On the class of controlled branching processes with random control functions

2002 ◽  
Vol 39 (4) ◽  
pp. 804-815 ◽  
Author(s):  
M. González ◽  
M. Molina ◽  
I. Del Puerto
Keyword(s):  
Population Size ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Control Functions ◽  
Controlled Branching Processes ◽  
Necessary And Sufficient ◽  
Random Control

In this paper, the class of controlled branching processes with random control functions introduced by Yanev (1976) is considered. For this class, necessary and sufficient conditions are established for the process to become extinct with probability 1 and the limit probabilistic behaviour of the population size, suitably normed, is investigated.

On the class of controlled branching processes with random control functions

2002 ◽  
Vol 39 (04) ◽  
pp. 804-815 ◽  
Author(s):  
M. González ◽  
M. Molina ◽  
I. Del Puerto
Keyword(s):  
Population Size ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Control Functions ◽  
Controlled Branching Processes ◽  
Necessary And Sufficient ◽  
Random Control

In this paper, the class of controlled branching processes with random control functions introduced by Yanev (1976) is considered. For this class, necessary and sufficient conditions are established for the process to become extinct with probability 1 and the limit probabilistic behaviour of the population size, suitably normed, is investigated.

scholarly journals Asymptotic behaviour of critical controlled branching processes with random control functions

2005 ◽  
Vol 42 (2) ◽  
pp. 463-477 ◽  
Author(s):  
M. González ◽  
M. Molina ◽  
I. del Puerto
Keyword(s):  
Asymptotic Behaviour ◽  
Critical Case ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Control Functions ◽  
Limiting Behaviour ◽  
Controlled Branching Processes ◽  
Random Control

In this paper, we investigate the asymptotic behaviour of controlled branching processes with random control functions. In a critical case, we establish sufficient conditions for both their almost-sure extinction and for their nonextinction with a positive probability. For some suitably chosen norming constants, we also determine different kinds of limiting behaviour for this class of processes.

scholarly journals Asymptotic behaviour of critical controlled branching processes with random control functions

2005 ◽  
Vol 42 (02) ◽  
pp. 463-477 ◽  
Author(s):  
M. González ◽  
M. Molina ◽  
I. del Puerto
Keyword(s):  
Asymptotic Behaviour ◽  
Critical Case ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Control Functions ◽  
Limiting Behaviour ◽  
Controlled Branching Processes ◽  
Random Control

In this paper, we investigate the asymptotic behaviour of controlled branching processes with random control functions. In a critical case, we establish sufficient conditions for both their almost-sure extinction and for their nonextinction with a positive probability. For some suitably chosen norming constants, we also determine different kinds of limiting behaviour for this class of processes.

On the geometric growth in controlled branching processes with random control function

2003 ◽  
Vol 40 (04) ◽  
pp. 995-1006 ◽  
Author(s):  
M. González ◽  
M. Molina ◽  
I. del Puerto
Keyword(s):  
Branching Process ◽  
Control Function ◽  
Branching Processes ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Limit Behaviour ◽  
Controlled Branching Processes ◽  
Necessary And Sufficient ◽  
Random Control ◽  
Geometric Growth

The limit behaviour of a controlled branching process with random control function is investigated. A necessary condition and a sufficient condition for the geometric growth of such a process are established by considering the L 1-convergence. Finally, taking into account the classical X log+ X criterion in branching processes, a necessary and sufficient condition is provided.

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 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.

Sums of i.i.d. random variables and an application to the explosion criterion for markov branching processes

1982 ◽  
Vol 19 (01) ◽  
pp. 29-38 ◽  
Author(s):  
H.-J. Schuh
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Probabilistic Proof ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for in terms of , where Sn is the sum of n i.i.d. random variables with values in]0, ∞[, and A ≧ 0. We use these results to give a probabilistic proof of the ‘explosion criterion' for continuous-time Markov branching processes, which is usually shown analytically.

Sums of i.i.d. random variables and an application to the explosion criterion for markov branching processes

1982 ◽  
Vol 19 (1) ◽  
pp. 29-38 ◽  
Author(s):  
H.-J. Schuh
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Probabilistic Proof ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for in terms of , where Sn is the sum of n i.i.d. random variables with values in]0, ∞[, and A ≧ 0. We use these results to give a probabilistic proof of the ‘explosion criterion' for continuous-time Markov branching processes, which is usually shown analytically.

On the geometric growth in controlled branching processes with random control function

2003 ◽  
Vol 40 (4) ◽  
pp. 995-1006 ◽  
Author(s):  
M. González ◽  
M. Molina ◽  
I. del Puerto
Keyword(s):  
Branching Process ◽  
Control Function ◽  
Branching Processes ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Limit Behaviour ◽  
Controlled Branching Processes ◽  
Necessary And Sufficient ◽  
Random Control ◽  
Geometric Growth

The limit behaviour of a controlled branching process with random control function is investigated. A necessary condition and a sufficient condition for the geometric growth of such a process are established by considering the L1-convergence. Finally, taking into account the classical X log+X criterion in branching processes, a necessary and sufficient condition is provided.

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

2002 ◽  
Vol 39 (03) ◽  
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.

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