Turing-Hopf instability in biochemical reaction networks arising from pairs of subnetworks

Network conditions for Turing instability in biochemical systems with two biochemical species are well known and involve autocatalysis or self-activation. On the other hand general network conditions for potential Turing instabilities in large biochemical reaction networks are not well developed. A biochemical reaction network with any number of species where only one species moves is represented by a simple digraph and is modeled by a reaction-diffusion system with non-mass action kinetics. A graph-theoretic condition for potential Turing-Hopf instability that arises when a spatially homogeneous equilibrium loses its stability via a single pair of complex eigenvalues is obtained. This novel graph-theoretic condition is closely related to the negative cycle condition for oscillations in ordinary differential equation models and its generalizations, and requires the existence of a pair of subnetworks, each containing an even number of positive cycles. The technique is illustrated with a double-cycle Goodwin type model.