Open Access 2012 | OriginalPaper | Buchkapitel
A Non-Markovian Model for Cell Population Growth: Tail Behavior and Duration of the Growth Process
verfasst von : Mathisca C. M. de Gunst, Willem R. van Zwet
Erschienen in: Selected Works of Willem van Zwet
Verlag: Springer New York
De Gunst has formulated a stochastic model for the growth of a certain type of plant cell population that initially consists of
n
cells. The total cell number
N
n
(
t
) as predicted by the model is a non-Markovian counting process. The relative growth of the population,
n
−1
(
N
n
(
t
) - n), converges almost surely uniformly to a nonrandom function
X.
In the present paper we investigate the behavior of the limit process X(t) as
t
tends to infinity and determine the order of magnitude of the duration of the process
N
n
(
t
). There are two possible causes for the process
N
n
to stop growing, and correspondingly, the limit process
X
(
t
) has a derivative
X’
(
t
) that is the product of two factors, one or both of which may tend to zero as
t
tends to infinity. It turns out that there is a remarkable discontinuity in the tail behavior of the processes. We find that if only one factor of
X’
(
t
) tends to zero, then the rate at which the limit process reaches its final limit is much faster and the order of magnitude of the duration of the process
N
n
is much smaller than when both occur approximately at the same time.