Branching Processes in Biology - Rice University

Branching Processes in Biology

Branching Processes in Biology

Marek Kimmel

Department of Statistics

Rice University

Marek Kimmel

Sept/Oct 2004

Branching Processes in Biology

Branching Processes in Biology

Marek Kimmel

Based on Kimmel and Axelrod ¡±Branching Processes in Biology¡±

Springer 2002

Marek Kimmel

Sept/Oct 2004

Branching Processes in Biology

What are branching processes?

? Family Z (t, ¦Ø) , t ¡Ý 0} of nonnegative integer-valued random

variables defined on a common probability space ?

? Process is initiated at time t = 0 by birth of a single ancestor

particle with random lifelength ¦Ó (¦Ø)

? Ancestor produces X (¦Ø) progeny at death

? Each progeny is the ancestor of its own independent process, which

is a component of our branching process.

? The number of individuals present in the process at time t is equal

to the sum of numbers of the individuals present in all these

subprocesses:

Marek Kimmel

Sept/Oct 2004

Branching Processes in Biology

Marek Kimmel

Sept/Oct 2004

Branching Processes in Biology

?

X(¦Ø)

?

? P Z (i) (t, ¦Ó (¦Ø), ¦Ø), t ¡Ý ¦Ó (¦Ø),

Z(t, ¦Ø) =

i=1

?

?

1,

t < ¦Ó (¦Ø),

where Z (i) (t, ¦Ó (¦Ø), ¦Ø) denotes the number of individuals time t in the

i-th independent identically distributed subprocess started by the a

single ancestor born at time ¦Ó (¦Ø)

Marek Kimmel

Sept/Oct 2004

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