General theory of competitive coexistence in spatially-varying environments.

A general model of competitive and apparent competitive interactions in a spatially-variable environment is developed and analyzed to extend findings on coexistence in a temporally-variable environment to the spatial case and to elucidate new principles. In particular, coexistence mechanisms are divided into variation-dependent and variation-independent mechanisms with variation-dependent mechanisms including spatial generalizations of relative nonlinearity and the storage effect. Although directly analogous to the corresponding temporal mechanisms, these spatial mechanisms involve different life history traits which suggest that the spatial storage effect should arise more commonly than the temporal storage effect and spatial relative nonlinearity should arise less commonly than temporal relative nonlinearity. Additional mechanisms occur in the spatial case due to spatial covariance between the finite rate of increase of a local population and its local abundance, which has no clear temporal analogue. A limited analysis of these additional mechanisms shows that they have similar properties to the storage effect and relative nonlinearity and potentially may be considered as enlargements of the earlier mechanisms. The rate of increase of a species perturbed to low density is used to quantify coexistence. A general quadratic approximation, which is exact in some important cases, divides this rate of increase into contributions from the various mechanisms above and admits no other mechanisms, suggesting that opportunities for coexistence in a spatially-variable environment are fully characterized by these mechanisms within this general model. Three spatially-implicit models are analyzed as illustrations of the general findings and of techniques using small variance approximations. The contributions to coexistence of the various mechanisms are expressed in terms of simple interpretable formulae. These spatially-implicit models include a model of an annual plant community, a spatial multispecies version of the lottery model, and a multispecies model of an insect community competing for spatially-patchy and ephemeral food.