  A class of chemical reaction networks is described with the property that
each positive equilibrium is locally asymptotically stable relative to its
stoichiometry class, an invariant subspace on which it lies. The reaction
systems treated are characterised primarily by the existence of a certain
factorisation of their stoichiometric matrix, and strong connectedness of an
associated graph. Only very mild assumptions are made about the rates of
reactions, and in particular, mass action kinetics are not assumed. In many
cases, local asymptotic stability can be extended to global asymptotic
stability of each positive equilibrium relative to its stoichiometry class. The
results are proved via the construction of Lyapunov functions whose existence
follows from the fact that the reaction networks define monotone dynamical
systems with increasing integrals.
