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

Abstract

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.

Waves and diffusion on metric graphs with general vertex conditions

“…Matrix mixed conditions. Motivated by applications in population dynamics, in [7,8] a diffusion problem on a compact network with the general boundary condition…”

A singular limit for an age structured mutation problem

“…Let us fix ω > v max B . Using (8) and (14), we see that there is a constant L, independent of ǫ such that…”

Some transport and diffusion processes on networks and their graph realizability

“…As with the transport problem, there is no mathematical reason why the matrices K ω , ω ∈ Ω = {00, 01, 10, 11}, in (3.10) should be restricted to matrices given by (3.9). It follows that the wellposedness of (3.10) can be studied by the same methods for any real m × m matrices K ω [11]. Then we arrive at the same problem as for (2.2)-under what conditions on K ω general system (3.10) is a diffusion system on a graph.…”

“…In fact, B can be an arbitrary nonnegative matrix and thus, as we shall see later, despite formal similarity with (2.1), in general (2.3) does not describe any transport process on a graph. Well-posedness of (2.2) in a general context (K even need not to be positive) is studied in [11] by methods developed in [5]. Here we address a natural question: under what conditions system (2.2) with general matrix K describes a graph transport model; that is, such that the exchange between subgroups only can occur if these are 'physically' connected by a node.…”