Some intuition for why cooperative systems "look 1-dimensional" and 2-cooperative systems "look 2-dimensional"

It is known that cooperative systems, and more generally systems monotone with respect to cones, behave under appropriate irreducibility conditions like one-dimensional systems, as evidenced by results such as Hirsch’s generic convergence theorem. It is also known, following work of Sanchez and others, that two-cooperative systems, those preserving a generally nonconvex 2-dimensional cone (tested through the diminishing of sign variations), behave like two-dimensional systems, as evidenced by Poincare–Bendixson-type theorems. In these notes I attempt to give some geometric intuition for these dimensionality reductions, based on Birkhoff–Hilbert contractions of the projective metric on a positive cone.