Spatial point processes and moment dynamics in the life sciences: a parsimonious derivation and some extensions

Mathematical models of dynamical systems in the life sciences typically assume that biological systems are spatially well mixed (the mean-field assumption). Even spatially explicit differential equation models typically make a local mean-field assumption. In effect, the assumption is that diffusive movement is strong enough to destroy spatial structure or that interactions between individuals are sufficiently long-range that the effects of spatial structure are weak. However, many important biophysical processes, such as chemical reactions of biomolecules within cells, disease transmission among humans, and dispersal of plants, have characteristic spatial scales that can generate strong spatial structure at the scale of individuals, with important effects on the behaviour of biological systems. This calls for mathematical methods that incorporate spatial structure. Here, we focus on one method, spatial-moment dynamics, which is based on the idea that important information about a spatial point process is held in its low-order spatial moments. The method goes beyond the dynamics of the first moment, i.e. the mean density or concentration of agents in space, in which no information about spatial structure is retained. By including the dynamics of at least the second moment, the method retains some information about spatial structure. Whereas mean-field models effectively use a closure assumption for the second moment, spatial-moment models use a closure assumption for the third (or a higher-order) moment. The aim of the paper was to provide a parsimonious and intuitive derivation of spatial-moment dynamic equations that is accessible to non-specialists. The derivation builds naturally from the first moment to the second, and we show how it can be extended to higher-order moments. Rather than tying the model to a specific biological example, we formulate a general model of movement, birth, and death of multiple types of interacting agents. This model can be applied to problems from a range of disciplines, some of which we discuss. The derivation is performed in a spatially non-homogeneous setting, to facilitate future investigations of biological scenarios, such as invasions, in which the spatial patterns are non-stationary over space.