Mathematical methods for analyzing biological interaction networks

Mathematical models of biological interaction networks give rise to a very large family of nonlinear dynamical systems. Some of the most common such models are based on mass-action kinetics and give rise to polynomial dynamical systems.

We will discuss how the analysis of these models over the last few decades has resulted in a wealth of new ideas and open problems. For example: (i) the study of uniqueness of equilibria in reaction network models led to general theorems on global injectivity of polynomial functions; (ii) the study of persistence properties of these models (i.e., finding conditions that imply that no species in an ecosystem goes extinct) led to the introduction of new mathematical tools, such as toric differential inclusions; and (iii) the construction of convex invariant regions for these models led to theorems on existence of solutions for nonlinear reaction-diffusions PDEs.

We will also mention some connections to the Jacobian Conjecture, and to classical results in thermodynamics, such as Boltzmann's H-theorem.