Modelling with and analysis of dynamical systems

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Differential equations are used to model many different phenomena from many different areas. Two examples that I study are food webs in ecology and ion channels in neurobiology. Mathematical models allow us to investigate the behaviour of complex systems without necessarily observing the real system directly, and allow us to alter the system and ask the question, "What would happen if this aspect changed?", something that may be impossible to do on the real system itself. This research proposal is to study models based on systems of ordinary differential equations and particularly to look at the relationship between these models and the data available from the real system. In order for mathematical models to be useful they must be calibrated to the real systems as closely as possible. Typically this means determining the appropriate values of some model parameters based on observable data from the real world. However, this fitting procedure can be extremely difficult and computationally intensive. In addition there is always the question of whether the model is too simplistic to capture the important features in the data or whether it is more complicated than is warranted by the data available. Part of this research aims to develop computational algorithms which can confine the ranges of these parameter values by extracting information from time series data. If the ranges remain excessively large, this is an indication that a simpler model might fit the data equally well. This research will be applied to at least two specific areas. Various models of ion channel activation will be investigated, comparing their qualitative and quantitative features and developing methods for neuroscientists to use in choosing appropriate models based on the data they have available. The research also aims to identify different experimental protocols that neuroscientists could use to gather data which are more discriminatory between different models. Models of food web dynamics help answer such ecological questions as whether top predators which typically span larger geographic areas have a stabilizing or destabilizing influence on their prey populations, and whether asymmetry in food webs enhances persistence.