Stability criteria for off-diagonally monotone nonlinear dynamical systems

This paper discusses the global stability of a nonlinear dynamical system
\dot{x}=f(x)
in which
f
is a locally Lipschitz continuous off-diagonally monotone function and
f(\theta) > \theta
. Two results are proved: 1) if
f
is piecewise-linear function and if
-f
is an
M
-function, then a unique equilibrium point exists and it is globally asymptotically stable; 2) if
f
is a nonlinear function with separate variables in the sense that
f
is given by
f_{1}(x)= \Sigma^{n}_{j-1}f_{ij}(x_j)
for all
i
, and if
-f
is an
M
-function satisfying
f(x^{\ast})= \theta
for some nonnegative vector
x^{\ast}
, then
x^{\ast}
is globally asymptotically stable. These results are applied to the stability analyses of a large scale composite system and a compartmental system.