Toward a “modern coexistence theory” for the discrete and spatial

The usual theoretical condition for coexistence is that each species in a community can increase when it is rare (mutual invasibility). Traditional coexistence theory implicitly assumes that the invading species is common enough that we can ignore demographic stochasticity but rare enough that it does not compete with itself, even after it has reached a stationary spatial distribution. However, short-distance dispersal of discrete individuals leads to locally dense population clusters, and existing theory breaks down. We have an intuition that when we account for invader–invader competition, shorter-range dispersal should reduce the invader's ability to escape competition, but exactly how does this translate into lower population growth? And how will invader discreteness affect outcomes? We need a way of partitioning the contributions to coexistence, but current modern coexistence theory (MCT) does not apply under these conditions. Here we present a computationally based partitioning method to quantify the contributions to coexistence from different mechanisms, as in MCT. We also build up an intuition for how invader clumping and discreteness will affect these contributions by analyzing a case study, a lattice-based spatial lottery model. We first consider fluctuation-dependent coexistence, partitioning the contributions of variable environment, variable competition, demographic stochasticity, and their correlations and interactions. Our second example examines fluctuation-independent coexistence maintained by a fecundity–survival trade-off, and partitions the contributions to coexistence from interspecific differences in fecundity, in mortality, and in dispersal. We find that demographic stochasticity harms an invader, but only slightly. Localized invader dispersal, on the other hand, can have a strong effect. When invaders are more clumped, they compete with each other more intensely when rare, so they too become limited by environment-competition covariance. More invader clumping also means that variation in competition changes from helping the invader to harming it. More broadly, invader clumping is likely to weaken any coexistence mechanism that relies on the invader escaping competition from the resident, because invader clumping means that the resident is no longer the only source of competition.