Diffusion Approximations for Ecological Models

Diffusion models are widely used in ecology, and in more general population biology contexts, for predicting population-size distributions and extinction times. They are often used because they are particularly simple to analyse and give rise to explicit formulae for most of the quantities of interest. However, whilst diffusion models are ubiquitous in the literature on population models, their use is frequently inappropriate and often leads to inaccurate predictions of critical quantities such as persistence times. This paper examines diffusion models in the context in which they most naturally arise: as approximations to discrete-state Markovian models, which themselves are often more appropriate in describing the behaviour of the populations in question, yet are difficult to analyse from both an analytical and a computational point of view. We will identify a class of Markovian models (called asymptotically density dependent models) that permit a diffusion approximation through a simple limiting procedure. This procedure allows us to immediately identify the most appropriate approximating diffusion and to decide whether the diffusion approximation, and hence a diffusion model, is appropriate for describing the population in question. This will be made possible through the remarkable work of Tom Kurtz and Andrew Barbour, which is frequently cited in the applied probability literature, but is apparently not widely accessible to practitioners. Their results will be presented here in a form that most easily allows their direct application to population models. We will also present results that allow one to assess the accuracy of diffusion approximations by specifying for how long and over what ranges the underlying Markovian model is faithfully approximated. We will explain why diffusion models are not generally useful for estimating extinction times, a serious shortcoming that has been identified by other authors using empirical means.