Proof. Assume $(G,\kappa ^*)$ is a full-dimensional mass-action system in $n$ species, and let $N$ be as in (7).

First, we prove (1). Assume $\operatorname{rank}(N) \leq n-1$ , and let $x^*$ be a positive steady state. It follows that the polynomials $f_i$ , as in (1), are linearly dependent (over $\mathbb R$ ). Hence, the Jacobian matrix – even before evaluating at $x^*$ – has rank less than $n.$ Thus, the image of the Jacobian matrix, after evaluating at $x^*$ , has dimension less than $n$ , i.e. $\text{Im}(df(x^*)|_S)\neq \mathbb{R}^n =S$ . Hence, $x^*$ is degenerate.

Next, we prove (2). As in equation (7), we write the mass-action ODEs for $(G,\kappa ^*)$ as

\begin{align*} \begin{bmatrix} dx_1/dt \\[5pt] dx_2/dt \\[5pt] \vdots \\[5pt] dx_n/dt \\[5pt] \end{bmatrix} = N \begin{bmatrix} m_1 \\[5pt] \vdots \\[5pt] m_n \\[5pt] m_{n+1}\\[5pt] \end{bmatrix}, \end{align*}

where $N$ is $n \times (n+1)$ and the $m_i$ ’s are distinct monic monomials in $x_1,x_2, \ldots,x_n$ .

As $G$ is full-dimensional and $\operatorname{rank}(N) =n$ , we can relabel the $m_i$ ’s, if needed, so that the square sub-matrix of $N$ formed by the first $n$ columns has rank $n$ . Thus, by row-reducing $N$ , we obtain a matrix of the following form (where $\beta _1, \beta _2, \ldots,\beta _n \in \mathbb{R}$ ):

\begin{align*} N^{\prime} \,:\!=\, \left [\begin{array}{c@{\quad}c@{\quad}c|c} & & & -\beta _1\\[5pt] &{I_n} & & -\beta _2 \\[5pt] & & & \vdots \\[5pt] & & & -\beta _n\\[5pt] \end{array}\right ]. \end{align*}

We conclude from the above discussion that the positive steady states of $(G,\kappa ^*)$ are the positive roots of the following $n$ binomial equations (which are in the desired form):

(8) \begin{align} m_i - \beta _i m_{n+1} = 0 \quad \quad \textrm{for\,}i=1,2,\ldots, n. \end{align}

Before moving on to part (3), we summarise what we know (so we can use it later). The positive steady states are the roots of the binomials (8), which we rewrite using Laurent monomials (our interest is in positive roots, so there is no issue of dividing by zero):

(9) \begin{align} x_1^{a_{i1}} x_2^{a_{i2}} \dots x_n^{a_{in}} \,:\!=\, \frac{m_i}{m_{n+1}} = \beta _i \quad \quad \textrm{for\,}i=1,2,\ldots, n. \end{align}

We apply the natural $\log$ to (9) and obtain the following, which involves the $n \times n$ matrix $A\,:\!=\,(a_{ij})$ :

(10) \begin{align} A \begin{pmatrix} \ln (x_1) \\[5pt] \ln (x_2) \\[5pt] \vdots \\[5pt] \ln (x_n) \end{pmatrix} = \begin{pmatrix} \ln (\beta _1) \\[5pt] \ln (\beta _2) \\[5pt] \vdots \\[5pt] \ln (\beta _n) \end{pmatrix} \,=\!:\, \ln (\beta ). \end{align}

Now we prove (3). Assume $x^*$ is a nondegenerate, positive steady state. (We must show that no other positive steady states exist.) By part (1), the $n \times (n+1)$ matrix $N$ has rank $n$ , so the proof of part (2) above applies. Assume for contradiction that $x^{**}$ is a positive steady state, with $x^{**} \neq x^*$ . Then, by (10), the linear system $Ay=\ln (\beta )$ has more than one solution, and so $\textrm{rank}(A) \leq n-1$ . It follows that the set of positive steady states, $\left\{\left(e^{y_1}, e^{y_2}, \dots, e^{y_n}\right) \mid Ay=\ln (\beta )\right\}$ , is positive-dimensional and so (by the Inverse Function Theorem and the fact that $G$ is full-dimensional) all positive steady states of $(G,\kappa ^*)$ are degenerate. This is a contradiction, as $x^*$ is nondegenerate.

