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Nondegenerate multistationarity in small reaction networks

  • * Corresponding author: Anne Shiu

    * Corresponding author: Anne Shiu 
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  • 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.

    Mathematics Subject Classification: Primary: 12D10, 80A30; Secondary: 37C10, 37C25, 14P05 34A34, 65H04.

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  • Figure 1.  Stoichiometric compatibility classes for the network in Example 2.2.

    Table 1.  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 propertyNondegenerately 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 pairNo (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 pairSee Theorem 3.5
    $s=2$ and 2 reversible-reaction pairsSee Theorem 3.6
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