Birth and death processes in phylogenetics and population genetics.

Area of Science:

• Mathematical Biology
• Evolutionary Biology
• Stochastic Processes

Background:

• Linear birth-and-death processes (LBDPs) model population dynamics and evolutionary relationships.
• Ancestral trees are crucial for understanding how evolving entities (species, cells) are related over time.
• Phylogenetic models have a rich history, with roots in mathematical theories of evolution dating back a century.

Purpose of the Study:

• To provide a comprehensive review of linear birth-and-death processes (LBDPs) and their applications.
• To detail the stochastic structure of ancestral trees derived from these processes.
• To present a modern method for approximate Bayesian computation in evolutionary models.

Main Methods:

• Description of the Yule (linear birth) process and its properties.
• Analysis of the birth-and-death process (BDP) with immigration.
• Application of random forests for approximate Bayesian computation in a cell evolution model.

Main Results:

• Detailed characterization of the population-size process and ancestral trees in LBDPs.
• Stochastic structure of reconstructed trees, accounting for non-surviving lineages.
• Successful inference of split and mutation rates from DNA sequence data using Bayesian methods.

Conclusions:

• LBDPs provide a robust mathematical framework for studying population evolution.
• Reconstructed ancestral trees offer insights into evolutionary history.
• Random forest-based Bayesian computation enables effective inference in complex evolutionary models.