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 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.

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 branching processes to model extinction conditions for Y-linked genes

In a two-sex monogamic population, the evolution of the number of carriers of the two alleles of a Y-linked gene is considered. To this end, a multitype bisexual branching model is presented in which it is assumed that the gene has no influence on the mating process. It is deduced from this model that the average numbers of female and male descendants per mating unit constitute the key to determining the extinction or survival of each allele. Moreover, the destiny of each allele in the population is found not to depend on the behavior of the other.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

Bisexual branching processes in a genetic context: the extinction problem for Y-linked genes

In this paper the inheritance of a Y-linked gene with alleles R and r in a population with both females and males is modelled using a two-type bisexual branching process. It is assumed that the reproductive distribution associated with the R allele can differ from the associated with the r allele and that females prefer to mate with a male having the R allele rather than with a male with the r allele. Under these assumptions, we provide some conditions for the extinction and/or survival of both alleles in the population. These conditions depend on the magnitudes of the average number of females and males per mating unit. Moreover, the almost sure extinction of the r allele is independent of the behaviour of the R allele. On the other hand, the survival of the R allele with positive probability may depend strongly on the reproductive behaviour of the other allele. Theoretical results are illustrated by means of simulated examples and some open problems are proposed to the reader as conjectures.

Population dynamics under parasitic sex ratio distortion

We analyse the population dynamic effects of sex ratio distortion by vertically transmitted, feminizing parasites. We show that, for diploid hosts, sex ratio distortion may lead to extinction as males become too rare to maintain the host population through reproduction. Feminizers can magnify Allee effects, broadening the range of conditions leading to extinction of small populations. Depending on male mating constraints and strength of density dependence, feminizers may either increase or decrease the equilibrium host density. Under conditions leading to deterministic host extinction, stochastic elimination of the parasite may allow the host population to recover. Hence, infection by parasitic sex ratio distorters may be transient in finite populations. We consider the implications of this process for parasite prevalence, host population regulation, and sex ratio evolution.

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

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.