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
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
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
- 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)