In a previous post I wrote about chemical reaction network theory and, in particular, about a result belonging to this theory called the deficiency zero theorem. Now I have realized that in that post I claimed more than was justified. I will correct this point here. The assumptions of the deficiency zero theorem are that we have a chemical reaction network which is weakly reversible and of deficiency zero. (For the terminology I refer to the previous post on CRNT.) One conclusion for the associated dynamical system is that there is a unique stationary solution (in each stoichiometric compatibility class) where all concentrations are positive. A second conclusion is that is asymptotically stable (a local statement). The Lyapunov function used to prove the second statement also allows some further conclusions to be drawn. For solutions with positive concentrations is strictly decreasing along solutions away from . This means that a positive solution can have no positive -limit points other than . In addition tends to infinity at infinity, thus showing that each solution stays in a compact set. It can be concluded that the -limit set is compact and that unless it is it must consist of points where at least one concentration is zero. In the post just quoted I claimed that every solution converges to . Reading the original three basic papers on this subject by Horn, Jackson and Feinberg might easily give the impression that this is the case. Looking at the proofs in detail, as I have done in the meantime, reveals that this statement is not proved in those papers. There are also many later papers on CRNT where this issue is not raised. This does not mean that the problem escaped attention completely, even in the early days. In a paper by Horn from 1974 he explicitly states that the proof of the result on global stability in his paper with Jackson was not correct. He expresses the opinion that the statement is nevertheless probably true. In a paper from 2001 on the kinetic proofreading model Eduardo Sontag proved a result of this kind under some extra conditions.
Recently this issue has received renewed attention and has been given the name ‘global attractor conjecture’ by Craciun et. al. (J. Symbolic. Comput. 44, 1551). In a 2011 paper of Anderson (SIAM J. Appl. Math. 71, 1487) he writes that it ‘is considered to be one of the most important open problems in the field of chemical reaction network theory’. In that paper he proves the conjecture in the case of systems with a single linkage class and so perhaps the question is close to being resolved.