Solving the stochastic dynamics of population growth

Abstract

Population growth is a fundamental process in ecology, evolution, and epidemiology. The population size dynamics during growth are often described by deterministic equations derived from kinetic models. Here, we simulate several population growth models and compare the size averaged over many stochastic realizations with the deterministic predictions. We show that these deterministic equations are generically bad predictors of the average stochastic population dynamics. Specifically, deterministic predictions overestimate the simulated population sizes, especially those of populations starting with a small number of individuals. Describing population growth as a stochastic birth process, we prove that the discrepancy between deterministic predictions and simulated data is due to unclosed-moment dynamics. In other words, the deterministic approach does not take into account the variability of birth times, which is particularly important at small population sizes. We evaluate different moment-closure approximations and show that they do not satisfactorily reduce the error between analytical predictions and simulated data. We present two novel solutions to the stochastic growth dynamics, one of which applies to any population growth model. We show that our solution exactly quantifies the dynamics of a community composed of different strains and correctly predicts the fixation probability of a strain in a serial dilution experiment. Our work sets the foundations for a more faithful modeling of community dynamics. It provides tools for a more accurate analysis of experimental results, including the inference of important growth parameters.