Analyse und asymptotische Analyse von Kompartimentsystemen
- Analysis and asymptotic analysis of compartmental systems
Lax, Christian; Walcher, Sebastian (Thesis advisor); Frank, Martin (Thesis advisor)
Dissertation / PhD Thesis
Dissertation, RWTH Aachen University, 2016
This thesis deals with singularly disturbed systems of differential equations. The primary goal is the computation of asymptotic reductions which help with the analysis of those systems. The reductions are based on the classical theories by Tikhonov and Fenichel and thus are referred to as Tikhonov-Fenichel reductions.The thesis consists of two parts. The first part discusses autonomous ODEs and focuses on modelling chemical reactions subjected to quasi steady state assumptions.Therefore, most of the results refer to rational or polynomial systems. Building on the work of Lena Nöthen and Alexandra Goeke, three main results are proven: First of all, Hoppensteadt's theorem can be generalized to systems which are not in Tikhonov normal form. Moreover, a reformulation of the assumptions of the theorem helps to gain a better applicability. Secondly, we can compute Tikhonov-Fenichel reductions for all chemical reaction networks which can be divided into slow and fast reactions, as long as the fast part consists of weakly reversible reactions of first order. The third result refers to compartmental systems, i.e. systems that are governed by transport between subsystems and interaction within these subsystems. It turns out that Tikhonov-Fenichel reductions of those systems can be derived from the individual interaction terms alone.The second part of the thesis discusses asymptotic reductions of reaction diffusion systems. As there is no counterpart to Tikhonov's theorem in infinite dimensions, the main goal is finding and computing explicit reductions of reaction diffusion systems (without giving general convergence results). We do this by considering spatially discretized reaction-diffusion systems as compartmental systems. Our method is backed by various results: First and foremost, we show the consistency of the proposed reduction. Furthermore, a convergence result is proved for reaction diffusion systems for which every reaction is of first order and the fast part satisfies the principle of detailed balance. Lastly, we discuss some examples of which the Michaelis-Menten reaction is the most prominent one. We compare our heuristical reduction of these examples to known results in the literature and discuss systems where no previous results seem to be known. In the latter case, numerical simulations exhibit nice convergence properties.