Mathematical Modelling of Metapopulation Dynamics: Revisiting its Meaning

In this paper, we revisit the metapopulation dynamics model of typical Levins type, and reconsider its mathematical modeling. For the metapopulation dynamics with three states for the patch of a habitat composed of a number patches available for the reproduction, 'vacant', 'small' (i.e., threatened to the extinction) and 'large' (i.e., far from the extinction risk) in terms of population size in the patch, we reconstruct the mathematical model in a general form, making use of the difference in time scale between the state transition and the dispersal of individuals within the patchy habitat. The typical Levins type of metapopulation dynamics model appears only for a specific case with some additional assumptions for mathematical simplification. Especially we discuss the rationality of mass-action terms for the patch state transition in the Levins model, and find that such mass-action term could be rational for the modeling of metapopulation dynamics only in some ideal condition.