Birth–death branching models. Application to African elephant populations

Branching models have a long history of biological applications, particularly in population dynamics. In this work, our interest is the development of mathematical models to describe the demographic dynamics of socially structured animal populations, focusing our attention on lineages, usually matrilines, as the basic structure in the population. Significant efforts have been made to develop models based on the assumption that all individuals behave identically with respect to reproduction. However, the reproduction phase has a large random component that involves not only demographic but also environmental factors that change across range distribution of species. In the present work, we introduce new classes of birth–death branching models which take such factors into account. We assume that both, the offspring probability distribution and the death probabilities may be different in each generation, changing either predictably or unpredictably in relation to habitat features. We consider the genealogical tree generated by observation of the process until a pre-set generation. We determine the probability distributions of the random variables representing the number of dead or living individuals having at least one ancestor alive, living individuals whose ancestors are all dead, and dead individuals whose ancestors are all dead, explicitly obtaining their principal moments. Also, we derive the probability distributions corresponding to the partial and total numbers of such biological variables, obtaining in particular the distribution of the total number of matriarchs in the genealogical tree. We apply the proposed models to describe the demographic dynamics of African elephant populations living in different habitats.