# Stability and Instability in Polynomial Equations Arising from Complex Chemical Reaction Networks: The Big Picture

Monday, March 5, 2007 - 1:30pm - 2:20pm

EE/CS 3-180

Martin Feinberg (The Ohio State University)

In nature there are millions of distinct networks of chemical reactions that might present themselves for study at one time or another. Written at the level of elementary reactions taken with classical mass action kinetics, each new network gives rise to its own (usually large) system of polynomial equations for the species concentrations. In this way, chemistry presents a huge and bewildering array of polynomial systems, each determined in a precise way by the underlying network up to parameter values (e.g., rate constants). Polynomial systems in general, even simple ones, are known to be rich sources of interesting and sometimes wild dynamical behavior. It would appear, then, that chemistry too should be a rich source of dynamical exotica.

Yet there is a remarkable amount of stability in chemistry. Indeed, chemists and chemical engineers generally expect homogeneous isothermal reactors, even complex ones, to admit precisely one (globally attractive) equilibrium. Although this tacit doctrine is supported by a long observational record, there are certainly instances of homogeneous isothermal reactors that give rise, for example, to multiple equilibria. The vast landscape of chemical reaction networks, then, appears to have wide regions of intrinsic stability (regardless of parameter values) punctuated by far smaller regions in which instability might be extant (for at least certain parameter values).

In this talk, I will present some recent joint work with Gheorghe Craciun that goes a long way toward explaining this landscape — in particular, toward explaining how biological chemistry escapes the stability doctrine to (literally) make life interesting. A subsequent talk by Craciun will emphasize more mathematical detail.

Yet there is a remarkable amount of stability in chemistry. Indeed, chemists and chemical engineers generally expect homogeneous isothermal reactors, even complex ones, to admit precisely one (globally attractive) equilibrium. Although this tacit doctrine is supported by a long observational record, there are certainly instances of homogeneous isothermal reactors that give rise, for example, to multiple equilibria. The vast landscape of chemical reaction networks, then, appears to have wide regions of intrinsic stability (regardless of parameter values) punctuated by far smaller regions in which instability might be extant (for at least certain parameter values).

In this talk, I will present some recent joint work with Gheorghe Craciun that goes a long way toward explaining this landscape — in particular, toward explaining how biological chemistry escapes the stability doctrine to (literally) make life interesting. A subsequent talk by Craciun will emphasize more mathematical detail.

MSC Code:

65H04

Keywords: