The use of the vec-permutation matrix in spatial matrix population models

Matrix models for a metapopulation can be formulated in two ways: in terms of the stage distribution within each spatial patch, or in terms of the spatial distribution within each stage. In either case, the entries of the projection matrix combine demographic and dispersal information in potentially complicated ways. We show how to construct such models from a simple block-diagonal formulation of the demographic and dispersal processes, using a special permutation matrix called the vec-permutation matrix. This formulation makes it easy to calculate and interpret the sensitivity and elasticity of λ to changes in stage- and patch-specific demographic and dispersal parameters.