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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
Open Access
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

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 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Mathisca C. M. de Gunst
    • 1
    • 2
    • 3
    • 4
    • 5
    • 6
  • Willem R. van Zwet
    • 1
    • 2
    • 3
    • 4
    • 5
    • 6
  1. 1.Free University of AmsterdamAmsterdamThe Netherlands
  2. 2.University of LeidenLeidenThe Netherlands
  3. 3.University of North CarolinaNorth CarolinaUSA
  4. 4.Department of MathematicsFree UniversityAmsterdamThe Netherlands
  5. 5.Department of MathematicsUniversity of LeidenLeidenThe Netherlands
  6. 6.Department of StatisticsUniversity of North CarolinaChapel HillUSA

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