Branching Processes and Non-Commuting Random Variables in Population Biology

Branching processes are a well-established tool in mathematical biology used to study the dynamics of rarefied populations where agents act independently and small stochastic density-independent changes in population sizes. However, they are often avoided by non-mathematicians because of their reliance on generating functions. Generating functions are powerful computational aids but are often difficult to motivate. In this paper, I review branching process theory using a non-commuting random variable description of multiplication as mnemonic for generating functions. Starting from the elementary definition of multiplication, I show how uncertainty leads to a natural generalization of integer multiplication without the commutative property, and how this in-turn is connected to the well-established study of generating functions. Non-commuting random-variable methods are described in detail and illustrated using examples.