ABSTRACT
A power law relationship between the mean and variance of ecological time series has been shown to hold for a vast number of species. Here we examine the behaviour of single-species stochastic models and concentrate in particular on the mean-variance relationship as the carrying capacity becomes large. Single-species stochastic models can be written as Markov chains, and the long-term distribution of population sizes and power-law scaling can be found analytically. Finally we extend our analysis to consider moment closure models, spatial heterogeneity and multi-species models.