Abstract:
Many physiological systems have the property that some processes evolve much faster than others, and mathematical models can be constructed with multiple timescales to re ect this property. A gonadotropin-releasing hormone (GnRH) neuron model developed by Duan et al. [Theoretical Biology 276 (2011), 22-34] provides an ex- ample of such a system and has variables that evolve on at least three timescales. Techniques such as geometric singular perturbation theory (GSPT) have previously been developed to analyse systems with two timescales, but methods for analysing models with three or more timescales are still limited. This thesis aims to advance understanding of systems with three timescales by studying a selection of different types of three timescale models. First we investigate a three-dimensional system with three timescales, speci cally the food chain model of Rosenzweig et al. [The American Naturalist 97 (1963), 209-223]. Second we construct a more general three-dimensional three timescale system than the food chain model, to illustrate the required conditions for generating a certain type of oscillation seen in the food chain model. Third, we couple two two-dimensional fast-slow systems in a con guration based on the structure of the GnRH neuron model. In each case, we apply methods from GSPT, extending the techniques where necessary to allow for the presence of three timescales. We investigate a selection of complex oscillations seen in these models, and explain the origin of the various features seen. We nd that the shapes and relative positions of two invariant manifolds of related singular systems, speci cally the critical manifold and the superslow manifold, are crucial to an explanation of the complicated patterns of oscillation seen in the three timescale systems. Also, we nd that the existing theory for two timescale systems is not appropriate for explaining all the folded singularities observed in three timescale systems. Finally, we reassess the oscillations in the models of interest, with the aim of identifying which patterns are intrinsically three timescale phenomena and which could also be seen in models with only two timescales. We nd examples of both types of oscillation and in particular show that three timescales are needed to generate a time series like the solution of interest in the GnRH neuron model.