On a multi-type critical age-dependent branching process

1970 ◽  
Vol 7 (3) ◽  
pp. 523-543 ◽  
Author(s):  
H. J. Weiner
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Continuous Distribution ◽  
Cell Type ◽  
Distinguishable Particle ◽  
Age Dependent ◽  
Critical Age ◽  
Image Position

We will consider a branching process with m > 1 distinguishable particle types as follows. At time 0, one newly born cell of type i is born (i = 1, 2, ···, m). Cell type i lives a random lifetime with continuous distribution function Gi(t), Gi(0+) = 0. At the end of its life, cell i is replaced by j1 new cells of type 1, j2 new cells of type 2, ···, jm new cells of type m with probability , and we define the generating functions for i = 1,···,m, where and . Each new daughter cell proceeds independently of the state of the system, with each cell type j governed by Gj(t) and hj(s).

On a multi-type critical age-dependent branching process

1970 ◽  
Vol 7 (03) ◽  
pp. 523-543 ◽  
Author(s):  
H. J. Weiner
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Continuous Distribution ◽  
Cell Type ◽  
Distinguishable Particle ◽  
Age Dependent ◽  
Critical Age ◽  
Image Position

We will consider a branching process with m > 1 distinguishable particle types as follows. At time 0, one newly born cell of type i is born (i = 1, 2, ···, m). Cell type i lives a random lifetime with continuous distribution function Gi (t), Gi (0+) = 0. At the end of its life, cell i is replaced by j 1 new cells of type 1, j 2 new cells of type 2, ···, jm new cells of type m with probability , and we define the generating functions for i = 1,···,m, where and . Each new daughter cell proceeds independently of the state of the system, with each cell type j governed by Gj(t) and hj(s).

A multi-type critical age-dependent branching process with immigration

1972 ◽  
Vol 9 (4) ◽  
pp. 697-706 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Gamma Distribution ◽  
Generating Functions ◽  
Branching Process ◽  
Cell Types ◽  
Distribution Law ◽  
One Dimensional ◽  
Age Dependent ◽  
Critical Age ◽  
Branching Process With Immigration

In a multi-type critical age-dependent branching process with immigration, the numbers of cell types alive at time t, each divided by t, as t becomes large, tends to a one-dimensional gamma distribution law. The method of proof employs generating functions and compution of asymptotic moments. Connections with earlier results and extensions are indicated.

A multi-type critical age-dependent branching process with immigration

1972 ◽  
Vol 9 (04) ◽  
pp. 697-706 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Gamma Distribution ◽  
Generating Functions ◽  
Branching Process ◽  
Cell Types ◽  
Distribution Law ◽  
One Dimensional ◽  
Age Dependent ◽  
Critical Age ◽  
Branching Process With Immigration

In a multi-type critical age-dependent branching process with immigration, the numbers of cell types alive at time t, each divided by t, as t becomes large, tends to a one-dimensional gamma distribution law. The method of proof employs generating functions and compution of asymptotic moments. Connections with earlier results and extensions are indicated.

Total progeny in a multitype critical age dependent branching process with immigration

1974 ◽  
Vol 11 (3) ◽  
pp. 458-470 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Cell Types ◽  
Dimensional Case ◽  
One Dimensional ◽  
Age Dependent ◽  
Total Progeny ◽  
Critical Age ◽  
The One ◽  
Branching Process With Immigration

In a multitype critical age dependent branching process with immigration, the numbers of cell types born by t, divided by t2, tends in law to a one-dimensional (degenerate) law whose Laplace transform is explicitily given. The method of proof makes a correspondence between the moments in the m-dimensional case and the one-dimensional case, for which the corresponding limit theorem is known. Other applications are given, a possible relaxation of moment assumptions, and extensions are indicated.

Asymptotic probabilities in a critical age-dependent branching process

1976 ◽  
Vol 13 (3) ◽  
pp. 476-485 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Branching Process ◽  
Comparison Method ◽  
Lifetime Distribution ◽  
Order Moment ◽  
Moment Condition ◽  
Age Dependent ◽  
Total Progeny ◽  
Critical Age ◽  
The Mean ◽  
Critical Process

Let Z(t) denote the number of cells alive at time t in a critical Bellman-Harris age-dependent branching process, that is, where the mean number of offspring per parent is one. A comparison method is used to show for k ≧ 1, and a high-order moment condition on G(t), where G(t) is the cell lifetime distribution, that lim t→∞t2P[Z(t) = k] = ak > 0, where {ak} are constants.The method is also applied to the total progeny in the critical process.

