scholarly journals Extinction Probability of Interacting Branching Collision Processes

2012 ◽  
Vol 44 (1) ◽  
pp. 226-259 ◽  
Author(s):  
Anyue Chen ◽  
Junping Li ◽  
Yiqing Chen ◽  
Dingxuan Zhou
Keyword(s):  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Extinction Probability ◽  
Hitting Times ◽  
Collision Process ◽  
Markov Branching Process ◽  
Extinction Probabilities ◽  
Collision Processes ◽  
Necessary And Sufficient

We consider the uniqueness and extinction properties of the interacting branching collision process (IBCP), which consists of two strongly interacting components: an ordinary Markov branching process and a collision branching process. We establish that there is a unique IBCP, and derive necessary and sufficient conditions for it to be nonexplosive that are easily checked. Explicit expressions are obtained for the extinction probabilities for both regular and irregular cases. The associated expected hitting times are also considered. Examples are provided to illustrate our results.

Extinction Probability of Interacting Branching Collision Processes

2012 ◽  
Vol 44 (01) ◽  
pp. 226-259 ◽  
Author(s):  
Anyue Chen ◽  
Junping Li ◽  
Yiqing Chen ◽  
Dingxuan Zhou
Keyword(s):  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Extinction Probability ◽  
Hitting Times ◽  
Collision Process ◽  
Markov Branching Process ◽  
Extinction Probabilities ◽  
Collision Processes ◽  
Necessary And Sufficient

We consider the uniqueness and extinction properties of the interacting branching collision process (IBCP), which consists of two strongly interacting components: an ordinary Markov branching process and a collision branching process. We establish that there is a unique IBCP, and derive necessary and sufficient conditions for it to be nonexplosive that are easily checked. Explicit expressions are obtained for the extinction probabilities for both regular and irregular cases. The associated expected hitting times are also considered. Examples are provided to illustrate our results.

The collision branching process

2004 ◽  
Vol 41 (4) ◽  
pp. 1033-1048 ◽  
Author(s):  
Anyue Chen ◽  
Phil Pollett ◽  
Hanjun Zhang ◽  
Junping Li
Keyword(s):  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Hitting Times ◽  
Branching Model ◽  
Extinction Probabilities ◽  
Free Case ◽  
Necessary And Sufficient ◽  
Explicit Expressions

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.

The collision branching process

2004 ◽  
Vol 41 (04) ◽  
pp. 1033-1048 ◽  
Author(s):  
Anyue Chen ◽  
Phil Pollett ◽  
Hanjun Zhang ◽  
Junping Li
Keyword(s):  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Hitting Times ◽  
Branching Model ◽  
Extinction Probabilities ◽  
Free Case ◽  
Necessary And Sufficient ◽  
Explicit Expressions

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.

On the L 2-convergence of a superadditive bisexual Galton-Watson branching process

1997 ◽  
Vol 34 (03) ◽  
pp. 575-582 ◽  
Author(s):  
M. González ◽  
M. Molina
Keyword(s):  
Branching Process ◽  
Final Section ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

In this paper the L 2-convergence of a superadditive bisexual Galton–Watson branching process is studied. Necessary and sufficient conditions for the convergence of the suitably normed process are given. In the final section, a result about one of the most important bisexual models is proved.

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.

Bisexual Galton-Watson branching process with population-size-dependent mating

2002 ◽  
Vol 39 (3) ◽  
pp. 479-490 ◽  
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Population Size ◽  
Generating Functions ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Size Dependent ◽  
Probability Generating Functions ◽  
Necessary And Sufficient ◽  
Mating Function

In this paper, we introduce a bisexual Galton-Watson branching process with mating function dependent on the population size in each generation. Necessary and sufficient conditions for the process to become extinct with probability 1 are investigated for two possible conditions on the sequence of mating functions. Some results for the probability generating functions associated with the process are also given.

On the L2-convergence of a superadditive bisexual Galton-Watson branching process

1997 ◽  
Vol 34 (3) ◽  
pp. 575-582 ◽  
Author(s):  
M. González ◽  
M. Molina
Keyword(s):  
Branching Process ◽  
Final Section ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

In this paper the L2-convergence of a superadditive bisexual Galton–Watson branching process is studied. Necessary and sufficient conditions for the convergence of the suitably normed process are given. In the final section, a result about one of the most important bisexual models is proved.

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.

A branching process with disasters

1975 ◽  
Vol 12 (1) ◽  
pp. 47-59 ◽  
Author(s):  
Norman Kaplan ◽  
Aidan Sudbury ◽  
Trygve S. Nilsen
Keyword(s):  
Asymptotic Behavior ◽  
Renewal Process ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Population Process ◽  
Age Dependent ◽  
The Mean ◽  
Necessary And Sufficient

A population process is considered where particles reproduce according to an age-dependent branching process, and are subjected to disasters which occur at the epochs of an independent renewal process. Each particle alive at the time of a disaster, survives it with probability p and the survival of any particle is assumed independent of the survival of any other particle. The asymptotic behavior of the mean of the process is determined and as a consequence, necessary and sufficient conditions are given for extinction.

A branching process with disasters

1975 ◽  
Vol 12 (01) ◽  
pp. 47-59 ◽  
Author(s):  
Norman Kaplan ◽  
Aidan Sudbury ◽  
Trygve S. Nilsen
Keyword(s):  
Asymptotic Behavior ◽  
Renewal Process ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Population Process ◽  
Age Dependent ◽  
The Mean ◽  
Necessary And Sufficient

A population process is considered where particles reproduce according to an age-dependent branching process, and are subjected to disasters which occur at the epochs of an independent renewal process. Each particle alive at the time of a disaster, survives it with probability p and the survival of any particle is assumed independent of the survival of any other particle. The asymptotic behavior of the mean of the process is determined and as a consequence, necessary and sufficient conditions are given for extinction.

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