Simple stochastic models and their power-law type behaviour

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 hence power-law scaling can be found analytically. The various power-law scalings that arise have very different biological implications for the effects of stochasticity and the departure from the deterministic paradigm. Finally we extend our analysis to consider the complicating factors of spatial heterogeneity, nontrivial deterministic dynamics, and multispecies models.