P matrix 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 negative feedback. The techniques presented are applied to examples published in the mathematical and biological literature

P matrix 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 negative feedback. The techniques presented are applied to examples published in the mathematical and biological literature

A survey of methods for deciding whether a reaction network is multistationary

Which reaction networks, when taken with mass-action kinetics, have the capacity for multiple steady states? There is no complete answer to this question, but over the last 40 years various criteria have been developed that can answer this question in certain cases. This work surveys these developments, with an emphasis on recent results that connect the capacity for multistationarity of one network to that of another. In this latter setting, we consider a network $N$ that is embedded in a larger network $G$, which means that $N$ is obtained from $G$ by removing some subsets of chemical species and reactions. This embedding relation is a significant generalization of the subnetwork relation. For arbitrary networks, it is not true that if $N$ is embedded in $G$, then the steady states of $N$ lift to $G$. Nonetheless, this does hold for certain classes of networks; one such class is that of fully open networks. This motivates the search for embedding-minimal multistationary networks: those networks which admit multiple steady states but no proper, embedded networks admit multiple steady states. We present results about such minimal networks, including several new constructions of infinite families of these networks

Power-law Kinetics and Determinant Criteria for the Preclusion of Multistationarity in Networks of Interacting Species

We present determinant criteria for the preclusion of non-degenerate multiple steady states in networks of interacting species. A network is modeled as a system of ordinary differential equations in which the form of the species formation rate function is restricted by the reactions of the network and how the species influence each reaction. We characterize families of so-called power-law kinetics for which the associated species formation rate function is injective within each stoichiometric class and thus the network cannot exhibit multistationarity. The criterion for power-law kinetics is derived from the determinant of the Jacobian of the species formation rate function. Using this characterization we further derive similar determinant criteria applicable to general sets of kinetics. The criteria are conceptually simple, computationally tractable and easily implemented. Our approach embraces and extends previous work on multistationarity, such as work in relation to chemical reaction networks with dynamics defined by mass-action or non-catalytic kinetics, and also work based on graphical analysis of the interaction graph associated to the system. Further, we interpret the criteria in terms of circuits in the so-called DSR-graphComment: To appear in SIAM Journal on Applied Dynamical System

Graph-theoretic conditions for injectivity of functions on rectangular domains

This paper presents sufficient graph-theoretic conditions for injectivity of collections of differentiable functions on rectangular subsets of R^n. The results have implications for the possibility of multiple fixed points of maps and flows. Well-known results on systems with signed Jacobians are shown to be easy corollaries of more general results presented here.Comment: 16 pages, 5 figure