On dispersal and population growth for multistate matrix models

To describe the dynamics of stage-structured populations with m stages living in n patches, we consider matrix models of the form SD, where S is a block diagonal matrix with n× n column substochastic matrices S1, . . . , Sm along the diagonal and D is a block matrix whose blocks aren× n nonnegative diagonal matrices. The matrix S describes movement between patches and the matrix D describes growth and reproduction within the patches. Consider the multiple arc directed graph G consisting of the directed graphs corresponding to the matrices S1, . . . , Sm, where each directed graph is drawn in a different color. We say G has a polychromatic cycle if G has a directed cycle that includes arcs of more than one color. We prove that ρ(SD) ρ(D) for all block matrices D with nonnegative diagonal blocks if and only if G has no polychromatic cycle. Applications to ecological models are presented. © 2006 Elsevier Inc. All rights reserved. AMS classification: 15A51; 15A18; 92D25