# American Institute of Mathematical Sciences

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June  2019, 24(6): 2683-2700. doi: 10.3934/dcdsb.2018270

## Nondegenerate multistationarity in small reaction networks

 1 Texas A&M University, Department of Mathematics, Mailstop 3368, College Station, TX 77843-3368, USA 2 Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany

* Corresponding author: Anne Shiu

Received  January 2018 Revised  May 2018 Published  October 2018

Much attention has been focused in recent years on the following algebraic problem arising from applications: which chemical reaction networks, when taken with mass-action kinetics, admit multiple positive steady states? The interest behind this question is in steady states that are stable. As a step toward this difficult question, here we address the question of multiple nondegenerate positive steady states. Mathematically, this asks whether certain families of parametrized, real, sparse polynomial systems ever admit multiple positive real roots that are simple. Our main results settle this problem for certain types of small networks, and our techniques may point the way forward for larger networks.

Citation: Anne Shiu, Timo de Wolff. Nondegenerate multistationarity in small reaction networks. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2683-2700. doi: 10.3934/dcdsb.2018270
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##### References:
Stoichiometric compatibility classes for the network in Example 2.2.
Summary of results on nondegenerate multistationarity for small reactions. Here r denotes the number of reactions and s the number of species. See Section 2.
 Network property Nondegenerately multistationary? Network with only 1 species ($s=1$) If and only if some subnetwork is 2-alternating (Proposition 1.1) [17] Network consists of 1 reaction ($r=1$) or 1 reversible-reaction pair No (Proposition 1.2) [17] Network consists of 2 reactions ($r=2$) See Proposition 1.3 [17] $r+s \leq 3$ No ([17,Corollary 3.8]) $s=2$ and 1 irreversible reaction and 1 reversible-reaction pair See Theorem 3.5 $s=2$ and 2 reversible-reaction pairs See Theorem 3.6
 Network property Nondegenerately multistationary? Network with only 1 species ($s=1$) If and only if some subnetwork is 2-alternating (Proposition 1.1) [17] Network consists of 1 reaction ($r=1$) or 1 reversible-reaction pair No (Proposition 1.2) [17] Network consists of 2 reactions ($r=2$) See Proposition 1.3 [17] $r+s \leq 3$ No ([17,Corollary 3.8]) $s=2$ and 1 irreversible reaction and 1 reversible-reaction pair See Theorem 3.5 $s=2$ and 2 reversible-reaction pairs See Theorem 3.6

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