Title: Compartment models with memory
Abstract: The foundation for very many computer simulations (e.g., of solute transport, global carbon cycling, epidemic diseases like Covid-19, etc.) is called “compartment modeling” because it starts with a conceptual model of “Boxes” or Compartments, between which stuff moves. In the case of diffusion between biofilms and a bulk aqueous phase the compartments are, well, the biofilm and the bulk aqueous phase, and perhaps a boundary layer. In global cycling of carbon the very simplest models involve oceanic, air, biotic, and subsurface compartments and carbon moves among these four places at agreed-upon rates of exchange. Unfortunately the state-of-the-mathematics of “compartment modeling” (systems of ordinary differential equations, ODEs) does not allow agreed-upon rates of exchange, but requires one embarrassingly simple and often inadequate type of exchange. Here we give a generalization of compartment modeling to allow essentially any practical exchange models to be included, by installing mathematical chronometers of residence time in compartment and writing the exchange rates as functions of that residence time. In math terms, the ODEs are generalized to partial differential equations that are readily converted to Volterra integral equations. Applications involving biofilms, global carbon cycling, Covid disease dynamics, and solute transport in river corridors will be touched on at varying depths.
Bio: From rural Northern Virginia Tim Ginn attended University of Virginia in Classics and Environmental Science, then obtained graduate degrees in Civil Engineering at Purdue and almost a pilot’s license. He migrated to Washington to take a position at Battelle Pacific NW Lab 1989-1996, before moving to a faculty position at UC Davis 1996-2015. Happily joining WSU Civil and Environmental Engineering in January 2016, he is now Interim Associate Chair and Boeing Distinguished Professor in Environmental Engineering. He develops mathematical models for analyses of data in reactive transport occurring mainly in water.
