The smallest bimolecular mass action reaction networks admitting Andronov-Hopf bifurcation

TL;DR

This paper classifies small bimolecular chemical reaction networks with three species and four reactions that can exhibit Hopf bifurcation, revealing complex dynamics and providing tools for predicting such behavior in larger networks.

Contribution

It fully classifies three-species, four-reaction bimolecular CRNs regarding Hopf bifurcation, identifying 86 minimal networks and analyzing their bifurcation types and dynamics.

Findings

86 minimal networks admit Hopf bifurcation

25 networks admit both supercritical and subcritical bifurcations

29 networks can have stable equilibrium and periodic orbit coexistence

Abstract

We address the question of which small, bimolecular, mass action chemical reaction networks (CRNs) are capable of Andronov-Hopf bifurcation (from here on abbreviated to "Hopf bifurcation"). It is easily shown that any such network must have at least three species and at least four irreversible reactions, and one example of such a network with exactly three species and four reactions was previously known due to Wilhelm. In this paper, we develop both theory and computational tools to fully classify three-species, four-reaction, bimolecular CRNs, according to whether they admit or forbid Hopf bifurcation. We show that there are, up to a natural equivalence, 86 minimal networks which admit nondegenerate Hopf bifurcation. Amongst these, we are able to decide which admit supercritical and subcritical bifurcations. Indeed, there are 25 networks which admit both supercritical and subcritical bifurcations, and we can confirm that all 25 admit a nondegenerate Bautin bifurcation. A total of 31 networks can admit more than one nondegenerate periodic orbit. Moreover, 29 of these networks admit the coexistence of a stable equilibrium with a stable periodic orbit. Thus, fairly complex behaviours are not very rare in these small, bimolecular networks. Finally, we can use previously developed theory on the inheritance of dynamical behaviours in CRNs to predict the occurrence of Hopf bifurcation in larger networks which include the networks we find here as subnetworks in a natural sense.