained in a linear case including the calculation of the correlation coefficient for the life spans of sisters.

The linear cell-size-dependent branching process

1972 ◽  
Vol 9 (4) ◽  
pp. 687-696 ◽  
Author(s):  
Peter Clifford ◽  
Aidan Sudbury
Keyword(s):  
Cell Size ◽  
Correlation Coefficient ◽  
Branching Process ◽  
Branching Processes ◽  
Linear Case ◽  
Size Dependent ◽  
Rate Of Growth ◽  
Linear Cell

In this paper is developed a theory of branching processes in which the division probability and rate of growth of cells depend only on their ‘size’ and in which ‘size’ is shared between daughters. Specific results are obtained in a linear case including the calculation of the correlation coefficient for the life spans of sisters.

Population-size-dependent branching process with linear rate of growth

1983 ◽  
Vol 20 (02) ◽  
pp. 242-250 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Positive Probability ◽  
Linear Rate ◽  
Size Dependent ◽  
Rate Of Growth ◽  
Binary Splitting

The process we consider is a binary splitting, where the probability of division, , depends on the population size, 2i. We show that Zn converges to ∞ almost surely on a set of positive probability. Zn /n converges in distribution to a proper limit, diverges almost surely on converges almost surely on and there are no constants cn such that Zn /cn converges in probability to a non-degenerate limit.

The expected population size in a cell-size-dependent branching process

1981 ◽  
Vol 18 (01) ◽  
pp. 65-75 ◽  
Author(s):  
Aidan Sudbury
Keyword(s):  
Cell Size ◽  
Exponential Growth ◽  
Branching Process ◽  
Expected Number ◽  
Size Dependent ◽  
Daughter Cells ◽  
Rate Of Increase ◽  
A Cell ◽  
Dependent Growth ◽  
Number Of Cells

In cell-size-dependent growth the probabilistic rate of division of a cell into daughter-cells and the rate of increase of its size depend on its size. In this paper the expected number of cells in the population at time t is calculated for a variety of models, and it is shown that population growths slower and faster than exponential are both possible. When the cell sizes are bounded conditions are given for exponential growth.

The xlogx condition for general branching processes

1998 ◽  
Vol 35 (03) ◽  
pp. 537-544
Author(s):  
Peter Olofsson
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Rate Of Growth

The xlogx condition is a fundamental criterion for the rate of growth of a general branching process, being equivalent to non-degeneracy of the limiting random variable. In this paper we adopt the ideas from Lyons, Pemantle and Peres (1995) to present a new proof of this well-known theorem. The idea is to compare the ordinary branching measure on the space of population trees with another measure, the size-biased measure.

Asymptotic behaviour of population-size-dependent branching processes in Markovian random environments

1999 ◽  
Vol 36 (2) ◽  
pp. 611-619 ◽  
Author(s):  
Han-Xing Wang ◽  
Dafan Fang
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Process ◽  
Branching Processes ◽  
Random Environments ◽  
Real Numbers ◽  
Size Dependent ◽  
Recurrent Markov Chain ◽  
The Mean ◽  
Positive Recurrent

A population-size-dependent branching process {Zn} is considered where the population's evolution is controlled by a Markovian environment process {ξn}. For this model, let mk,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Zn(ω) = 0 for some n} and q = P(B). The asymptotic behaviour of limnZn and is studied in the case where supθ|mk,θ − mθ| → 0 for some real numbers {mθ} such that infθmθ > 1. When the environmental sequence {ξn} is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q < 1) are studied.

Some results for general cell-size-dependent branching processes

1973 ◽  
Vol 10 (2) ◽  
pp. 289-298 ◽  
Author(s):  
Aidan Sudbury ◽  
Peter Clifford
Keyword(s):  
Growth Rate ◽  
Cell Size ◽  
Wide Class ◽  
General Model ◽  
Branching Processes ◽  
Size Dependent ◽  
Sister Cells

A general model for the growth and division of cells in which the growth rate and division probability at any instant depend only on their size at that time is introduced. Conditions under which (a) the distribution of cell-size at division converges ergodically, (b) the sizes tend to 0 or ∞, are exhibited, and bounds to the correlation between the sizes at division of sister cells are given in a wide class of cases.

Geometric rate of growth in population-size-dependent branching processes

1984 ◽  
Vol 21 (01) ◽  
pp. 40-49 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Population Size ◽  
Process Model ◽  
Branching Process ◽  
Necessary Conditions ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Size Dependent ◽  
Offspring Distribution ◽  
The Mean

We consider a branching-process model {Zn }, where the law of offspring distribution depends on the population size. We consider the case when the means mn (mn is the mean of offspring distribution when the population size is equal to n) tend to a limit m &gt; 1 as n →∞. For a certain class of processes {Zn } necessary conditions for convergence in L 1 and L 2 and sufficient conditions for almost sure convergence and convergence in L 2 of Wn = Zn/mn are given.

Extinction of population-size-dependent branching processes in random environments

1999 ◽  
Vol 36 (1) ◽  
pp. 146-154 ◽  
Author(s):  
Han-xing Wang
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Size Dependent ◽  
Branching Model

We generalize a population-size-dependent branching process to a more general branching model called the population-size-dependent branching process in random environments. For the model where {Zn}n≥0 is associated with the stationary environment ξ− = {ξn}n≥0, let B = {ω : Zn(ω) = for some n}, and q(ξ−) = P(B | ξ−, Z0 = 1). The result is that P(q(̅ξ) = 1) is either 1 or 0, and sufficient conditions for certain extinction (i.e. P(q(ξ−) = 1) = 1) and for non-certain extinction (i.e. P(q(ξ−) < 1) = 1) are obtained for the model.

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