Population and replicate variability in an exponential growth model

We have studied variability and predictability of population behaviour in a simple model of exponential growth. Population variability is related to uncertainty of prediction for the dynamics conditioned upon the initial state only. We contrasted it with replicate variability, defined in terms of short-term predictability along a single realisation of a stochastic process. We show that for exponential growth, the population variance increases proportionally to the square of the current population size, whereas the replicate variance is a linear function of the population size. Thus, for large population sizes, the relative predictability for a single population is much better than for an ensemble of realisations. This stands in contrast with the behaviour of a simple stochastic process (Ornstein-Uhlenbeck process), where the population and the replicate variances have similar behaviour. The results have profound consequences for parameter estimation and prediction for many stochastic population models based on the exponential formula.