Abstract for the talk on 30.03.2021 (17:00 h)Nonlinear Algebra Seminar Online (NASO)
Georg Regensburger (Johannes Kepler Universität Linz)
Positive solutions of generalized polynomial systems with real exponents and chemical reaction networks
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Reaction networks with mass-action kinetics give rise to high-dimensional polynomial ODE systems with positive parameters. Chemical reaction network theory provides statements about uniqueness, existence, and stability of positive steady states for all rate constants and initial conditions depending on the underlying network structure alone. However, mass-action kinetics holds only for elementary reactions in homogeneous and dilute solutions. In joint work with Stefan Müller, we propose the notion of generalized mass-action systems, which can serve as a more realistic model for biochemical reaction networks. It turns out that in this setting, uniqueness and existence of steady states for all parameters additionally depend on sign vectors of related real subspaces. In terms of the corresponding generalized polynomial equations, our results guarantee existence and uniqueness of positive solutions for all positive parameters.
In this talk, we describe some of our results for positive solutions of generalized polynomial equations and the motivation coming from the study of reaction networks. We also outline recent joint work on when the existence of a unique positive solution is robust with respect to small perturbations of the real exponents and on the stability of the resulting steady state. Finally, we discuss directions of ongoing and future research for positive solutions of generalized polynomial equations and applications to reaction networks.