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

TY - JOUR

T1 - Power-Law Kinetics and Determinant Criteria for the Preclusion of Multistationarity in Networks of Interacting Species

AU - Wiuf, Carsten Henrik

AU - Feliu, Elisenda

PY - 2013/10/16

Y1 - 2013/10/16

N2 - We present determinant criteria for the preclusion of nondegenerate 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 noncatalytic kinetics, and also work based on graphical analysis of the interaction graph associated with the system. Further, we interpret the criteria in terms of circuits in the so-called DSR-graph.

AB - We present determinant criteria for the preclusion of nondegenerate 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 noncatalytic kinetics, and also work based on graphical analysis of the interaction graph associated with the system. Further, we interpret the criteria in terms of circuits in the so-called DSR-graph.

U2 - 10.1137/120873388

DO - 10.1137/120873388

M3 - Journal article

SN - 1536-0040

VL - 12

SP - 1685

EP - 1721

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

IS - 4

ER -