Remark 3.2. For algebraically inclined readers, observe that the equations in Proposition 3.1(2) define a toric variety. Additionally, every such variety has at most one irreducible component that intersects the positive orthant [Reference Conradi, Iosif and Kahle8, Proposition 5.2]. This fact can be used to give a more direct proof of Proposition 3.1(3).

Remark 3.3. The end of the proof of Proposition 3.1 concerns nondegenerate positive steady states and their relation to the dimension of the set of positive steady states. More ideas in this direction are explored in the recent work of Feliu, Henriksson, and Pascual-Escudero [Reference Feliu, Henriksson and Pascual-Escudero14].

Proof. This result follows directly from Proposition 3.1(1) and the fact that, in this case, the rank of $N$ , as in (7), is strictly less than $n$ .

Proof. Assume $G$ has $n$ species, which we denote by $X_1, X_2, \dots, X_n$ , with exactly $j$ reactant complexes, for some $1\leq j \leq n$ . Let $\kappa ^*$ be a vector of positive rate constants. As in (7), we write the mass-action ODE system arising from $(G, \kappa ^*)$ as follows:

(11) \begin{align} \begin{bmatrix} dx_1/dt \\[5pt] \vdots \\[5pt] dx_n/dt \\[5pt] \end{bmatrix} = N \begin{bmatrix} m_1 \\[5pt] \vdots \\[5pt] m_j \\[5pt] \end{bmatrix}\, \,=\!:\, \begin{bmatrix} f_1 \\[5pt] \vdots \\[5pt] f_n \\[5pt] \end{bmatrix}, \end{align}

where $N\,:\!=\,(N_{ij})$ is an $(n \times j)$ -matrix (with entries in $\mathbb R$ ) and $m_1, \dots, m_j$ are distinct monic monomials in $x_1, \dots, x_n$ (as $G$ has $n$ species and $j$ reactant complexes).

We first prove part (1). In this case, the right-hand sides of the ODEs have the form $f_i = c_i \prod _{k=1}^{n}x_k^{a_k}$ , with at least one $c_i\neq 0$ . It follows that there are no positive steady states.

We prove part (2). Assume that $G$ is full-dimensional (the stoichiometric subspace is $\mathbb{R}^n$ ) and that $2 \leq j \leq n$ . Let $x^*=(x_1^*,x_2^*,\ldots,x_n^*)$ be a positive steady state. We must show $x^*$ is degenerate.

We first consider the subcase when the rank of the matrix $N$ is at most $(n-1)$ . By Proposition 3.1(1), every positive steady state is degenerate.

Now we handle the remaining subcase, when $N$ has rank $n$ (and hence, $N$ is $n\times n$ ). Now, solving the steady-state equations $f_1= \dots = f_n=0$ can be accomplished by multiplying the expression in (11) by $N^{-1}$ , which implies that every monomial $m_1,\dots, m_n$ evaluates to zero at steady state. Hence, no positive steady states exist.

Proof. Assume $G$ is full-dimensional, with $n$ species, $r$ reactions (denoted by $y_1 \to y^{\prime}_1, \dots y_r \to y^{\prime}_r$ ), and exactly $j$ reactant complexes, where $j \geq n$ .

We begin with part (1). Let $\kappa =(\kappa _1,\dots, \kappa _r)$ denote the vector of unknown rate constants (each $\kappa _i$ is a variable). Let $\widetilde N$ be the $(n \times j)$ matrix for $(G,\kappa )$ in the sense of $N$ in (7). More precisely, the entries of $\widetilde N$ are $\mathbb Z$ -linear combinations of the $\kappa _i$ ’s, such that, for every vector of positive rate constants $\kappa ^* \in \mathbb R_{\gt 0}^r$ , the evaluation $\widetilde N|_{\kappa =\kappa ^*}$ is the matrix $N$ as in (7) for $(G, \kappa ^*)$ .

