Bisexual Galton Watson Branching Process Research Articles

A NOTE ON BISEXUAL GALTON-WATSON BRANCHING PROCESSES IN RANDOM ENVIRONMENTS

A bisexual Galton-Watson branching process is a two-type branching model, in which matings in one generation give rise to random numbers of both males and females in the next. The mating function describes how many mating units are formed from given numbers of males and females. In this paper we consider the case that the distributions of the random numbers of males and females produced by the mating units depend on some fertility parameters evolving randomly in time. By means of a main stochastic comparison result, we show that the total population increases, in some stochastic sense, as the positive dependence between the fertility indexes increases. Simple examples of applications of this result are provided, together with other similar results for a different model of population growth.

Weighted conditional least square estimators for bisexual branching processes with immigration

The aim of this paper is to study some inferential problems arising from the class of bisexual Galton–Watson branching processes with immigration of females and males (BGWPI). Immigrants are assumed to be unobservable, and it is only possible to sample the number of females, males, and couples (mating units) in each generation. Under these conditions, weighted conditional least square estimators are proposed for the offspring and immigration mean vectors. Asymptotic properties of such estimators are investigated when the process is subcritical and supercritical, paying especial attention to their strong consistency and limit distributions. Weighted conditional least square estimators are also developed for the offspring and immigration variance vectors, and their asymptotic properties are studied. Some comments on the critical case are also given to possibly provide a unified estimation theory for BGWPI.

Bayesian estimation in the class of bisexual branching processes with population-size dependent mating

This paper deals with the class of bisexual branching processes with population-size dependent mating introduced in Molina et al. (2002, Bisexual Galton–Watson branching process with population-size dependent mating. J Appl Probab 39:479–490). We determine Bayes estimators for the main moments of such processes in both situations: assuming offspring distribution belonging to the bivariate power series family and considering that no assumption is made about the underlying offspring distribution. As illustration we provide several examples, in particular, considering some classical offspring distributions, we explicitly determine the proposed Bayes estimators and, through simulation, we show the evolution of their estimates and of the corresponding high posterior density credibility sets.

On the asymptotic growth for a bisexual Galton-Watson branching process in varying environments

On the asymptotic growth for a bisexual Galton-Watson branching process in varying environments

THE ASYMPTOTIC PROPERTIES OF SUPERCRITICAL BISEXUAL GALTON-WATSON BRANCHING PROCESSES WITH IMMIGRATION OF MATING UNITS

In this article the supercritical bisexual Galton-Watson branching processes with the immigration of mating units is considered. A necessary condition for the almost sure convergence, and a sufficient condition for the L 1 convergence are given for the process with the suitably normed condition.

Bisexual Galton-Watson Branching Processes in Random Environments

In this paper, we consider a bisexual Galton-Watson branching process whose offspring probability distribution is controlled by a random environment process. Some results for the probability generating functions associated with the process are obtained and sufficient conditions for certain extinction and for non-certain extinction are established.

The bisexual branching process with population-size dependent mating as a mathematical model to describe phenomena concerning to inhabit or re-inhabit environments with animal species

We consider the bisexual Galton–Watson branching process with population-size dependent mating as a mathematical model adequate for the description of some natural phenomena. More specifically we are interested in studying some questions about the problem of populating an environmental with new animal species or re-populating it with species which have previously disappeared.

On the Extinction of a Class of Population-Size-Dependent Bisexual Branching Processes

In this paper, we study a class of bisexual Galton-Watson branching processes in which the law of offspring distribution is dependent on the population size. Under a suitable condition on the offspring distribution, we prove that the limit of mean growth-rate per mating unit exists. Based on this limit, we give a criterion to identify whether the process admits ultimate extinction with probability one.

On the Extinction of a Class of Population-Size-Dependent Bisexual Branching Processes

In this paper, we study a class of bisexual Galton-Watson branching processes in which the law of offspring distribution is dependent on the population size. Under a suitable condition on the offspring distribution, we prove that the limit of mean growth-rate per mating unit exists. Based on this limit, we give a criterion to identify whether the process admits ultimate extinction with probability one.

Limit behaviour for a supercritical bisexual Galton–Watson branching process with population-size-dependent mating

A supercritical bisexual Galton–Watson branching process with population-size-dependent mating is considered and some necessary and sufficient conditions for the almost sure and L 1-convergence of the suitably normalized process are given. Similar results for the underlying sequences of females and males are provided.

Bisexual Galton‐Watson Branching Process in Varying Environments

In this paper we introduce a bisexual Galton‐Watson branching process (BGWP) in which the offspring probability distribution is different in each generation. We obtain some relations among the probability generating functions (pgf) involved in the model and, making use of mean growth rates and fractional linear functions (flf), we provide sufficient and necessary conditions for its almost sure extinction.

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

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.

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

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 limit behavior of a supercritical bisexual Galton–Watson branching process with immigration of mating units1

In this paper a bisexual Galton–Watson branching process allowing the immigration of mating units is introduced and its limit behaviour is investigated. For the supercritical case, i.e., asymptotic growth rate greater than one, necessary and sufficient conditions for the almost sure and L 1 convergence of the suitably normed underlying sequences are given. *Research supported by the Consejerı´a de Educación y Juventud de la Junta de Extremadura and the Fondo Social Europeo, grant IPR98A023.

A reconsideration of Lotka's extinction probability using a bisexual branching process

It is generally recognized that Alfred Lotka made the first application of standard Galton-Watson branching process theory to calculate an extinction probability in a specific population (using asexual reproduction). This note applies bisexual Galton-Watson branching process theory to the calculation of an extinction probability from Lotka's data, yielding a somewhat higher value.

A reconsideration of Lotka's extinction probability using a bisexual branching process

It is generally recognized that Alfred Lotka made the first application of standard Galton-Watson branching process theory to calculate an extinction probability in a specific population (using asexual reproduction). This note applies bisexual Galton-Watson branching process theory to the calculation of an extinction probability from Lotka's data, yielding a somewhat higher value.

Limit behaviour for a subcritical bisexual Galton–Watson branching process with immigration

A bisexual Galton–Watson branching process allowing immigration of females and males is considered and for the subcritical case, i.e. growth rate less than one, the limit behaviour of related sequences is investigated.

A Reconsideration of Galton's Problem (Using a Two-Sex Population)

The main purposes of this paper are to promote and expound the bisexual Galton–Watson branching process as a relevant model for the consideration of Francis Galton's problem regarding the extinction of surnames of “men of note.” A scheme for adapting the bisexual process to consider Galton's problem is introduced. A necessary and sufficient condition for the certain extinction of a male-induced property in a two-sex species is presented. An approach for calculating the extinction of a male-generated characteristic in the two-sex species is proposed. That approach is then used to find the probability of the extinction of surnames in a bisexual population for Alfred Lotka's data based on an United States Census. Finally, these results are then compared with the classic extinction probabilities (from Lotka) associated with the traditional Galton–Watson branching process using asexual reproduction.

On the partial and total progeny of a bisexual Galton–Watson branching process

On the partial and total progeny of a bisexual Galton–Watson branching process

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

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.