Dynamical systems: Analysis, Modelling, and Numerics

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This research program will investigate mathematical models of biological processes in several different ways. ******Often the parameters in models of biological processes are unmeasurable or vary over wide ranges. Identifying correct values for these parameters from recorded experimental data can be a very difficult task. This research will continue the development and implementation of a numerical method for quickly eliminating large portions of the parameter ranges that are inconsistent with the data. This will allow for more accurate models that can be used for predictive purposes. ******Some biological systems exhibit a behaviour whereby they rapidly collapse from an initial state to a "steady state", and then slowly move along this steady state as elements of the system evolve. Mathematical models of such systems can be simplified if the steady state can be identified in the model and if appropriate methods for projecting the full system onto the steady state can be employed. This research will investigate models with specific constraints looking for identifiable characteristics that these constraints impose on the steady state, which will allow for easier identification of these states and reduction of the full system.******Mathematical models of ion channels are a useful tool for neurobiologists to characterize these proteins. This research will enhance the software NEUROFIT by implementing additional types of mathematical ion channel models into this software and using the parameter range reduction scheme mentioned above to fit these models to experimental data.******Many systems have some sort of symmetry, for example, networks of identical neurons that are coupled to one another in the same way. This research will investigate such systems, especially those with oscillating behaviour, in order to determine and classify all possible dynamic behaviour of such systems near the point where these oscillations first appear.******This research will also build mathematical models for various biological processes. For example, farmers need to know the appropriate length of time one should wait between manure application to crops and harvest, in order to be sure that any pathogens in the manure are not transferred to the food. We will model the die-off rates of various pathogens in manure after it is spread on fields of different types and under different weather conditions in order to answer this question.