As $G$ is full-dimensional, there are no $\mathbb{R}$ -linear relations among the $n$ rows of $\widetilde N$ . Hence, the size- $n$ minors of $\widetilde N$ define a (possibly empty) measure-zero subset $V \subseteq \mathbb R^r_{\gt 0}$ . Thus, $\mathbb R^r_{\gt 0} \smallsetminus V$ is nonempty, and every $\kappa ^* \in \mathbb R^r_{\gt 0} \smallsetminus V$ yields a matrix $N=\widetilde N|_{\kappa =\kappa ^*}$ with rank $n$ . This proves part (1).

For part (2), suppose that there exists $\kappa ^* \in \mathbb R_{\gt 0}^r$ such that the resulting matrix $N$ has rank strictly less than $n$ . It follows that there is a linear relation:

(12) \begin{align} c_1f_{\kappa ^*,1}+\dots + c_nf_{\kappa ^*,n} = 0, \end{align}

where $c_1, \dots, c_n$ are real numbers – not all $0$ – and the $f_{\kappa ^*,i}$ denote the right-hand sides of the mass-action ODEs for $(G,\kappa ^*)$ .

On the other hand, for unknown rate constants $\kappa$ , as in the proof above for part (1), $c_1f_{\kappa,1}+\dots + c_nf_{\kappa,n}$ is not the zero polynomial. Thus, when we rewrite this expression as a sum over $r$ reactions $y_i \to y^{\prime}_i$ as follows: $c_1f_{\kappa,1}+\dots + c_nf_{\kappa,n} = d_1 \kappa _1 x^{y_1} + \dots + d_r \kappa _r x^{y_r}$ , where $d_i \in \mathbb{Z}$ for all $i$ , we conclude that $d_i \neq 0$ for some $i$ . By relabelling reactions, if needed, we may assume that $i=1$ .

Now consider the following vector of positive rate constants $ \kappa _\epsilon ^* \,:\!=\, \left(\kappa _1^* + \epsilon, \kappa _2^*, \dots, \kappa _r^*\right)$ , for some $\epsilon \gt 0$ . Assume for contradiction that $(G,\kappa _\epsilon ^*)$ has a positive steady state $x^*$ . At steady state, $f_{\kappa ^*_\epsilon,i}$ evaluates to $0$ , for all $i$ , and this yields the first equality here:

\begin{equation*} 0 = \left ( c_1f_{\kappa ^*_\epsilon,1}+\dots + c_nf_{\kappa ^*_\epsilon,n} \right ) |_{x=x^*} = c_1f_{\kappa ^*,1}|_{x=x^*}+\dots + c_nf_{\kappa ^*,n}|_{x=x^*} + \epsilon d_1 x^{y_1}|_{x=x^*} = \epsilon d_1 x^{y_1}|_{x=x^*}, \end{equation*}

and the second and third equalities come from the fact that the mass-action ODEs are linear in the rate constants and from equation (12), respectively. We obtain $x^{y_1}|_{x=x^*}=0$ , which contradicts the fact that $x^*$ is a positive steady state. This concludes the proof.

Proof. Case 1: $\operatorname{rank}(A) = n$ . Fix an arbitrary vector of positive rate constants $\kappa ^*$ . We must show that $(G, \kappa ^*)$ is not nondegenerately multistationary. Let $N$ denote the $n \times (n+1)$ matrix defined by $(G, \kappa ^*)$ , as in (7). We consider two subcases.

Subcase: $\operatorname{rank}(N) \leq n-1$ . In this subcase, Proposition 3.1(1) implies that every positive steady state of $(G, \kappa ^*)$ is degenerate, and so $(G, \kappa ^*)$ is not nondegenerately multistationary.

