This chapter is a slightly more gentle introduction to moment dynamics and the importance of considering local interactions for plant populations and communities

Basic application of spatial moment closure (power-1) to dynamics of a simple epidemic. Considers the covariance dynamics in a Poisson-distributed and aggregated host populations, and looks briefly at epidemics with removal/recovery.

Spatial moment equations (power-1 closure) for the spatial logistic model, one-species competition. Analytical methods for solving for equilibrium are presented (but these methods are somewhat clumsy, and are superseded by those in Bolker and Pacala 1999). Predicts when equilibrium population patterns will be even vs. aggregated.

Analyzes the spatial Lotka-Volterra model: two-species competition in a point process model. Decomposes spatial covariances into a series of terms that affect competitive invasion, attributing different terms to competition-colonization tradeoffs, successional niches, or phalanx growth. Uses power-1 closure and Bessel-function competition and dispersal kernels to get analytically tractable invasion criteria. (However, power-1 closure means that invasion criteria only act sensibly when the invader would lose in the non-spatial case.)

non-spatial moment closure for the dynamics of the distribution of host susceptibility. Incorporates a particularly clever scaled closure that interpolates between invasion-phase and equilibrium-phase statistics

A hybrid approximation method that elaborates on the common pair approximation is proposed and shown to give very accurate predictions for the order parameter (epidemic size) over the whole parameter-space, including the critical region.

Here we review and report recent progress on closure approximations applicable to lattice models with nearest-neighbour interactions, including cluster approximations and elaborations on the pair (or pairwise) approximation. (SIS model) A hybrid pairwise approximation is shown to provide the best predictions of transient and long-term, stationary behaviour over the whole parameter range of the model.

A class of simple spatio-temporal stochastic models for the spread and control of plant disease is investigated. We consider a lattice-based susceptible-infected model in which the infection of a host occurs through two distinct processes: a background infective challenge representing primary infection from external sources, and a short-range :interaction representing the secondary infection of susceptibles by infectives within the population. Recent data-modelling studies have suggested that the above model may describe the spread of aphid-borne virus diseases in orchards. In addition, we extend the model to represent the effects of different control strategies involving replantation (or recovery). The Contact Process is a particular case of this model. The behaviour of the model has been studied using cellular-automata simulations. An alternative approach is to formulate a set of deterministic differential equations that captures the essential dynamics of the stochastic system. Approximate solutions to this set of equations, describing the time evolution over the whole parameter range, have been obtained using the pairwise approximation (PA) as well as the most commonly used mean-field approximation (MF). Comparison with simulation results shows that PA is significantly superior to MF, predicting accurately both transient and long-run, stationary behaviour over relevant parts of the parameter space. The conditions for the validity of the approximations to the present model and extensions thereof are discussed.

This paper uses a moment approximation to an individual-based model of animals moving through a heterogeneous landscape taken from a satellite image of the UK. Note that the moment closure used here works only when the first moments (population densities) have no dynamics; i.e. when only the spatial structure of the population changes.

Uses the pairwise approach to look at the effects of long range (global) dispersal and short range dispersal on the evolutionary dynamics of infectious diseases

The contact process is used as a simple spatial model in many disciplines, yet because of the buildup of spatial correlations, its dynamics remain difficult to capture analytically. We introduce an empirically based, approximate method of characterizing the spatial correlations with only a single adjustable parameter. This approximation allows us to recast the contact process in terms of a stochastic birth-death process, converting a spatiotemporal problem into a simpler temporal one. We obtain considerably more accurate predictions of equilibrium population than those given by pair approximations, as well as good predictions of population variance and first passage time distributions to a given (low) threshold. A similar approach is applicable to any model with a combination of global and nearest-neighbor interactions.