Selected Works of Willem van Zwet pp 315-347 | Cite as

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

## 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.

### Key words and phrases

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

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