Asymptotic state lumping in transport and diffusion problems on networks with applications to population problems

In this paper we consider a general macro-model describing a metapopulation consisting of several interacting with each other subpopulations connected through a network, with the rules of interactions given by a system of ordinary differential equations. For such a model we construct two different micro-models in which each subpopulation has its own structure and dynamics. Precisely, each subpopulation occupies an edge of a graph and its dynamics is driven, respectively, by diffusion or transport along the edge. The interactions between the subpopulations are described by interface conditions at the nodes which the edges are incident to. We prove that with an appropriate scaling, roughly speaking with, respectively, fast diffusion or fast transport combined with slow exchange at the nodes, the solutions of the micro-models can be approximated by the solution to the macro-model.