In the first part of this thesis, we will explore an iterative procedure to determine the detailed asymptotic behaviour of
solutions of a certain class of nonlinear vector differential equations which approach a nonlinear sink as time tends to infinity.
This procedure is indifferent to resonance in the eigenvalues. Some attention will be given to finding approximations to solutions which are themselves flows. Moreover, we will address the writing of one component in terms of another in the case of a planar system.
In the second part of this thesis, we will explore the Michaelis-Menten mechanism of a single enzyme-substrate reaction. The focus is an analysis of the planar reduction in phase space or, equivalently, solutions of the scalar reduction. In particular, we
will prove the existence and uniqueness of a slow manifold between the horizontal and vertical isoclines. Also, we will determine the concavity of all solutions in the first quadrant. Moreover, we will establish the asymptotic behaviour of all solutions near the origin, which generally is not given by a Taylor series. Finally, we will
determine the asymptotic behaviour of the slow manifold at infinity. Additionally, we will study the planar reduction. In particular, we will find non-trivial bounds on the length of the pre-steady-state
period, determine the asymptotic behaviour of solutions as time tends to infinity, and determine bounds on the solutions valid for
all time.
In the third part of this thesis, we explore the (nonlinear) Lindemann mechanism of unimolecular decay. The analysis will be
similar to that for the Michaelis-Menten mechanism with an emphasis on the differences. In the fourth and final part of this thesis, we will present some open problems.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OWTU.10012/4535 |
| Date | January 2009 |
| Creators | Calder, Matthew Stephen |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
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