  For a general time-dependent linear competitive-cooperative tridiagonal
system of differential equations, we obtain canonical Floquet invariant bundles
which are exponentially separated in the framework of skew-product flows. Such
Floquet bundles naturally reduce to the standard Floquet space when the system
is assumed to be time-periodic. The obtained Floquet theory is applied to study
the dynamics on the hyperbolic omega-limit sets for the nonlinear
competitive-cooperative tridiagonal systems in time-recurrent structures
including almost periodicity and almost automorphy.
