DISCRETE AND CONTINUOUS TIME POPULATION MODELS, A COMPARISON CONCERNING PROLIFERATION BY FISSION

Abstract

We present two approaches, discrete time and continuous time models, for individuals which propagate through binary fission. The volumes of the two daughters are a fixed part of that of the mother, not necessarily the half, and their growth rates may differ. The discrete time approach gives more insight into the results obtained with the continuous time model. We define classes in the continuous time model such that the total number of individuals in these classes at specific moments in time is equal to the unknown number in a discrete time model. Then the discrete time model is homologous to the continuous one in the sense of having the same solutions at specific moments. Population matrix theory applies when the ratio of the inter-division times of the two daughters is rational. There is inter-class convergence but no intra-class convergence. The latter feature implies that there is no convergence of the size distribution in the continuous time model either. When the ratio is irrational the continuous time model holds and there is convergence but the rate of convergence can become infinitesimally small. This phenomenon is linked with quasi-periodicity on a 2-dimensional torus.