Simple stochastic models and their power-law type behaviour

A power-law relationship between the mean and variance of ecological time s eries has been shown to hold for a vast number of species. Here we examine the behaviour of single-species stochastic models and concentrate in partic ular on the mean-variance relationship as the carrying capacity becomes lar ge. Single-species stochastic models can be written as Markov chains, and t he long-term distribution of population sizes and hence power-law scaling c an be found analytically. The,various power-law scalings that arise have ve ry different biological implications for the effects of stochasticity and t he departure from the deterministic paradigm. Finally we extend our analysi s to consider the complicating factors of spatial heterogeneity, nontrivial deterministic dynamics, and multispecies models. (C) 2000 Academic Press.