Chemical reaction networks model the interactions of chemical substances. An important aim is to understand the long-term behaviors of these chemical reactions, which are called steady states. Interesting behaviors have implications in biology, such as in cell-signaling or regulatory processes. This motivates the need to predict if and when a network can exhibit interesting long-term behavior. Exact values of the physical parameters that describe these networks are often unknown. Because of this, the goal is to prove results based on a network's structure alone, independent of any specific numerical values. This dissertation harnesses theoretical mathematical techniques to investigate the number and stability of steady states of a chemical reaction network. In this dissertation, we answer the following: Does a given chemical reaction network have the capacity for Hopf bifurcations (an important unstable steady state)? How many steady states can it have? Our first contribution is a novel procedure for constructing a Hopf bifurcation of a chemical reaction network. This algorithm gives an easy-to-check condition for the existence of a Hopf bifurcation and explicitly constructs one if it exists. Our second set of contributions are new upper bounds on the number of steady states of a chemical reaction network. These new numerical invariants are both quick to compute and are surprisingly good bounds on the number of steady states. As the main application of our new tools, we analyze an important biological cell-signaling network called the Extracellular signal Regulated Kinase (ERK) network. Malfunctions in the ERK network are linked with human diseases, including cancers and developmental abnormalities, making it crucial to understand the ERK network's long-term behavior. We show how the ERK network has the capacity for different dynamic regimes, including multiple steady states, two stable steady states, simple Hopf bifurcations, and a unique, stable steady state. Applying our new tools, we directly relate each dynamic behavior to the network structure, specifically the presence of certain species or reactions.
