The primary goal of this project is to interpret analytically the effects of spatial environmental heterogeneity on the community dynamics and biodiversity of ecological systems. Simulation models and experimental studies have convincingly demonstrated the importance of endogenous and exogenous spatial heterogeneity, such as variation in resource supply rate or topography, to the coexistence and the maintenance of diversity in ecological communities. However, both mathematicians and ecologists still lack analytic understanding of the mechanisms operating, particularly in fully spatial, stochastic environments. The investigators extend existing analyses of the effects of heterogeneity by using a wide range of models (from metapopulation models to interacting particle systems and point processes); they develop moment approximations, which express spatial population dynamics in terms of mean densities and covariances, to simplify the analysis while preserving both exogenous and endogenous heterogeneity. The study uses moment approximations, and more standard techniques, to analyze the effects of environmental heterogeneity, endogenous heterogeneity, and their interaction on population dynamics and competitive outcomes in ecological models of plant and marine intertidal communities. Field ecologists have long known that small-scale variability in environmental factors such as soils or temperature can affect the structure and diversity of ecological communities --- which and how many species can live together in a region. Ecological modellers, however, have often found simple explanations for small-scale spatial patterns of biodiversity, caused by interactions between plants or animals (such as competition or predation), that ignore environmental variability. Computer simulation models have recently gone a long way toward reconciling these two camps, but (1) simulations show that patterns occur, but do not always explain why; and (2) even powerful computers have limits (it could be foolhardy, for example, to attempt to simulate every tree on a continent in order to predict the effects of global climate change). This study develops simple mathematical models that combine spatial variability caused by interactions between organisms with spatial variability caused by the environment to explore how these two kinds of variability affect the structure and diversity of communities. The results from these simple models will eventually show how to build more complex and realistic models that can predict patterns of biodiversity and ecosystem productivity, and will provide a much more general understanding of how variability in the environment leads to diversity in ecological communities.
