  We consider generators of positive $C_0$-semigroups and, more generally,
resolvent positive operators $A$ on ordered Banach spaces and seek for
conditions ensuring the negativity of their spectral bound $s(A)$. Our main
result characterizes $s(A) < 0$ in terms of so-called \emph{small-gain
conditions} that describe the behaviour of $Ax$ for positive vectors $x$. This
is new even in case that the underlying space is an $L^p$-space or a space of
continuous functions.
  We also demonstrate that it becomes considerably easier to characterize the
property $s(A) < 0$ if the cone of the underlying Banach space has non-empty
interior or if the essential spectral bound of $A$ is negative. To treat the
latter case, we discuss a counterpart of a Krein-Rutman theorem for resolvent
positive operators. When $A$ is the generator of a positive $C_0$-semigroup,
our results can be interpreted as stability results for the semigroup, and as
such, they complement similar results recently proved for the discrete-time
case.
  In the same vein, we prove a Collatz--Wielandt type formula and a logarithmic
formula for the spectral bound of generators of positive semigroups.
