Gompertzian population growth under some deterministic and stochastic jump schedules

Abstract

Many cell populations, exemplified by certain tumors, grow approximately according to a Gompertzian growth model which has a slower approach to an upper limit than that of logistic growth. Certain populations of animals and other organisms have also recently been analyzed with the Gompertz model. This article addresses the question of how long it takes to reduce the population from one level to a lower one under a schedule of sudden decrements, each of which removes a given fraction of the cell mass or population. A deterministic periodic schedule is first examined and yields exact results for the eradication or extinction time which is defined as that required to reduce the number of cells to less than unity. The decrements in cell mass at each hit could correspond to an approximation to reduction of a tumor by external beam radiation therapy. The effects of variations in magnitude of successive decrements, the time interval between them, the initial population size and the intrinsic growth rate are calculated and results presented graphically. With a schedule governed by a Poisson process, the number of organisms or cells satisfies a stochastic differential equation whose solution sample paths have downward jumps as random times. The moments of the exit time then satisfy a system of recurrent differential-difference equations. A simple transformation results in a simpler system which has been studied both analytically and numerically in the context of interspike intervals of a model neuron. Results are presented for the mean eradication time for various frequencies and magnitudes of hits and for various eventual and initial population sizes. The standard deviation of the eradication time is also investigated.