Dynamical Systems: Modelling, Analysis, and Numerics

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This proposal is to study dynamical models of biological and physical processes and especially the relationship of experimental data with both qualitative and quantitative features of these models. Energy balance models of climate are an example. For these models to be useful, their parameters must be calibrated to physical measurements in order for the model to replicate observed phenomena. Here the data informs the model. In the other direction, an analysis of the model may reveal sudden transitions between different model states as a result of small changes in parameters. Here the model predicts the reality. Preliminary results have shown that the Arctic climate may have such a bifurcation and that, without significant CO2 reduction, may switch, irreversibly, from the present cold climate to a temperate climate as existed there millions of years ago. Enhancing the energy balance climate model, understanding its predictions, and ascertaining their validity is part of present proposal. Other examples are models of biological processes. In a collaboration with a pathobiology lab that studies viral treatments for cancer, this proposal will develop mathematical models of the immune response to viral treatment, including cytokine production and infection dynamics. It will use data from the lab and help provide insight into the fundamental processes involved in the body's response to viral treatments. Other biological phenomena will also be modelled and studied. Modelling of physical and biological processes prompts questions about fitting models to experimental data and the numerical algorithms employed to do so. Many biological models involve parameters for which little or no prior information is available. If the number of parameters is large or the amount of available data is small, fitting the model can be a very difficult task. This proposal also aims to expand on the tools available for model fitting by continuing to develop a parameter range reduction scheme that utilizes monotonicity properties of the model and numerical discretizations of the the model equations. The method is especially useful as a pre-conditioner for more traditional point estimation schemes when the initial parameter ranges are large, since it identifies a much smaller region in parameter space where the optimal estimate must be located, and it also provides a good estimate of this optimal value from which the traditional scheme will most likely converge rapidly to the optimal value.