Subcase: $\operatorname{rank}(N) = n$ . Part (2) of Proposition 3.1 pertains to this setting, so we can follow that proof. In particular, equation (9) – the $(n \times n)$ matrix $A$ there exactly matches the matrix $A$ here – implies that the positive steady states are defined by a linear system of the form $Ay=\ln (\beta )$ , where $y=(\!\ln (x_1), \dots, \ln (x_n))^{\top }$ . Hence, as $\operatorname{rank}(A) = n$ , we have at most one positive steady state and so $(G, \kappa ^*)$ is not multistationary.

Case 2: $\operatorname{rank}(A) \leq n-1$ . We must show that there exists a choice of rate constants so that the resulting system has no positive steady states.

Proposition 3.6(1) implies that there exists $\kappa ^*$ such that the following holds:

(13) \begin{align} \text{the matrix $N$ defined by $(G,\kappa ^*)$ has (full) rank $n$.} \end{align}

Fix such a choice of $\kappa ^*$ . If $(G, \kappa ^*)$ has no positive steady states, then we are done. Therefore, for the rest of the proof, we assume that $(G, \kappa ^*)$ admits a positive steady state.

In what follows, we need to consider additional vectors of positive rate constants (besides $\kappa ^*$ ) and their corresponding matrices $N$ , as in (7). Therefore, as in the proof of Proposition 3.6(1), let $\kappa =(\kappa _1,\dots, \kappa _r)$ (where $r$ is the number of reactions) denote the vector of unknown rate constants, and let $\widetilde N$ be the $n \times (n+1)$ matrix for $(G,\kappa )$ in the sense of $N$ in (7), so that for every vector of positive rate constants $\kappa ^* \in \mathbb R_{\gt 0}^r$ , the evaluation $\widetilde N|_{\kappa =\kappa ^*}$ is the matrix $N$ as in (7).

We now follow the ideas in the proof of Proposition 3.1, part (2), with the difference being that we now consider unknown rate constants $\kappa$ . The mass-action ODEs for $(G,\kappa )$ are given by:

\begin{align*} \begin{bmatrix} dx_1/dt \\[5pt] \vdots \\[5pt] dx_n/dt \\[5pt] \end{bmatrix} = \widetilde N \begin{bmatrix} m_1 \\[5pt] \vdots \\[5pt] m_{n+1}\\[5pt] \end{bmatrix}, \end{align*}

where $m_1,\dots, m_{n+1}$ are distinct monic monomials in $x_1,x_2, \ldots,x_n$ .

Our next aim is to row-reduce $\widetilde N$ (over the field $\mathbb{Q}(\kappa _1,\dots, \kappa _r)$ ). Accordingly, for $1\leq k \leq n+1$ , let $[B_k]$ denote the determinant of the matrix obtained from $\widetilde N$ by removing the $k$ th column. By construction, each $[B_k]$ is in $\mathbb{Z}[\kappa _1,\dots, \kappa _r]$ .

We claim that, for all $1 \leq k \leq n+1$ , the polynomial $[B_k]$ is nonzero. By symmetry among the monomials $m_i$ , it suffices to show that $[B_{n+1}]$ is nonzero. To show this, assume for contradiction that $[B_{n+1}]=0$ . Then, $\widetilde N$ can be row-reduced to a matrix in which the last row has the form $(0,0,\dots, 0, \omega )$ , where $0 \neq \omega \in \mathbb{Q}(\kappa _1,\dots, \kappa _r)$ . Now consider the evaluation at $\kappa =\kappa ^*$ . By (13), the matrix $N = \widetilde N|_{\kappa =\kappa ^*}$ has (full) rank $n$ , so $\omega |_{\kappa =\kappa ^*}$ is nonzero. However, this implies that positive steady states of $(G,\kappa ^*)$ satisfy $\omega |_{\kappa =\kappa ^*} m_{n+1} = 0$ , much like in (8). Thus, $(G,\kappa ^*)$ has no positive steady states, which is a contradiction, and hence our claim holds.

