UCL Discovery

Abstract

This thesis describes three applications of the theory of continuous autonomous dynamical systems. The focus of the thesis is on qualitative, as opposed to numerical, analysis. The applications examined are biological and chemical, and as such there are signicant uncertainties in any mathematical representation of them. While the qualitative relationships that dene a biological or chemical system may be well understood, it is often dicult to obtain accurate measurements of the parameters that govern each interaction, due to inherent variability and/or experimental constraints. For this reason, a model that avoids dependence on numerical values while still accurately reecting the qualitative structure of the system it represents is potentially of use in gaining a greater understanding of how the system can behave. Conversely, if a purely qualitative model allows certain behaviour that is never experimentally observed, this may highlight the importance of certain parameter values for the system's real world behaviour. The rst application presented is a model of electron transport in mitochondria, the second is a model of an inter-cellular gap junction, and the third represents a set of reactions occurring in a continuous ow stirred tank reactor. For each application, a reasonable set of qualitative assumptions is found under which there is a unique steady state to which all initial conditions converge, regardless of precise numerical values. Uniqueness of steady states is proved using results on the injectivity of functions, and degree theory. The convergence criteria are constructed using two dierent areas of dynamical systems theory. The rst of these is the theory of monotone ows, while the second is a group of results known as autonomous convergence theorems. The theory of monotone ows is fairly well known, and relies on nding conditions under which trajectories of a dynamical system preserve a partial ordering, thereby limiting the possibly asymptotic behaviour of the system. The autonomous convergence theorems appear much less well known; they work by nding a norm under which trajectories approach each other, either in phase space or in a related exterior algebra space. Both theories are discussed in detail, along with some extensions.

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