Volume 17, Number 2, Summer 2009
BRANCHING PROCESSES AND
NONCOMMUTING RANDOM VARIABLES IN
POPULATION BIOLOGY
TIMOTHY C. RELUGA
Abstract. Branching processes are a wellestablished
tool in mathematical biology used to study the dynamics of
rarefied populations where agents act independently and small
stochastic densityindependent 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 inturn is connected to the
wellestablished study of generating functions. Noncommuting
randomvariable methods are described in detail and illustrated
using examples.
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