Next, as $[B_{n+1}]$ is nonzero, we can apply a version of Cramer’s rule to row-reduce $\widetilde N$ to the following matrix (where $I_n$ denotes the size- $n$ identity matrix):

\begin{align*}{\widetilde N}^{\prime} = \left [ \begin{array}{c@{\quad}c@{\quad}c|c} & & & ({-}1)^{n-1} \tfrac{[B_1]}{[B_{n+1}]} \\[9pt] & I_n & & ({-}1)^{n-2} \tfrac{[B_2]}{[B_{n+1}]} \\[5pt] & & & \vdots \\[5pt] & & & ({-}1)^{0} \tfrac{[B_n]}{[B_{n+1}]}\\[5pt] \end{array} \right ]. \end{align*}

Thus, as in (8), the positive steady states are the positive roots of the equations $m_i - \beta _i m_{n+1}=0$ (for $i=1,2,\dots,n$ ), where:

\begin{align*} \beta _i \,:\!=\, ({-}1)^{n-i+1} \frac{[B_i]}{[B_{n+1}]} \quad \textrm{for}\, i=1,2,\dots,n. \end{align*}

Thus, $\beta _i|_{\kappa = \kappa ^*} \gt 0$ (for all $i=1,2,\dots,n$ ), since $(G,\kappa ^*)$ admits a positive steady state. We conclude from this fact, plus the claim proven earlier (namely, that $[B_\ell ] \neq 0$ for all $\ell$ ), that the following is an open subset of $\mathbb{R}^r_{\gt 0}$ that contains $\kappa ^*$ :

\begin{align*} \Sigma \,:\!=\, \left \{ \bar \kappa \in \mathbb{R}^r_{\gt 0} \,:\, \beta _1|_{\kappa = \bar \kappa } \gt 0,\,\dots,\, \beta _n|_{\kappa = \bar \kappa } \gt 0,\, [B_1]|_{\kappa =\bar \kappa } \neq 0, \dots, [B_{n+1}]|_{\kappa = \bar \kappa } \neq 0 \right \}. \end{align*}

For the rest of the proof, we restrict our attention to rate constants, like $\kappa ^*$ , that are in $\Sigma$ . For such rate constants, like in (8)–(10), the positive steady states are the roots of the following equation

(14) \begin{align} A \begin{pmatrix} \ln (x_1) \\[5pt] \ln (x_2) \\[5pt] \vdots \\[5pt] \ln (x_n) \end{pmatrix} = \begin{pmatrix} \ln (\beta _1) \\[5pt] \ln (\beta _2) \\[5pt] \vdots \\[5pt] \ln (\beta _n) \end{pmatrix} \,=\!:\, \ln (\beta ). \end{align}

Next, as $\operatorname{rank}(A) \leq n-1$ , there exists a nonzero vector $\gamma \in \mathbb{R}^n$ in the orthogonal complement of the column space of $A$ . By relabelling the $m_i$ ’s (which permutes the columns of $\widetilde N$ ), if needed, we may assume that $\gamma _1 \neq 0$ . By construction of $\gamma$ and equation (14), we have $\langle \gamma,\, \ln (\beta ) \rangle = 0$ , which is readily rewritten as follows:

(15) \begin{align} \left ( \frac{[B_1]}{[B_{n+1}]}\right )^{\gamma _1} \dots \left (({-}1)^{k+1} \frac{[B_k]}{ [B_{n+1}] }\right )^{\gamma _k} \dots \left (({-}1)^{n+1} \frac{ [B_n]}{ [B_{n+1}] }\right )^{\gamma _n} = 1. \end{align}

For $\varepsilon \gt 0$ , let $\kappa ^*_{\epsilon }$ denote the vector of rate constants obtained from $\kappa ^*$ by scaling by $(1+ \varepsilon )$ all rate constants of reactions in which the reactant generates the monomial $m_1$ . As $\Sigma$ is an open set, $\kappa ^*_{\epsilon }\in \Sigma$ for $\varepsilon$ sufficiently small. Also, by construction, the matrix $\widetilde N |_{\kappa = \kappa ^*_{\varepsilon }}$ is obtained from $\widetilde N |_{\kappa = \kappa ^*}$ by scaling the first column by $(1+ \varepsilon )$ . So, for $2 \leq i \leq n+1$ , we have $[B_i] |_{\kappa = \kappa ^*_{\varepsilon }} = (1+ \epsilon ) [B_i] |_{\kappa = \kappa ^*}$ .

