Quasistationarität und fast-invariante Mengen gewöhnlicher Differentialgleichungen

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Nöthen, Anna Lena; Walcher, Sebastian (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2008, 2009)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2008


The objective of the present paper is to analyse the concept of quasi-stationarity, which is motivated from bio-chemistry, and to investigate it from a matematical perspective. In Chapter 1 the concept of quasi-stationarity is presented in the context of chemical reactions. Examples, particularly the Michaelis-Menten-reaction, are given. We then give a first overview of relevant approaches to analyse given differential equation systems for chemical reactions. Chapter 2 presents approaches via singular perturbation theory, particularly Tikhonov's theorem. Given a differential equation system of particular form, which depends on a small parameter, then Tikhonov's theorem provides (asymptotically) a reduction to smaller dimension. Fenichel's theorem deepens this result in a broader context. The application of these theories to chemical systems is not without problems as these systems generally do not appear in the specific form required. Moreover the small parameter should be motivated from chemical consideration. Next, we present an iterative method due to Fraser and Roussel who approximate an invariant manifold for a reduction of the differential equation. Their original system is in standard form for application of singular perturbation theory. Chapter 2 concludes with the methods of Heinrich and Schauer (resp. Stiefenhofer) for reacting systems split up into slow and fast reactions. They perform a transformation of the differential equation to obtain the reduction via Tikhonov. None of the methods presented above is particularly appropriate for a systematic analysis of quasi-stationarity. In Chapter 3 the chemical notion of quasi-stationarity is combined with the concept of near-invariance. We obtain a method which, starting from a biochemically motivated assumption, provides locally necessary conditions on the parameters for quasi-steady state behaviour. These conditions can be determined by using the near-invariance of a certain set. In addition "small parameters" can be detected. Moreover various approaches to reduction are presented which are motivated by quasi-stationarity. The iterative method of Chapter 2 is discussed again, now from the perspective of near-invariance. In Chapter 4 we analyse the long-term behaviour of solutions. For chemical reactions we frequently find a dynamical equilibrium in the limit. Mathematically this equilibrium is reflected by a stationary point in the differential equation system. It will be shown that the long-term behaviour in an asymptotically stable stationary point is determined by the linear part alone. In examples we analyse whether the long-term behaviour is consistent with the nearly-invariant set discussed in Chapter 3. Finally in Chapter 5 we discuss how the theory of Tikhonov and Fenichel can be applied to chemical reaction systems with a quasi-steady state assumption. Identifying a small parameter is essential, and this can be done via nearly-invariant sets as discussed in Chapter 3. Then we show how a transformation should be designed to obtain from a given system a system in standard form of singular perturbation theory. Conditions for applicability of Tikhonov and Fenichel can be tested and, if applicable, a reduced system can be identified. The motivation for this approach goes back to Heinrich and Schauer as outlined in Chapter 2. However, the formulation here is more general and the range of applications is larger, particularly by including quasi-steady state behaviour. With this approach the local statements from Chapter 3 are becoming global statements. We illustrate this systematic approach by considering several examples.