In a previous post I discussed the global attractor conjecture which concerns the asymptotic behaviour of solutions of the mass action equations describing weakly reversible chemical reaction networks of deficiency zero (or more generally complex balanced systems). Systems of the latter class are sometimes called toric dynamical systems because of relations to the theory of toric varieties in algebraic geometry. I just saw a paper by Gheorghe Craciun which he put on ArXiv last January (arxiv.org/pdf/1501.0286) where he proves this conjecture, thus solving a problem which has been open for more than 40 years. The result says that for reaction networks of this class the long-time behaviour is very simple. For given values of the conserved quantities there is exactly one positive stationary solution and all other solutions converge to it. What needs to be done beyond the classical results on this problem is to show that a positive solution can have no -limit points where some concentration vanishes. This property is sometimes known as persistence.

A central concept used in the proof of the result is that of a toric differential inclusion. This says that the time rate of change of the concentrations is contained in a special type of subset. The paper contains a lot of intricate geometric constructions. These are explained consecutively in dimensions one, two, three and four in the paper. This should hopefully provide a good ladder to understanding.

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