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Spatial patterns in biological populations and the effect of spatial patterns on ecological interactions are central topics in mathematical ecology. Various approaches to modeling have been developed to enable us to understand spatial patterns ranging from plant distributions to plankton aggregation. We present a new approach to modeling spatial interactions by deriving approximations for the time evolution of the moments (mean and spatial covariance) of ensembles of distributions of organisms; the analysis is made possible by "moment closure," neglecting higher-order spatial structure in the population. We use the growth and competition of plants in an explicitly spatial environment as a starting point for exploring the properties of second-order moment equations and comparing them to realizations of spatial stochastic models. We find that for a wide range of effective neighborhood sizes (each plant interacting with several to dozens of neighbors), the mean-covariance model provides a useful and analytically tractable approximation to the stochastic spatial model, and combines useful features of stochastic models and traditional reaction-diffusion-like models. Copyright 1997 Academic Press. Copyright 1997 Academic Press

Population dynamic models are developed for communities of annual plants in spatially heterogeneous environments. These models are constructed from submodels of the survivorship, fecundity, germination, and dispersal of individual plants. The submodels include the effects of spatially local interactions on plant performance and the spatial variation in performance caused by spatial heterogeneity in the physical environment. It is possible to estimate the submodels from data on experimental communities in either the field or greenhouse and so it is possible to empirically calibrate the population dynamic models developed. Each population dynamic model explicitly includes the spatial distribution of individuals in a plant community. Several two-species models for plants in patchy environments are studied to examine the community-level consequences of spatial heterogeneity in the physical environment. The results fall into two classes. First, community structure is in part determined by a relation between patch size and mean seed dispersal distance. Specifically, coexistence is, in some cases, possible only if patches are sufficiently larger than the mean dispersal distance. Second, community structure is also affected by relations between patch size and the maximum distance over which two plants interact (termed the neighborhood radius). In some cases, coexistence is possible only if patch size is sufficiently larger than the neighborhood radius. In others, the species coexist only if patch size is sufficiently smaller than the neighborhood radius. In still other cases, coexistence is possible only if patch sizes are within critical bounds, where the sizes of the critical bounds are in units of the neighborhood radius. All results involving relations between the neighborhood radius and patch size are direct consequences of the sedentary nature of plants and the fact that individual plants interact primarily with nearby plants.

Models are developed for the dynamics of multi-species communities of annual plants that lack seed dormancy. These models explicitly include plastic plant growth, the spatial distribution of individuals, and the fact that individuals interact primarily with nearby individuals. Because the models are based on submodels of individual plants (fecundity, survivorship and dispersal, and how these are affected by inter-individual interactions), they provide explanations of community-level phenomena in terms of the biology of individuals. All model parameters and functional forms may be estimated from data obtained in simple experiments of a single years's duration. The models are used to examine the community-level consequences of some types of inter-individual interactions that have been reported in the ecological literature. In addition, the models are used to demonstrate that dispersal may markedly influence the outcome of competition among plant species, even in a physically homogeneous environment, due to an effect of dispersal on the spatial distribution of individuals.

A model of two competing species is presented in which each species is able to disperse over a single spatial axis. The spatial axis is composed of two intervals with different carrying capacities. We ask the question: If species one is alone and at population dynamic equilibrium, then when can species two successfully invade when rare? We say that an interval is “suitable” if the interval can be invaded by species two in the absence of dispersal by both species, and we say an interval is “unsuitable” if the interval cannot be invaded by species two in the absence of dispersal by both species. We offer three findings: (I) If one interval is suitable and the other is unsuitable, then the success of invasion depends upon the length of the suitable interval. Invasion succeeds if the suitable interval is larger than a threshold minimum and fails otherwise. (II) It is possible for species two to invade even though both intervals are unsuitable. (III) It is possible for species two to fail to invade even though both intervals are suitable.

The evolution of resource partitioning in a multidimensional resource space is studied for two and three competing species. Optimal patterns of resource partitioning are determined by simultaneously maximizing the fitness of each species with respect to its own niche position, conditional on the positions of all other species. We find that there are only a finite number of possible solutions, and several of these may be optimal simultaneously. Some solutions of the three-species model involve partitioning along more resource axes than any solutions of the two-species model. The results are related to empirical resource partitioning phenomena in Anolis lizard populations.