On Scaling Up from Individual-Based Processes to Macroscopic Ecological Dynamics in Spatially-Extended Communities

How should ecologists scale up from the microscopic events affecting individuals, to the macroscopic processes affecting populations and communities? This question is becoming important in theoretical ecology due to the increasing use of individual-based models of spatially-extended populations and communities. We give here a dynamical system, derived from an individual-based stochastic process, that describes the principal features of such a stochastic process. The stochastic process models a multispecies community of organisms living in a spatial domain, containing organisms that (1) give birth and die with probabilistic rates which depend on other individuals in a specified neighborhood, and (2) move from one location to another. The dynamical system describes the change in the first and second spatial moments of the stochastic process, the first moments being the densities of species averaged over space, and the second moments being measures of the average spatial structure of the community in the vicinity of an individual. We show, by means of an example of two competing plant species, that the dynamics given by a simpler non-spatial model are qualitatively incorrect, whereas the dynamical system presented here gives a close approximation to the first and second moments of the underlying stochastic process.