  This paper concerns the dynamical behavior of weakly reversible,
deterministically modeled population processes near the facets (codimension-one
faces) of their invariant manifolds and proves that the facets of such systems
are "repelling." It has been conjectured that any population process whose
network graph is weakly reversible (has strongly connected components) is
persistent. We prove this conjecture to be true for the subclass of weakly
reversible systems for which only facets of the invariant manifold are
associated with semilocking sets, or siphons. An important application of this
work pertains to chemical reaction systems that are complex-balancing. For
these systems it is known that within the interior of each invariant manifold
there is a unique equilibrium. The Global Attractor Conjecture states that each
of these equilibria is globally asymptotically stable relative to the interior
of the invariant manifold in which it lies. Our results pertaining to weakly
reversible systems imply that this conjecture holds for all complex-balancing
systems whose boundary equilibria lie in the relative interior of the boundary
facets. As a corollary, we show that the Global Attractor Conjecture holds for
those systems for which the associated invariant manifolds are two-dimensional.
