PMatrix Properties, Injectivity, and Stability in Chemical Reaction Systems

In this paper we examine matrices which arise naturally as Jacobians in chemical dynamics. We are particularly interested in when these Jacobians are P matrices (up to a sign change), ensuring certain bounds on their eigenvalues, precluding certain behaviour such as multiple equilibria, and sometimes implying stability. We first explore reaction systems and derive results which provide a deep connection between system structure and the P matrix property. We then examine a class of systems consisting of reactions coupled to an external rate-dependent negative feedback process, and characterise conditions which ensure the P matrix property survives the nega- tive feedback. The techniques presented are applied to examples published in the mathematical and biological literature. 0 matrices, to be defined below. This condition is algorithmically easy to check, and immediately implies the absence of multiple equilibria as long as there are appropriate outflow conditions. A weaker condition is then derived specifically for mass action reaction systems, which ensures that they have Jacobians in this class, and hence, under appropriate outflow conditions, cannot have multiple equilibria. These conditions are shown to be not only sufficient to preclude multiple equilibria, but also necessary to ensure that the Jacobians can never be singular. Finally a class of systems of particular importance in biochemistry is examined. These systems involve reactions interacting with some external quantity giving rise to a negative feedback process. Necessary and sufficient conditions are derived which ensure that the P matrix properties of the system without feedback persist with the feedback.