Interplay between Contractivity and Monotonicity for Reaction Networks

Abstract: We establish a new relationship between monotonicity and contractivity and use this connection to describe a new general class of weakly contractive reaction networks. The new class is characterized by the stoichiometry matrix of the reaction network admitting a precise matrix factorization that can be verified computationally. Reaction networks in this class are weakly contractive, implying global convergence to equilibria under appropriate technical conditions. Furthermore, we describe the novel subclass of cross-polytope networks. We also show that our results provide a unified proof of global convergence for several classes of networks previously studied in the literature. The practical relevance of the results is demonstrated by examples from systems biology and signaling pathways.