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This research focuses on stochastic modeling of the evolution over time of biological populations through branching processes. We introduce a new class of discrete time two-sex branching processes with random mating and overlapping generations. Mating and reproduction are considered to be influenced by the numbers of females and males existing in the population. This evolution over time with generational overlap is a novel research in two-sex branching process literature. We study general probabilistic properties and establish some limiting results of biological interest. [ABSTRACT FROM AUTHOR]

In Jacob et al. [A General Class of Population-Dependent Two-Sex Processes with Random Mating. Bernoulli 2017, 23, 1737–1758], a new class of two-sex branching processes in discrete time was introduced. These processes present the novelty that, in each generation, mating between females and males is randomly governed by a set of Bernoulli distributions allowing polygamous behavior with only perfect fidelity on the part of female individuals. Moreover, mating as well as reproduction can be influenced by the number of females and males in the population. In that article, the authors study conditions leading to the almost sure extinction or to the possible survival with positive probability of such processes. In this work, we continue the research about this class of two-sex processes by investigating the rate of growth of the population in case of survival. In particular, we provide conditions for a simultaneous geometric growth of the number of females and males and conditions for the geometric growth of females with a stable population of males. [ABSTRACT FROM AUTHOR]

Lower and upper bounds for the cumulative distribution function (cdf) of the time to extinction in a subcritical two-sex branching process are derived. A recursive procedure for approximating this cdf is also utilized. The results are illustrated with some simulations. [ABSTRACT FROM AUTHOR]

We consider a discrete-time branching process in which the offspring distribution is generation-dependent and the number of reproductive individuals is controlled by a random mechanism. This model is a Markov chain, but, in general, the transition probabilities are nonstationary. Under not too restrictive hypotheses, this model presents the classical duality of branching processes: it either becomes extinct or grows to infinity. Sufficient conditions for the almost sure extinction and for a positive probability of indefinite growth are given. Finally, the rates of growth of the process are studied, provided that there is no extinction. [ABSTRACT FROM AUTHOR]

A class of homogeneous multitype Markov chains whose states have non-negative integer coordinates is considered in a situation similar to the supercritical case for branching processes. We establish the geometric growth of the chain, suitably normed, by investigating its almost sure and L2-convergence. Both sufficient and necessary conditions for these convergences are provided depending on certain constraints on the transition mean vectors and the transition covariance matrices of the chain. Finally, we apply the obtained results to a general class of controlled multitype branching processes. [ABSTRACT FROM AUTHOR]

• Problems on population dynamics of interest in biology and ecology are investigated. • A mathematical model to describe the evolution of long-lived raptor species is introduced. • Estimates for the reproductive parameters involved in the model are provided. • An algorithm based on approximate Bayesian computation methods is proposed. • The dynamics of the black vulture colony located at National Park of Monfragüe (Spain) is described. Population viability analysis of threatened large and long-lived raptor species has strong limitations due to the restricted demographic information available for these species. In this work, we mathematically model the demographic dynamics of these raptor species through time-indexed branching processes. By assuming the more general non-parametric statistical setting, we determine accurate estimates for the most relevant reproductive parameters involved in the model. To this end, we propose an algorithm based on approximate Bayesian computations methods. As illustration, by using real data of counts of the number of pairs in the population, we apply the proposed statistical and computational methods to describe the demographic dynamics of the Eurasian black vulture colony located at National Park of Monfragüe (Spain), which appears to be both the largest and densest breeding colony worldwide. In the scenario of these data-poor species, the class of time-indexed branching processes introduced appears to be appropriate and a more cost-effective method to evaluate dynamics and viability of the populations, applicable to the conservation of these taxa. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]