Decomposable Branching Process Research Articles

Atypical Population Size in a Two-Type Decomposable Branching Process

We consider a Galton--Watson branching process with particles of two types in which particles of type one produce both particles of types one and two, and particles of type two generate offsprings of only type two. It is known that if both types are critical, then, for a process that is initiated at time $0$ by a single type-one particle, the number of particles of type two at time $n$ (provided that the process is not degenerate by this time) is proportional to $n$. We find the asymptotics of the probability that the number of type-two particles at time $n$ is of the order $o(n) $ (provided that the process is not degenerate by this time).

Reduced processes evolving in a mixed environment

We consider a two-type decomposable branching process where type 1 particles may produce particles of types 1 and 2 while type 2 particles can give birth only to type 2 particles. Let i = 1, 2 be the number of type i particles existing in the process at moment m < n and having a positive number of descendants at moment n. Assuming that particles of the first type evolve in a random environment and particles of the second type evolve in a constant environment we investigate the distribution of the random vector when and

Large fluctuations in multi-scale modeling for rest hematopoiesis.

Hematopoiesis is a biological phenomenon (process) of production of mature blood cells by cellular differentiation. It is based on amplification steps due to an interplay between renewal and differentiation in the successive cell types from stem cells to mature blood cells. We will study this mechanism with a stochastic point of view to explain unexpected fluctuations on the mature blood cell number, as surprisingly observed by biologists and medical doctors in a rest hematopoiesis. We consider three cell types: stem cells, progenitors and mature blood cells. Each cell type is characterized by its own dynamics parameters, the division rate and the renewal and differentiation probabilities at each division event. We model the global population dynamics by a three-dimensional stochastic decomposable branching process. We show that the amplification mechanism is given by the inverse of the small difference between the differentiation and renewal probabilities. Introducing a parameter K which scales simultaneously the size of the first component, the differentiation and renewal probabilities and the mature blood cell death rate, we describe the asymptotic behavior of the process for large K. We show that each cell type has its own size and time scales. Focusing on the third component, we prove that the mature blood cell population size, conveniently renormalized (in time and size), is expanded in an unusual way inducing large fluctuations. The proofs are based on a fine study of the different scales involved in the model and on the use of different convergence and average techniques in the proofs.

A subcritical decomposable branching process in a mixed environment

AbstractA two-type decomposable branching process is considered in which particles of the first type may produce at the death moment offspring of both types while particles of the second type may produce at the death moment offspring of their own type only. The reproduction law of the first type particles is specified by a random environment. The reproduction law of the second type particles is one and the same for all generations.A limit theorem is proved describing the conditional distribution of the number of particles in the process at timent,t∈ (0,1], given the survival of the process up to momentn→ ∞.

Decomposable branching processes with two types of particles

Abstract A two-type critical decomposable branching process with discrete time is considered in which particles of the first type may produce at the death moment offspring of both types while particles of the second type may produce at the death moment offspring of their own type only. Assuming that the offspring distributions of particles of both types may have infinite variance, the asymptotic behavior of the tail distribution of the random variable Ξ 2, the total number of the second type particles ever born in the process is found. Limit theorems are proved describing (as N → ∞) the conditional distribution of the amount of the first type particles in different generations given that either Ξ 2 = N or Ξ 2 &gt; N.

A subcritical decomposable branching process in a mixed environment

Рассматривается разложимый ветвящийся процесс Гальтона-Ватсона с двумя типами частиц, в котором частицы первого типа производят в конце жизни как потомков первого типа, так и потомков второго типа, а частицы второго типа порождают в момент гибели лишь потомков своего типа. Размножение частиц первого типа задается случайной средой, а закон размножения частиц второго типа один и тот же для всех поколений. Доказана предельная теорема, описывающая условное распределение числа частиц в процессе в моменты времени $nt,t\in (0,1]$, в предположении, что процесс не выродился к моменту времени $n\rightarrow \infty$.

A conditional functional limit theorem for decomposable branching processes with two types of particles

Consider a critical decomposable branching process with two types of particles in which particles of the first type give birth, at the end of their life, to descendants of the first type, as well as to descendants of the second type, while particles of the second type produce only descendants of the same type at the time of their death. We prove a functional limit theorem describing the distribution for the total number of particles of the second type appearing in the process in time Nt, 0 ≤ t < ∞, given that the number of particles of the first type appearing in the process during its evolution is N.

Branching processes in continuous time as models of mutations: Computational approaches and algorithms

The appearance of mutations in cancer development plays a crucial role in the disease control and its medical treatment. Motivated by the practical significance, it is of interest to model the event of occurrence of a mutant cell that will possibly lead to a path of indefinite survival. A two-type branching process model in continuous time is proposed for describing the relationship between the waiting time till the first escaping extinction mutant cell is born and the lifespan distribution of cells, which due to the applied treatment have small reproductive ratio. A numerical method and related algorithm for solving the integral equations are developed, in order to estimate the distribution of the waiting time to the escaping extinction mutant cell is born. Numerical studies demonstrate that the proposed approximation algorithm reveals the substantial difference of the results in discrete-time setting. In addition, to study the time needed for the mutant cell population to reach high levels a simulation algorithm for continuous two-type decomposable branching process is proposed. Two different computational approaches together with the theoretical studies might be applied to different kinds of cancer and their proper treatment.

