Stochastic modelling of environmental variation for biological populations

We examine stochastic effects, in particular environmental variability, in population models of biological systems. Some simple models of environmenta l stochasticity are suggested, and we demonstrate a number of analytic appr oximations and simulation-based approaches that can usefully be applied to them. Initially, these techniques, including moment-closure approximations and local linearization, are explored in the context of a simple and relati vely tractable process. Our presentation seeks to introduce these technique s to a broad-based audience of applied modellers. Therefore, as a test case , we study a natural stochastic formulation of a non-linear deterministic m odel for nematode infections in ruminants, proposed by Roberts and Grenfell (1991). This system is particularly suitable for our purposes, since it ca ptures the essence of more complicated formulations of parasite demography and herd immunity found in the literature. We explore two modes of behaviou r. In the endemic regime the stochastic dynamic fluctuates widely around th e non-zero fixed points of the deterministic model. Enhancement of these fl uctuations in the presence of environmental stochasticity can lead to extin ction events. Using a simple model of environmental fluctuations we show th at the magnitude of this system response reflects not only the variance of environmental noise, but also its autocorrelation structure. In the managed regime host-replacement is modelled via periodic perturbation of the popul ation variables. In the absence of environmental variation stochastic effec ts are negligible, and we examine the system response to a realistic enviro nmental perturbation based on the effect of micro-climatic fluctuations on the contact rate. The resultant stochastic effects and the relevance of ana lytic approximations based on simple models of environmental stochasticity are discussed. (C) 2000 Academic Press.