FRG: Fluctuation Effects in Near-Continuum Descriptions of Discrete Dynamical Systems in Physics, Chemistry and Biology

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Abstract This Focused Research Group brings together researchers from the University of Michigan's Departments of Mathematics, Physics and Chemical Engineering to address important problems of modeling, simulation and analysis for dynamical processes where underlying discreteness plays a non-negligible role in large scale descriptions via deterministic continuum systems (generally systems of ordinary and partial differential equations). This Focused Research Group combines the investigators' expertise in theory, modeling, analysis and scientific computation to study a suite of problems from materials physics, chemical kinetics and the life sciences to elucidate the fundamental scientific issues and develop appropriate quantitative tools to analyze them. The specific problems to be studied are: (1) Mesoscopic mathematical models of wound healing with cell proliferation and migration, and including the biologically important effect of cell-cell adhesion; (2) The application of new and improved simulation techniques, direct solutions of the Becker-Doering equations, and simulation and analysis of stochastic models to investigate the role of microscopic correlations in Ostwald ripening; (3) The development ofanalytic asymptotic methods for accurate reduced descriptions of slow stochastic variables properly incorporating residual fluctuation effects with applications to (bio)chemical reaction networks possessing a wide spectrum of reaction rates; (4) An extension of modeling, analysis and simulation methods developed for simple systems to increasingly complex stochastic models in population biology and epidemiology including epidemics in structured populations and extinction of competing species; (5) Spatial inhomogeneities and reaction-rate variations in the stochastic Fisher-Kolmogorov equation, a fundamental paradigm of front propagation and pattern formation. Results from this project will lead to the development of effective mathematical descriptions and efficient computational schemes for problems of increasing importance for small-scale physical and chemical processes in materials science and nano-technology, and for quantitative modeling in the life sciences. With regard to the even broader impact of this project, it contributes to the development of the scientific workforce by providing advanced training for postdoctoral researchers and doctoral students in the natural, engineering and applied mathematical sciences.