On a decomposable branching process with two types of particles

A decomposable Galton–Watson branching process with two types of particles is considered. It is assumed that particles of the first type produce particles of both the first and the second types, and produce them in equal amounts, while particles of the second type only produce particles of the same type. An asymptotic formula is obtained for the probability that the total number of particles of the second type up to time N is greater than θN, where θ is a positive constant and N → ∞. A limit theorem is established for the total number of particles of the first type considered under the condition that the total number of particles of the second type up to time N is greater than θN.

Functional limit theorems for the decomposable branching process with two types of particles

Abstract A decomposable Galton - Watson process with two types of particles is considered. Particles of the first type produce equal random numbers of particles of both types, particles of the second type produce particles of the second type only. Under the condition that the total number of the first type particles is equal to N the functional limit theorems are proved for the numbers of particles of both types existing at times of the orders of N $\sqrt N $ , of N and of the intermediate orders.

Limit theorems for decomposable branching processes in a random environment

We study the asymptotics of the survival probability for the critical and decomposable branching processes in a random environment and prove Yaglom-type limit theorems for these processes. It is shown that such processes possess some properties having no analogues for the decomposable branching processes in a constant environment.

Limit theorems for decomposable branching processes in a random environment

We study the asymptotics of the survival probability for the critical and decomposable branching processes in a random environment and prove Yaglom-type limit theorems for these processes. It is shown that such processes possess some properties having no analogues for the decomposable branching processes in a constant environment.

Decomposable branching processes with a fixed extinction moment

The asymptotic behavior as n→∞ of the probability of the event that a decomposable critical branching process Z(m) = (Z 1(m),..., Z N(m)), m = 0, 1, 2,..., with N types of particles dies at moment n is investigated, and conditional limit theorems are proved that describe the distribution of the number of particles in the process Z(·) at moment m < n given that the extinction moment of the process is n. These limit theorems can be considered as statements describing the distribution of the number of vertices in the layers of certain classes of simply generated random trees of fixed height.

Functional limit theorems for the decomposable branching process with two types of particles

Рассматривается разложимый ветвящийся процесс Гальтона-Ватсона с двумя типами частиц. Предполагается, что частицы первого типа производят как частицы первого, так и второго типов, причем в одинаковых количествах, а частицы второго типа производят только частицы своего типа. Процесс рассматривается при условии, что полное число частиц первого типа равно $N$. Установлены функциональные предельные теоремы, в которых рассматриваются численности частиц как первого, так и второго типов в поколениях с номерами порядка $\sqrt{N}$, порядка $N$ и промежуточного порядка. Исследование выполнено за счет гранта Российского научного фонда (проект 14-50-00005).

Critical Branching Process with Two Types of Particles Evolving in Asynchronous Random Environments

A two-type pure decomposable branching process in a random environment is considered. Each particle of this process may produce offspring of its own type only. Let $\exp \{X_{k}(i)\}$ be the mean number of children produced by a particle of type $i=1,2$ of generation $k$. Assuming that $X_{k}(2)=-X_{k}(1)$ with probability 1 and that the random walk $S_{n}(1)=X_{1}(1)+\cdots +X_{n}(1)$ specified by the random environment is oscillating, we study the joint conditional distribution of the number of particles in the population at the moments $nt,0<t\le 1$ given that both processes survive up to the moment $n\to \infty $. Under the same conditioning we find asymptotic representation for the joint conditional distribution of the population sizes of both types at the moments when the environment is very unfavorable for particles of the first type. It is shown that the process has unusual properties which may be treated as bottlenecks and periods of growth in the model of predator-prey coexistence.

Decomposition of Supercritical Linear-Fractional Branching Processes

It is well known that a supercritical single-type Bienayme-Galton-Watson process can be viewed as a decomposable branching process formed by two subtypes of particles: those having infinite line of descent and those who have finite number of descendants. In this paper we analyze such a decomposition for the linear-fractional Bienayme-Galton-Watson processes with countably many types. We find explicit expressions for the main characteristics of the reproduction laws for so-called skeleton and doomed particles.

A multitype decomposable age-dependent branching process and its applications

Asymptotic formulas for means and variances of a multitype decomposable age-dependent supercritical branching process are derived. This process is a generalization of the Kendall–Neyman–Scott two-stage model for tumor growth. Both means and variances have exponential growth rates as in the case of the Markov branching process. But unlike Markov branching, these asymptotic moments depend on the age of the original individual at the start of the process and the life span distribution of the progenies.

ASYMPTOTIC BEHAVIOR OF THE SURVIVAL PROBABILITY FOR A DECOMPOSABLE BRANCHING PROCESS WITH REPLACEMENTS DEPENDING ON THE AGE OF THE PARTICLES

This article derives the asymptotic behavior of the survival probability for critical decomposable branching processes with replacements depending on the age of the particles.Bibliography: 9 titles.