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Persistence of deterministic population processes and the Global Attractor Conjecture

This paper gives sufficient conditions for persistence of deterministic population processes. It has been conjectured that any population process whose network graph is weakly reversible (has strongly connected components) is persistent. We prove this conjecture for a class of systems. An important application of this work pertains to chemical reaction systems that are “complex-balancing.” For these systems, it is known that each invariant manifold is a polyhedron and has a unique equilibrium in its interior. 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 main result implies that this conjecture holds for all complex-balancing systems whose boundary equilibria lie in relatively open facets of the boundary. As a corollary, we show that the Global Attractor Conjecture holds for those systems for which the associated invariant manifolds are two-dimensional.