Generalizing the Levins metapopulation model onto time varying colonization and extinction rates

The metapopulation theory explores the population persistence in fragmented habitats by considering the dynamics between the extinction of local populations and the recolonization of empty sites. The novelty of this paper is to assume that the parameters model are subject to deterministic variations whereas; at the same time; we propose theoretical tools to study it. We realize that the approach proposed by C. Puccia and R. Levins can be adapted to construct the upper and lower averages of the difference between the extinction and colonization rates, whose sign is useful to determine either the permanence or the extinction of the metapopulation. In fact, these averages can be seen as Bohl exponents which allow us to revisit the classical metapopulation model introduced by R. Levins. In fact, compared with the time invariant Levins model, the resulting dynamics of our time varying generalization shares the persistence/extinction scenario when the upper and lower averages have the same sign but also raises open and intriguing questions about persistence in the case of the averages have different signs.