On the accuracy of a diffusion approximation to a discrete state–space Markovian model of a population

The traditional Kolmogorov equations treat the size of a population as a discrete random variable. A model is introduced that extends these equations to incorporate environmental variability. Difficulties with this discrete model motivate approximating the population size as a continuous random variable through the use of diffusion processes. The set of cumulants for both the population size and the environmental factors affecting the population size characterize the population–environmental system. The evolution of this set, as predicted by the diffusion approximation, closely matches the corresponding predictions for the discrete model. It is also noted that the simulation estimates of the cumulants against which the predictions of the diffusion model are checked can vary considerably between simulations — despite averaging over a large number of simulation runs. The precision of the simulation estimates–both over time and with differing cumulant order–is discussed.