Nonmonotone systems decomposable into monotone systems with negative feedback

Motivated by the work of Angeli and Sontag [Monotone control systems, IEEE Trans. Automat. Control 48 (2003) 1684–1698] and Enciso and Sontag [On the global attractivity of abstract dynamical systems satisfying a small gain hypothesis, with applications to biological delay systems, Discrete Continuous Dynamical Systems, to appear] in control theory, we show that certain finite and infinite dimensional semi-dynamical systems with “negative feedback” can be decomposed into a monotone “open-loop” system with “inputs” and a decreasing “output” function. The original system is reconstituted by “plugging the output into the input”. Employing a technique of Gouzé [A criterion of global convergence to equilibrium for differential systems with an application to Lotka–Volterra systems, Rapport de Recherche 894, INRIA] and Cosner [Comparison principles for systems that embed in cooperative systems, with applications to diffusive Lotka–Volterra models, Dynam. Cont., Discrete Impulsive Systems 3 (1997) 283–303] of imbedding the system into a larger symmetric monotone system, we are able to obtain information on the asymptotic behavior of solutions, including existence of positively invariant sets and global convergence.