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.

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.

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.

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.

scholarly journals The Generalized Inverse A(2)T, Sof a Matrix Over an Associative Ring

2007 ◽  
Vol 83 (3) ◽  
pp. 423-438 ◽  
Author(s):  
Yaoming Yu ◽  
Guorong Wang
Keyword(s):  
Associative Ring ◽  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Arbitrary Matrix ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Explicit Expressions

AbstractIn this paper we establish the definition of the generalized inverse A(2)T, Swhich is a {2} inverse of a matrixAwith prescribed imageTand kernelsover an associative ring, and give necessary and sufficient conditions for the existence of the generalized inverseand some explicit expressions forof a matrix A over an associative ring, which reduce to the group inverse or {1} inverses. In addition, we show that for an arbitrary matrixAover an associative ring, the Drazin inverse Ad, the group inverse Agand the Moore-Penrose inverse. if they exist, are all the generalized inverse A(2)T, S.

scholarly journals A Predator–Prey Two-Sex Branching Process

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1408
Author(s):  
Cristina Gutiérrez ◽  
Carmen Minuesa
Keyword(s):  
Stochastic Process ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Predator Population ◽  
Predator Prey ◽  
Control Functions ◽  
Branching Model ◽  
Necessary And Sufficient ◽  
Number Of Individuals ◽  
Random Control

In this paper, we present the first stochastic process to describe the interaction of predator and prey populations with sexual reproduction. Specifically, we introduce a two-type two-sex controlled branching model. This process is a two-type branching process, where the first type corresponds to the predator population and the second one to the prey population. While each population is described via a two-sex branching model, the interaction and survival of both groups is modelled through control functions depending on the current number of individuals of each type in the ecosystem. In view of their potential for the conservation of species, we provide necessary and sufficient conditions for the ultimate extinction of both species, the fixation of one of them and the coexistence of both of them. Moreover, the description of the present predator–prey two-sex branching process on the fixation events can be performed in terms of the behaviour of a one-type two-sex branching process with a random control on the number of individuals, which is also introduced and analysed.

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.

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