Bifurcations in planar, quadratic mass-action networks with few reactions and low molecularity

TL;DR

This paper characterizes the types of bifurcations that can occur in simple planar quadratic mass-action networks with few reactions, providing a comprehensive classification and extending some results beyond the planar case.

Contribution

It fully classifies generic bifurcations in small, low-molecularity planar networks and extends understanding to broader classes of mass-action networks.

Findings

Identifies fold, Hopf, Bogdanov--Takens, and Bautin bifurcations in small networks.

Proves no other generic bifurcations occur in these networks.

Provides necessary conditions for certain bifurcations in general mass-action networks.

Abstract

In this paper we study bifurcations in mass-action networks with two chemical species and reactant complexes of molecularity no more than two. We refer to these as planar, quadratic networks as they give rise to (at most) quadratic differential equations on the nonnegative quadrant of the plane. Our aim is to study bifurcations in networks in this class with the fewest possible reactions, and the lowest possible product molecularity. We fully characterise generic bifurcations of positive equilibria in such networks with up to four reactions, and product molecularity no higher than three. In these networks we find fold, Andronov--Hopf, Bogdanov--Takens and Bautin bifurcations, and prove the non-occurrence of any other generic bifurcations of positive equilibria. In addition, we present a number of results which go beyond planar, quadratic networks. For example, we show that mass-action networks without conservation laws admit no bifurcations of codimension greater than $m-2$, where $m$ is the number of reactions; we fully characterise quadratic, rank-one mass-action networks admitting fold bifurcations; and we write down some necessary conditions for Andronov--Hopf and cusp bifurcations in mass-action networks. Finally, we draw connections with a number of previous results in the literature on nontrivial dynamics, bifurcations, and inheritance in mass-action networks.