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November, 1993 A Non-Markovian Model for Cell Population Growth: Tail Behavior and Duration of the Growth Process
Mathisca C. M. de Gunst, Willem R. van Zwet
Ann. Appl. Probab. 3(4): 1112-1144 (November, 1993). DOI: 10.1214/aoap/1177005275

Abstract

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.

Citation

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Mathisca C. M. de Gunst. Willem R. van Zwet. "A Non-Markovian Model for Cell Population Growth: Tail Behavior and Duration of the Growth Process." Ann. Appl. Probab. 3 (4) 1112 - 1144, November, 1993. https://doi.org/10.1214/aoap/1177005275

Information

Published: November, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0784.60051
MathSciNet: MR1241037
Digital Object Identifier: 10.1214/aoap/1177005275

Subjects:
Primary: 60G55
Secondary: 60F99 , 62P10

Keywords: duration , non-Markovian counting process , population growth , Stochastic model , tail behavior

Rights: Copyright © 1993 Institute of Mathematical Statistics

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Vol.3 • No. 4 • November, 1993
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