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Volume 17,     Number 2,     Summer 2009

 

BRANCHING PROCESSES AND
NONCOMMUTING RANDOM VARIABLES IN
POPULATION BIOLOGY
TIMOTHY C. RELUGA

Abstract. 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 nonmathematicians 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 noncommuting 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. Noncommuting random-variable methods are described in detail and illustrated using examples.

 

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