Dispersal-induced growth or decay in a time-periodic environment. The case of reducible migration matrices

This paper is a follow-up to a previous work where we considered populations with time-varying growth rates living in patches and irreducible migration matrix between the patches. Each population, when isolated, would become extinct. Dispersal-induced growth (DIG) occurs when the populations are able to persist and grow exponentially when dispersal among the populations is present. In this paper, we consider the situation where the migration matrix is not necessarily irreducible. We provide a mathematical analysis of the DIG phenomenon, in the context of a deterministic model with periodic variation of growth rates and migration. Our results apply in the case, important for applications, where there is migration in one direction in one season and in the other direction in another season. We also consider dispersal-induced decay (DID), where each population, when isolated, grows exponentially, while populations die out when dispersal between populations is present.