Conditional moments in a critical age-dependent branching process

1975 ◽  
Vol 12 (3) ◽  
pp. 581-587 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Asymptotic Behavior ◽  
Branching Process ◽  
Conditional Moments ◽  
Age Dependent ◽  
Critical Age ◽  
Number Of Cells

Let X(t), N(t) respectively denote the number of cells alive at t and the total number of cells born by t in a critical age-dependent Bellman-Harris branching process.The asymptotic behavior of the conditional moments, for 0 < α ≦ 1, E(Nn(αt) | X(t) > 0), E(Nn(t) |X(αt) > 0), is obtained.

scholarly journals Bounds for the Asymptotic Growth Rate of an Age-Dependent Branching Process

1969 ◽  
Vol 10 (1-2) ◽  
pp. 231-235 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Growth Rate ◽  
Population Size ◽  
Branching Process ◽  
Expected Number ◽  
Lattice Distribution ◽  
Asymptotic Growth ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
Asymptotic Growth Rate

Let M(t) denote the mean population size at time t (conditional on a single ancestor of age zero at time zero) of a branching process in which the distribution of the lifetime T of an individual is given by Pr {T≦t} =G(t), and in which each individual gives rise (at death) to an expected number A of offspring (1λ A λ ∞). expected number A of offspring (1 < A ∞). Then it is well-known (Harris [1], p. 143) that, provided G(O+)-G(O-) 0 and G is not a lattice distribution, M(t) is given asymptotically by where c is the unique positive value of p satisfying the equation .

scholarly journals An Instructive Component in T Helper Cell Type 2 (Th2) Development Mediated by Gata-3

2001 ◽  
Vol 193 (5) ◽  
pp. 643-650 ◽  
Author(s):  
J. David Farrar ◽  
Wenjun Ouyang ◽  
Max Löhning ◽  
Mario Assenmacher ◽  
Andreas Radbruch ◽  
...  
Keyword(s):  
Cell Fate ◽  
T Helper Cell ◽  
Helper Cell ◽  
Cell Type ◽  
T Helper ◽  
Cell Fate Decisions ◽  
Helper Cell Type ◽  
Th2 Development

Although interleukin (IL)-12 and IL-4 polarize naive CD4+ T cells toward T helper cell type 1 (Th1) or Th2 phenotypes, it is not known whether cytokines instruct the developmental fate in uncommitted progenitors or select for outgrowth of cells that have stochastically committed to a particular fate. To distinguish these instructive and selective models, we used surface affinity matrix technology to isolate committed progenitors based on cytokine secretion phenotype and developed retroviral-based tagging approaches to directly monitor individual progenitor fate decisions at the clonal and population levels. We observe IL-4–dependent redirection of phenotype in cells that have already committed to a non–IL-4–producing fate, inconsistent with predictions of the selective model. Further, retroviral tagging of naive progenitors with the Th2-specific transcription factor GATA-3 provided direct evidence for instructive differentiation, and no evidence for the selective outgrowth of cells committed to either the Th1 or Th2 fate. These data would seem to exclude selection as an exclusive mechanism in Th1/Th2 differentiation, and support an instructive model of cytokine-driven transcriptional programming of cell fate decisions.

scholarly journals A Decomposable Branching Process in a Markovian Environment

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Vladimir Vatutin ◽  
Elena Dyakonova ◽  
Peter Jagers ◽  
Serik Sagitov
Keyword(s):  
Survival Probability ◽  
Branching Process ◽  
The Other ◽  
Variable Environment ◽  
Markovian Environment ◽  
Constant Environment ◽  
Environment Type

A population has two types of individuals, with each occupying an island. One of those, where individuals of type 1 live, offers a variable environment. Type 2 individuals dwell on the other island, in a constant environment. Only one-way migration () is possible. We study then asymptotics of the survival probability in critical and subcritical cases.

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