We consider a class of continuous-time cooperative systems
evolving on the positive orthant. We show that if the
origin is globally attractive, then it is also globally stable
and, furthermore, there exists an unbounded invariant manifold
where trajectories strictly decay. This leads to a small-gain type
condition which is sufficient for global asymptotic stability
(GAS) of the origin.
We establish the following connection to large-scale
interconnections of (integral) input-to-state stable (ISS)
subsystems: If the cooperative system is (integral) ISS, and
arises as a comparison system associated with a large-scale
interconnection of (i)ISS systems, then the composite nominal
system is also (i)ISS. We provide a criterion in terms of a
Lyapunov function for (integral) input-to-state stability of the
comparison system. Furthermore, we show that if a small-gain
condition holds then the classes of systems participating in the
large-scale interconnection are restricted in the sense that
certain iISS systems cannot occur. Moreover, this small-gain
condition is essentially the same as the one obtained previously
by Dashkovskiy et al. for large-scale interconnections of ISS systems.