On moment closures for population dynamics in continuous space

A first-order moment closure, the mean-field assumption that organisms encounter one another in proportion to their spatial average densities, lies at the heart of much theoretical ecology. This assumption ignores all spatial information and, at the very least, needs to be replaced by a second-order closure to gain understanding of ecological dynamics in spatially structured populations. We describe a number of conditions that a second-order closure should satisfy and use these conditions to evaluate some closures currently available in the literature. Two conditions are particularly helpful in discriminating among the alternatives: that the closure should be positive, and that the dynamics should be unaltered when identical individuals are given different labels. On this basis, a class of closures we refer to as 'power-2' turns out to provide a good compromise between positivity and dynamical invariance under relabelling.