Moment Methods for Ecological Processes in Continuous Space

Introduction Spatial dynamics of populations have long been of interest to ecologists (Skellam 1951; Levin and Paine 1974; Andow et al. 1990), but recent advances in data collection and in computational power have put these concepts within the reach of many ecologists for the first time. Computational models (Wilson et al. 1993, 1995b; McCauley et al. 1993; Pacala and Deutschman 1995) suggest important and previously unexplored effects of space and discrete individuals on population dynamics. Analytic approaches that capture these effects are emerging, building on methods developed in other contexts. This chapter presents a general method for deriving approximate equations for spatial dynamics in continuous space and time that has advantages over classical and many modern approaches. We are interested in spatial pattern formation in plant communities and in the effects of pattern on plant competition. Our goal is to find general methods for exploring this problem that are analytically tractable, so that we can gain insight into qualitative behaviors of the system and analyze how they depend on the parameters; sufficiently general, so that some of the same tools can be applied to answer a range of different questions about spatial dynamics in ecology; and close enough to the characteristics of real populations that we can eventually fit the models to field data on individual behavior. We focus on spatial point processes (Diggle 1983; Gandhi et al. 1998), continuous-time dynamical systems for discrete individuals interacting in a continuous habitat.