Some bounds on positive equilibria in mass action networks

We present some results helpful for parameterising positive equilibria, and bounding the number of positive nondegenerate equilibria, in mass action networks. Any mass action network naturally gives rise to a set of polynomial equations whose positive solutions are precisely the positive equilibria of the network. Here we derive alternative systems of equations, often also polynomial, whose solutions are in smooth, one-to-one correspondence with positive equilibria of the network. Often these alternative systems are simpler than the original mass action equations, and allow us to infer useful bounds on the number of positive equilibria. The alternative equation systems can also be helpful for parameterising the equilibrium set explicitly, for deriving descriptions of the parameter regions for multistationarity, and for studying bifurcations. We present the main construction, some bounds which follow for particular classes of networks, numerous examples, and some open questions and conjectures.