The 'interacting-particle' framework allows the inclusion of local interactions in individual-based competition models with a discrete time and space representation. An interacting particle system is a stochastic Markov process in which patches are located on a bidimensional integer lattice. The dynamics of the populations are described by rate functions that indicate how a patch changes its state based on the current states of patches in a local neighborhood. These local rules may have for example the same components as the classical two-species Lotka-Volterra competition model. Six additional functions are needed in the spatial version of the Lotka-Volterra model to deal fully with local interactions and local dispersal, and each function has at least one parameter. To make mathematical analysis tractable, the first model presented by C. Neuhauser and S. W. Pacala in 1999 assumed symmetries in the spatial scales of dispersal and interactions. This simplification leads to the surprising conclusion that segregation of species in space does not promote coexistence: local competitive interactions between species may result in a reduction of the parameter region where coexistence or founder control occurs in the classical model, with spatial segregation of the two species in parts of the parameter region where the classical model predicts coexistence.
However, the assumption of symmetrical interactions leaves parts of the parameter space uninvestigated. In plant communities, dynamics depend very much on the neighborhoods over which propagules are dispersed, and on the neighborhoods over which plants compete. Remarkably little is known about the spatial competition kernels that determine the growth of competing plants. This calls for formal mathematical models that link the microscopic processes with the macroscopic dynamics.
A 'moment-based model' is a deterministic approximation of stochastic processes in a continuous time and continuous space representation, based on 'spatial moment equations'. The method involves the first two spatial moments of an individual-based model. The first spatial moments are simply the population abundances of the species averaged over space, and the second moments measure the variability of abundances over space, that is, the spatial auto-covariance within a single species and the spatial cross-covariance among species. By incorporating the second spatial moments, the model tracks important aspects of the community's spatial structure and couples this structure to the dynamics of mean abundance, thereby allowing a feedback between the two. Such coupling is crucial in plant communities: the fate of individuals is affected by local spatial structure, and this in turn causes the spatial structure to change. Such moment-based models have been recently applied to characterize spatial strategies, such as rapid colonization, rapid resource capture, or resistance to competition.
Using this moment-approximation method, M. D. Murrell and R. Law presented in 2003 another spatial version of the Lotka-Volterra competition equations to investigate effects of removing symmetries in the distances over which individuals disperse and compete. It is thus possible to vary the distances over which interactions between and within species occur.
Some spatial segregation of the species always occurs due to competition, and such endogenous spatial pattern formation does not necessarily lead to coexistence. But whenever interspecific competition occurs over shorter distances than intraspecific competition, spatial segregation becomes strong enough to enable weaker competitors to coexist with their stronger rivals. This asymmetry in interaction distances, called 'heteromyopia', reduces the strength ofinterspecific competition sufficiently to permit either species to invade the other.
Such coexistence is most likely when the species have similar dynamics, in contrast to the competition-colonization tradeoff derived from the multispecies metapopulation model that requires large competitive differences between species. Heteromyopia, by involving spatial aggregation of conspecifics and segregation of heterospecifics, appears to be an important feature of plant competition in determining community structure and dynamics. From regression analyses of tree competition in tropical rain forests and few experiments with artificial communities of annual plant species, there is indeed empirical evidence that weak competitors show increased biomass and seed production within neighborhoods of conspecifics, while stronger competitors show increased biomass and seed production within neighborhoods of heterospecifics.
However, more theoretical and experimental work is needed to assess the importance of neighborhood competition in plant communities made of clonal species - for which the concept of individual is not applicable - and in temporally heterogeneous environments.