  Long-time behavior is one of the most fundamental properties in dynamical
systems. The limit behaviors of flows on surfaces are captured by the
Poincar\'e-Bendixson theorem using the $\omega$-limit sets. This paper
demonstrates that the positive and negative long-time behaviors are not
independent. In fact, we show the dependence between the $\omega$-limit sets
and the $\alpha$-limit sets of points of flows on surfaces, which partially
generalizes the Poincar\'e-Bendixson theorem. Applying the dependency result to
solve what kinds of the $\omega$-limit sets appear in the area-preserving (or,
more generally, non-wandering) flows on compact surfaces, we show that the
$\omega$-limit set of any non-closed orbit of such a flow with arbitrarily many
singular points on a compact surface is either a subset of singular points or a
locally dense Q-set. Moreover, we show the wildness of surgeries to add totally
disconnected singular points and the tameness of those to add finitely many
singular points for flows on surfaces.
