Toric dynamical systems

A. DICKENSTEIN

INRIA Sophia Antipolis, Francia

Conferencia; Seminario GALAAD; 2007

INRIA

Toric dynamical systems are mass action kynetics systems for which the steady state locus is a (deformed) toric variety [K. Gatermann (2001)], which has a unique point within each invariant polyhedron. They are known as complex balancing mass action systems in the mathematical chemistry literature, where many of their remarkable properties have been established [M. Feinberg (1972), F.Horn (1972, 1973), G. Craciun and M. Feinberg (2005)]. They have a wide range of applications in physics, in theoretical computer science and also in biology. We develop the basic theory of toric dynamical systems in the context of computational algebraic geometry and show that the associated moduli space is also a toric variety.It is conjectured that the complex balancing state is a global attractor.This conjecture is open even for deficiency zero systems (for which the moduli ideal is zero). It has been recently proved for a certain classof ``monotone´´ deficiency zero systems [De Leenheer, Angeli, E.Sontag (2007)]. We prove this for detailed balancing systems whose invariant polyhedron is two-dimensional and bounded.This is joint work with Gheorghe Craciun, Anne Shiu and Bernd Sturmfels