## Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 3, Number 4 (1993), 1112-1144.

### A Non-Markovian Model for Cell Population Growth: Tail Behavior and Duration of the Growth Process

Mathisca C. M. de Gunst and Willem R. van Zwet

#### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 3, Number 4 (1993), 1112-1144.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005275

**Digital Object Identifier**

doi:10.1214/aoap/1177005275

**Mathematical Reviews number (MathSciNet)**

MR1241037

**Zentralblatt MATH identifier**

0784.60051

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G55: Point processes

Secondary: 60F99: None of the above, but in this section 62P10: Applications to biology and medical sciences

**Keywords**

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

#### Citation

de Gunst, Mathisca C. M.; van Zwet, Willem R. A Non-Markovian Model for Cell Population Growth: Tail Behavior and Duration of the Growth Process. Ann. Appl. Probab. 3 (1993), no. 4, 1112--1144. doi:10.1214/aoap/1177005275. https://projecteuclid.org/euclid.aoap/1177005275