  Aims To propose and analyze a general, dynamic, process-oriented theory of
the area of distribution. Methods The area of distribution is modelled by
combining (by multiplication) three matrices: one matrix represents movements,
another niche tolerances, and a third, biotic interactions. Results are derived
from general properties of this product and from simulation of a cellular
automaton defined in terms of the matrix operations. Everything is implemented
practically in an R package. Results Results are obtained by simulation and by
mathematical analysis. We show that the mid-domain effect is a direct
consequence of dispersal; that to include movements to Ecological Niche
Modeling significantly affects results, but cannot be done without choosing an
ancestral area of distribution. We discuss ways of estimating such ancestral
areas. We show that, in our approach, movements and niche effects are mixed in
ways almost impossible to disentangle, and show this is a consequence of the
singularity of a matrix. We introduce a tool (the
Connectivity-Suitability-Dispersal plot) to extend the results of simple niche
modeling to understand the effects of dispersal. Main conclusions The
conceptually straightforward scheme we present for the area of distribution
integrates, in a mathematically sound and computationally feasible way, several
key ideas in biogeography: the geographic and environmental matrix, the
Grinnellian niche, dispersal capacity and the ancestral area of origin of
groups of species. We show that although full simulations are indispensable to
obtain the dynamics of an area of distribution, interesting results can be
derived simply by analyzing the matrices representing the dynamics.
