Global Stability for Mixed Monotone Systems

The idea of embedding a dynamical system, whose generator has both increasing and decreasing monotonicity properties (positive and negative feedback), into a larger symmetric monotone dynamical system and exploiting the convergence properties of the latter is very old. For a discussion of history of the method, see [4,8]. The method is repeatedly rediscovered and its implications are often underestimated. In this paper, we review the main results of the embedding method following [8] and then we show how it leads immediately to an improved version of a nice result on global stability due to Kulenović and Merino [7] for componentwise monotone maps that leave invariant a hypercube in Euclidean space. We then ask whether embedding a system into a larger monotone system is really necessary to obtain global stability results. On the face of it, it seems unnatural to pass to a larger dimensional dynamical system in order to gain information on the dynamics of a smaller one. We show that for the class of mixed-monotone systems, one can obtain global stability results directly without the need of embedding. As noted in [8], the embedding method leads to a nice generalization of some results of Kulenović, Ladas and Sizer [5], also contained in the monograph of Kulenović and Ladas [6], on higher order difference equations with componentwise monotonicity.