Multiplicative moments and measures of persistence in ecology

Ecologists and epidemiologists have begun focusing on demographic stochasticity and spatial heterogeneity as important biological factors. With high-powered computers simulation of such systems is a common modelling technique; however we lack a detailed understanding of the processes involved. Moment closure approximations provide a simple method which can be used to capture the main features of a wide variety of stochastic models and to gain a more intuitive understanding. In this paper we give an alternative variation based on multiplicative moments which is equivalent to taking a novel third-order cumulant approximation. The differential equations for these multiplicative moments are far more robust than their additive counterparts. We use this technique to consider the behaviour and persistence of finite metapopulations for two common ecological systems.