On (Eventually) Monotone Dynamical Systems and Positive Koopman Semigroups

Monotone systems are dynamical systems whose solutions preserve a partial order in initial conditions for all times. It stands to reason that some systems may preserve a partial order only after an initial transient. These systems are usually called eventually monotone. While monotone systems have an easy characterization in terms of the sign pattern of the Jacobian matrix (i.e. Kamke-M"uller condition), eventually monotone systems have not been characterized in such an explicit manner. In order to provide such a characterization, we drew inspiration from the results for linear systems, where eventually monotone (positive) systems are studied using the spectral properties of the system (i.e. Perron-Frobenius property). In the case of nonlinear systems, a spectral characterization of nonlinear eventually monotone systems is not straightforward, but can be obtained in the framework of the so-called Koopman operator. Additionally, we explore connections between (eventual) monotonicity and (eventual) positivity of the Koopman semigroup. This allows to view our results as a generalization of the Perron-Frobenius theory to nonlinear dynamical systems. We consider a biologically inspired example to illustrate the applicability of eventual monotonicity.