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

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 Nn(t) as predicted by the model is a non-Markovian counting process. The relative growth of the population, n.1(Nn(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 Nn(t). There are two possible causes for the process Nn 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 Nn is much smaller than when both occur approximately at the same time.