  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.
