Persistence and Permanence of Mass-Action and Power-Law Dynamical Systems

Persistence and permanence are properties of dynamical systems that describe the long-term behavior of the solutions and in particular specify whether positive solutions approach the boundary of the positive orthant. Mass-action systems (or more generally power-law systems) are very common in chemistry, biology, and engineering and are often used to describe the dynamics in interaction networks. We prove that two-species mass-action systems derived from weakly reversible networks are both persistent and permanent, for any values of the reaction rate parameters. Moreover, we prove that a larger class of networks, called endotactic networks, also give rise to permanent systems, even if the reaction rate parameters vary in time (to allow for the influence of external signals). These results also apply to power-law systems and other nonlinear dynamical systems. In addition, ideas behind these results allow us to prove the global attractor conjecture for three-species systems.