scholarly journals Multitype branching processes in random environments

1971 ◽  
Vol 8 (1) ◽  
pp. 17-31 ◽  
Author(s):  
Edward W. Weissner
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Restrictive Assumption ◽  
Multitype Branching Processes ◽  
Watson Process ◽  
Necessary And Sufficient

Consider the classical Galton-Watson process described by Harris ((1963), Chapter 1). Recently this model has been generalized in Smith (1968), Smith and Wilkinson (1969), and Wilkinson (1967). They removed the restrictive assumption that the particles always divide in accordance with the same p.g.f. Instead, they assumed that at each unit of time, Nature be allowed to choose a p.g.f. from a class of p.g.f.'s, independently of the population, past and present, and the previously selected p.g.f.'s, which would then be assigned to the present population. Each particle of the present population would then split, independently of the others, in accordance with the selected p.g.f. This process, called a branching process in a random environment (BPRE), is clearly more applicable than the Galton-Watson process. Moreover, Smith and Wilkinson have found necessary and sufficient conditions for almost certain extinction of the BPRE which are almost as easy to verify as those for the Galton-Watson process.

scholarly journals Multitype branching processes in random environments

1971 ◽  
Vol 8 (01) ◽  
pp. 17-31 ◽  
Author(s):  
Edward W. Weissner
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Restrictive Assumption ◽  
Multitype Branching Processes ◽  
Watson Process ◽  
Necessary And Sufficient

Consider the classical Galton-Watson process described by Harris ((1963), Chapter 1). Recently this model has been generalized in Smith (1968), Smith and Wilkinson (1969), and Wilkinson (1967). They removed the restrictive assumption that the particles always divide in accordance with the same p.g.f. Instead, they assumed that at each unit of time, Nature be allowed to choose a p.g.f. from a class of p.g.f.'s, independently of the population, past and present, and the previously selected p.g.f.'s, which would then be assigned to the present population. Each particle of the present population would then split, independently of the others, in accordance with the selected p.g.f. This process, called a branching process in a random environment (BPRE), is clearly more applicable than the Galton-Watson process. Moreover, Smith and Wilkinson have found necessary and sufficient conditions for almost certain extinction of the BPRE which are almost as easy to verify as those for the Galton-Watson process.

scholarly journals On the extinction times of varying and random environment branching processes

1975 ◽  
Vol 12 (01) ◽  
pp. 39-46 ◽  
Author(s):  
Alan Agresti
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Extinction Time ◽  
Probability Of Extinction ◽  
Stochastically Bounded ◽  
Necessary And Sufficient ◽  
Varying Environment

Bounds are derived for the probability of extinction by the nth generation for a branching process in a varying environment. From these bounds, necessary and sufficient conditions are established for such a process to become extinct with probability one. The extinction time of a random environment branching process in which the environmental random variables are independent but not necessarily identically distributed is stochastically bounded by the extinction times of two varying environment processes.

scholarly journals On the extinction times of varying and random environment branching processes

1975 ◽  
Vol 12 (1) ◽  
pp. 39-46 ◽  
Author(s):  
Alan Agresti
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Extinction Time ◽  
Probability Of Extinction ◽  
Stochastically Bounded ◽  
Necessary And Sufficient ◽  
Varying Environment

Bounds are derived for the probability of extinction by the nth generation for a branching process in a varying environment. From these bounds, necessary and sufficient conditions are established for such a process to become extinct with probability one. The extinction time of a random environment branching process in which the environmental random variables are independent but not necessarily identically distributed is stochastically bounded by the extinction times of two varying environment processes.

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.

Conditions for extinction in certain bisexual Galton–Watson branching processes

1984 ◽  
Vol 21 (02) ◽  
pp. 414-418
Author(s):  
David M. Hull
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Multitype Branching Process ◽  
Extinction Probabilities ◽  
Multitype Branching Processes ◽  
Necessary And Sufficient

A multitype branching process, the n-family community mating process, is introduced for the purpose of comparing extinction probabilities with those of bisexual Galton–Watson branching processes. Consideration of known properties of standard multitype branching processes leads to conditions which are both necessary and sufficient for extinction in a bisexual Galton–Watson branching process. An application is then made to the counterexample of the author's earlier paper.

Branching processes in generalized autoregressive conditional environments

2016 ◽  
Vol 48 (4) ◽  
pp. 1211-1234 ◽  
Author(s):  
Irene Hueter
Keyword(s):  
Ebola Virus ◽  
Negative Binomial ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Offspring Number ◽  
Generalized Poisson ◽  
Novel Approach ◽  
Necessary And Sufficient ◽  
Near Extinction

AbstractBranching processes in random environments have been widely studied and applied to population growth systems to model the spread of epidemics, infectious diseases, cancerous tumor growth, and social network traffic. However, Ebola virus, tuberculosis infections, and avian flu grow or change at rates that vary with time—at peak rates during pandemic time periods, while at low rates when near extinction. The branching processes in generalized autoregressive conditional environments we propose provide a novel approach to branching processes that allows for such time-varying random environments and instances of peak growth and near extinction-type rates. Offspring distributions we consider to illustrate the model include the generalized Poisson, binomial, and negative binomial integer-valued GARCH models. We establish conditions on the environmental process that guarantee stationarity and ergodicity of the mean offspring number and environmental processes and provide equations from which their variances, autocorrelation, and cross-correlation functions can be deduced. Furthermore, we present results on fundamental questions of importance to these processes—the survival-extinction dichotomy, growth behavior, necessary and sufficient conditions for noncertain extinction, characterization of the phase transition between the subcritical and supercritical regimes, and survival behavior in each phase and at criticality.

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

1999 ◽  
Vol 36 (01) ◽  
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 {Z n } n≥0 is associated with the stationary environment ξ− = {ξ n } n≥0, let B = {ω : Z n (ω) = for some n}, and q(ξ−) = P(B | ξ−, Z 0 = 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−) &lt; 1) = 1) are obtained for the model.

A continuous time Markov branching model with random environments

1973 ◽  
Vol 5 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Population Model ◽  
Limit Theorems ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Markov Branching Process ◽  
Branching Model ◽  
Necessary And Sufficient

A population model is constructed which combines the ideas of a discrete time branching process with random environments and a continuous time non-homogeneous Markov branching process. The extinction problem is considered and necessary and sufficient conditions for extinction are determined. Also discussed are limit theorems for what corresponds to the supercritical case.

Conditions for extinction in certain bisexual Galton–Watson branching processes

1984 ◽  
Vol 21 (2) ◽  
pp. 414-418 ◽  
Author(s):  
David M. Hull
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Multitype Branching Process ◽  
Extinction Probabilities ◽  
Multitype Branching Processes ◽  
Necessary And Sufficient

A multitype branching process, the n-family community mating process, is introduced for the purpose of comparing extinction probabilities with those of bisexual Galton–Watson branching processes. Consideration of known properties of standard multitype branching processes leads to conditions which are both necessary and sufficient for extinction in a bisexual Galton–Watson branching process. An application is then made to the counterexample of the author's earlier paper.

A continuous time Markov branching model with random environments

1973 ◽  
Vol 5 (01) ◽  
pp. 37-54 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Population Model ◽  
Limit Theorems ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Markov Branching Process ◽  
Branching Model ◽  
Necessary And Sufficient

A population model is constructed which combines the ideas of a discrete time branching process with random environments and a continuous time non-homogeneous Markov branching process. The extinction problem is considered and necessary and sufficient conditions for extinction are determined. Also discussed are limit theorems for what corresponds to the supercritical case.

Sign in / Sign up

Export Citation Format

Share Document