  The persistence conjecture is a long-standing open problem in chemical
reaction network theory. It concerns the behavior of solutions to coupled ODE
systems that arise from applying mass-action kinetics to a network of chemical
reactions. The idea is that if all reactions are reversible in a weak sense,
then no species can go extinct. A notion that has been found useful in thinking
about persistence is that of "critical siphon." We explore the combinatorics of
critical siphons, with a view towards the persistence conjecture. We introduce
the notions of "drainable" and "self-replicable" (or autocatalytic) siphons. We
show that: every minimal critical siphon is either drainable or
self-replicable; reaction networks without drainable siphons are persistent;
and non-autocatalytic weakly-reversible networks are persistent. Our results
clarify that the difficulties in proving the persistence conjecture are
essentially due to competition between drainable and self-replicable siphons.
