Polynomial dynamical systems and reaction networks: persistence and global attractors

The mathematical analysis of global properties of polynomial dynamical systems can be very challenging (for example: the second part of Hilbert’s 16th problem about polynomial dynamical systems in 2D, or the analysis of chaotic dynamics in the Lorenz system).

On the other hand, any dynamical system with polynomial right-hand side can essentially be regarded as a model of a reaction network. Key properties of reaction systems are closely related to fundamental results about global stability in classical thermodynamics. For example, the Global Attractor Conjecture can be regarded as a finite dimensional version of Boltzmann’s H-theorem. We will discuss some of these connections, as well as the introduction of toric differential inclusions as a tool for proving the Global Attractor Conjecture.

We will also discuss some implications for the more general Persistence Conjecture (which says that solutions of weakly reversible systems cannot "go extinct"), as well as some applications to biochemical mechanisms that implement cellular homeostasis.