This research will use mathematics to understand how living cells make decisions, such as when to divide or when to die. In living cells, many chemical reactions take place all the time. Scientists think of these reactions as forming a network, like a wiring diagram, with connecting wires describing the movement of nutrients and metabolites through a cell's chemical reactions. Knowing the entire wiring diagram is not necessary for understanding the fate of individual nutrients, but the many interconnected circuits make it difficult to isolate the part of the diagram important for certain activities, such as making the decision to divide or to die. This research will use mathematics (specifically, algebraic geometry and dynamical systems) to find and analyze simple components of a reaction network that allow cells to exhibit certain important behaviors. In addition, the project will provide interdisciplinary training opportunities for early-career mathematicians as well as undergraduate students, and thereby supporting the next generation of mathematicians and the STEM workforce. The dynamics observed in living systems is much more than the sum of its parts. Systems biology, therefore, seeks to understand how biological components come together to generate emergent, systems-level behavior. A current bottleneck in systems biology is the need for mathematical theory specialized to the field. Accordingly, this research will develop theory for reaction systems tailored to biological networks. The project will prove new theorems that predict dynamics from reaction-network structure and generate new insights about the dynamical behavior of biologically significant networks. In particular, the results will yield insight into how the dynamics of caspase proteins are tightly regulated and bring about the irreversible process of apoptosis (programmed cell death). Additional results will deepen our understanding of how certain biologically significant properties, s