Reaction systems are continuos-time dynamical systems with polynomial right-hand side, and are very common in biochemistry, cell signaling, population dynamics, and many other biological applications. We discuss global stability (i.e., the existence of a globally attracting point) and persistence (i.e., robust absence of extinction) for large classes of reaction systems. In particular, we describe recent progress on the proof of the Global Attractor Conjecture (which says that vertex-balanced reaction systems are globally stable) and the Persistence Conjecture (which says that weakly-reversible reaction systems are persistent), and how these results can be extended outside their classical setting using the notion of “disguised reaction systems". We will also discuss analogous results for the case where reaction systems are replaced by generalized Lotka-Volterra systems of arbitrary degree. 

to follow