Multi-trait Branching Models with Applications to Species Evolution

Abstract: This thesis provides an analysis of the evolution of discrete traits and their effect on the birth and survival of species using the theory of supercritical, continuous time Markov branching processes. We present a branching modeling framework that incorporates multi-trait diversification processes associated with the emergence of new species, death of existing species, and transition of species carrying one type of a trait to another. The trait-dependent speciation, extinction, and transition help in interpreting the relationships between traits on one hand, and linking together the diversification process with molecular evolution on the other. Various multitype species branching models are applied in order to examine the evolutionary patterns in known data sets, such as the impact of outcrossing and selfing mating systems on the diversification rates of species, and the analysis of virulent behavior of pathogenic bacterial strains in different environments. Stochastic equations and limit theorems for branching processes help scrutinize the long time asymptotics of the models under an asymmetry in change of types, and under various schemes of rescaling. In addition, we explore diversity-dependent processes in which, instead of allowing supercritical growth of population sizes, the increase in species numbers is regulated by modifying the species branching rates. The use of probabilistic methods in a setting of population genetics leads to an analogy between biallelic frequency models and binary trait species tree models. To obtain an approximation for a Markov branching process of species evolution over a long geological time scale, we methodically utilize the theory of diffusion processes. Overall, our results show that branching models can be effectively used to seek to comprehend the diversification patterns in species during the process of evolution.