A graph-theoretic method for detecting potential Turing bifurcations

Maya Mincheva, Marc R. Roussel
2006 Journal of Chemical Physics  
The conditions for diffusion-driven ͑Turing͒ instabilities in systems with two reactive species are well known. General methods for detecting potential Turing bifurcations in larger reaction schemes are, on the other hand, not well developed. We prove a theorem for a graph-theoretic condition originally given by Volpert and Ivanova ͓Mathematical Modeling ͑Nauka, Moscow, 1987͒ ͑in Russian͒, p. 57͔ for Turing instabilities in a mass-action reaction-diffusion system involving n substances. The
more » ... substances. The method is based on the representation of a reaction mechanism as a bipartite graph with two types of nodes representing chemical species and reactions, respectively. The condition for diffusion-driven instability is related to the existence of a structure in the graph known as a critical fragment. The technique is illustrated using a substrate-inhibited bifunctional enzyme mechanism which involves seven chemical species.
doi:10.1063/1.2397073 pmid:17144685 fatcat:xtfdruzairgp3i6wbvggvyzole