  A nonlinear dynamical system is called eventually competitive (or
cooperative) provided that it preserves a partial order in backward (or
forward) time only after some reasonable initial transient. We presented in
this paper the Non-oscillation Principle for eventually competitive or
cooperative systems, by which the non-ordering of (both $\omega$- and
$\alpha$-) limit sets is obtained for such systems; and moreover, we
established the Poincar\'{e}-Bendixson Theorem and structural stability for
three-dimensional eventually competitive and cooperative systems.
