# Spatial moment equations

Spatial moment equations, also referred to as pair approximations or correlation equations, are a relatively new method in mathematical biology for understanding the behavior of groups of individuals evolving and interacting in a spatial arena. These models use a variety of methods to derive coupled equations for both the mean densities and the spatial pattern of single or multiple interacting populations; the spatial pattern is described in terms of the probabilities of two individuals inhabiting spatial locations a particular distance apart. These probabilities can be expressed in different ways, as correlations or as conditional probabilities or as covariances.

Moment equations can be applied to population dynamic models on discrete regular (usually square) lattices, in which case they are typically called pair approximation equations, on discrete regular or irregular networks, or to individuals located at points in a continuous spatial arena.

Moment equations have long been used in physics, but entered the ecological literature as recently as 1988. They are now used by a variety of researchers in mathematical ecology, epidemiology, and evolution as a way of approximating complex stochastic individual-based models in a way that simplifies calculation and gives analytical insights. At a recent meeting at the Isaac Newton Institute a group of those researchers met to discuss the future of these equations.

This page is intended to be a repository for information, references, and code on pair approximations on moment equations. I hope you find it useful; constructive feedback is greatly appreciated.

• a bibliography of references on spatial moment equations. (I hope to annotate this bibliography somewhat in the near future. If you have annotations or extra references to provide, please let me know.)
• a list of people working on moment equations: interests, contact information, and web links
• TBP (to be posted): auxiliary information
• electric monk: Mathematica code etc. for deriving point-process moment equations
• slides (PDF, 1.24M) from a recent seminar given at a joint math/biology seminar at UMass Boston (19 April 2002)
• moment equation derivations and discussion

Maintained by Ben Bolker. Last updated: 22 Feb 2002 (contributions from David Murrell)