# Multistationarity in Structured Reaction Networks

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## Abstract

Many dynamical systems arising in biology and other areas exhibit multistationarity (two or more positive steady states with the same conserved quantities). Although deciding multistationarity for a polynomial dynamical system is an effective question in real algebraic geometry, it is in general difficult to determine whether a given network can give rise to a multistationary system, and if so, to identify witnesses to multistationarity, that is, specific parameter values for which the system exhibits multiple steady states. Here we investigate both problems. First, we build on work of Conradi, Feliu, Mincheva, and Wiuf, who showed that for certain reaction networks whose steady states admit a positive parametrization, multistationarity is characterized by whether a certain “critical function” changes sign. Here, we allow for more general parametrizations, which make it much easier to determine the existence of a sign change. This is particularly simple when the steady-state equations are linearly equivalent to binomials; we give necessary conditions for this to happen, which hold for many networks studied in the literature. We also give a sufficient condition for multistationarity of networks whose steady-state equations can be replaced by equivalent triangular-form equations. Finally, we present methods for finding witnesses to multistationarity, which we show work well for certain structured reaction networks, including those common to biological signaling pathways. Our work relies on results from degree theory, on the existence of explicit rational parametrizations of the steady states, and on the specialization of Gröbner bases.

## Keywords

Reaction network Mass-action kinetics Multistationarity Parametrization Binomial ideal Brouwer degree Gröbner basis## Notes

### Acknowledgements

The authors thank Frank Sottile for helpful discussions, Alan Rendall for pointing us to the Calvin Cycle model, and Carsten Conradi for helpful discussions on the ERK network. The authors also thank three conscientious referees whose comments helped improve our work. AD and MPM were partially supported by UBACYT 20020170100048BA, CONICET PIP 11220150100473 and 11220150100483, and ANPCyT PICT 2016-0398, Argentina. AS partially supported by the NSF (DMS-1513364 and DMS-1752672) and the Simons Foundation (#521874). XT was partially supported by the NSF (DMS-1752672).

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