Semester
Fall
Date of Graduation
2022
Document Type
Dissertation
Degree Type
PhD
College
Eberly College of Arts and Sciences
Department
Mathematics
Committee Chair
Casian Pantea
Committee Co-Chair
Adam Halasz
Committee Member
Adrian Tudorascu
Committee Member
Charis Tsikkou,
Committee Member
Vyacheslav Akkerman,
Abstract
This project explores a topic in Chemical Reaction Network Theory. We analyze networks with one dimensional stoichiometric subspace using mass-action kinetics. For these types of networks, we study how the capacity for multiple positive equilibria and multiple positive nondegenerate equilibria can be determined using Euclidian embedded graphs. Our work adds to the catalog of the class of reaction networks with one-dimensional stoichiometric subspace answering in the affirmative a conjecture posed by Joshi and Shiu: Conjecture 0.1 (Question 6.1 [26]). A reaction network with one-dimensional stoichiometric subspace and more than one source complex has the capacity for multistationarity if and only if it has a one-species embedded subnetwork with the pattern (→,←), and another (possibly the same) with pattern (←,→). Additionally, we provide classifications of networks with one dimensional stoichiometric subspace according to their capacity for multiple positive nondegenerate equilbria. The paper is structured as follows. In Section 2 general reaction networks terminology is introduced. Also network projections, arrow diagrams and their connections with Euclidean embedded graphs are described. Note that the conclusion about multistationarity is built by inspecting arrow diagrams. In Section 3 we talk about mass-action kinetics, injective networks and inheritance theorems. Injectivity is a useful tool to rule out the capacity for multistationarity. Inheritance theorems can be used to lift nondegenerate multistationarity from a smaller subnetwork to the corresponding large network. Inheritance theorems are also useful for further applications of the obtained result. Section 4 sets the notation for networks with one-dimensional stoichiometric subspace and defines classes of networks that our main theorem (Theorem 5.1) depends on. The main result, examples, and connection with the literature are presented in Section 5. Section 6 contains proofs.
Recommended Citation
Voitiuk, Galyna, "Structure-dependent Characterizations of Multistationarity in Mass-action Reaction Networks" (2022). Graduate Theses, Dissertations, and Problem Reports. 11607.
https://researchrepository.wvu.edu/etd/11607