Thus, by replacing $\kappa ^*$ by $\kappa ^*_{\varepsilon }$ , the left-hand side of equation (15) is scaled by $(1+\varepsilon )^{-\gamma _1}$ , and so there exists $\varepsilon \gt 0$ for which equation (15) does not hold (when evaluated at $\kappa = \kappa ^*_{\varepsilon }$ ). Hence, this vector $\kappa ^*_{\varepsilon }$ yields a mass-action system $(G,\kappa ^*_{\varepsilon })$ with no positive steady states, as desired.

Proof. We prove the contrapositive. Assume that there is a species $X_i$ such that for every reactant complex $a_1X_1+ a_2 X_2 + \dots +a_nX_n$ we have $a_i = 0$ . Then, by Lemma 2.3, the right-hand side of the mass-action ODE for $X_i$ , which we denote by $f_i$ , is a sum of monomials, all of which have positive coefficients. But $(G,\kappa ^*)$ has a positive steady state, so $f_i$ must be $0$ . We conclude that the $i$ th row (of the $n$ rows) of $N$ is the zero row and so $\operatorname{rank}(N) \leq n-1$ .

Proof. We mimic the proofs of Propositions 3.1(1) and 3.5. Let $\kappa ^*$ be a vector of positive rate constants. Let $N$ be an $(n \times j)$ matrix defined, as in (7), by $(G,\kappa ^*)$ :

(16) \begin{align} \begin{bmatrix} dx_1/dt \\[5pt] \vdots \\[5pt] dx_n/dt \\[5pt] \end{bmatrix} = N \begin{bmatrix} m_1 \\[5pt] \vdots \\[5pt] m_j \\[5pt] \end{bmatrix}\, \,=\!:\, \begin{bmatrix} f_1 \\[5pt] \vdots \\[5pt] f_n \\[5pt] \end{bmatrix}, \end{align}

where $m_1, \dots, m_j$ are distinct monic monomials in $x_1, \dots, x_n$ .

We consider two cases. First assume that $\operatorname{rank}(N) \leq n-k-1$ . Then, the polynomials $f_i$ span a subspace of dimension $\leq n-k-1$ and hence the Jacobian matrix – even before evaluating at a positive steady state – has rank $\leq n-k-1$ . Every positive steady state is therefore degenerate.

Consider the remaining case: $\operatorname{rank}(N) = n-k$ (so, $j=n-k$ ). In this case, multiplication by $N$ defines an injective map $\mathbb{R}^{n-k} \to \mathbb{R}^n$ . Hence, by (16), the steady-state equations $f_1= \dots = f_{n}=0$ imply the monomial equations $m_1=\dots = m_{j}=0$ . Thus, there are no positive steady states.

Proof. Assume that $G$ is one-dimensional, with $n$ species. Thus, $G$ has $n-1$ linearly independent conservation laws. Let $\kappa ^*$ be a vector of positive rate constants for which there is ACR. We may assume that the ACR species is $X_1$ (by relabelling species, if needed). Let $f_1,\dots, f_n$ denote the right-hand sides of the mass-action ODEs arising from $(G,\kappa ^*)$ .

Let $x^*=(x_1^*, \dots, x_n^*)$ denote an arbitrary positive steady state of $(G,\kappa ^*)$ . (The ACR value is $x^*_1$ .) Let $P_{x^*}$ denote the (one-dimensional) stoichiometric compatibility class that contains $x^*$ . It suffices to show that (1) $x^*$ is the unique positive steady state in $P_{x^*}$ or (2) $x^*$ is degenerate.

We consider two cases.

Case (a): $X_1$ is not a catalyst-only species (in some reaction of $G$ ). This implies that $f_2,\dots, f_n$ are all scalar multiples of $f_1$ , and that the compatibility class $P_{x^*}$ is defined by $n-1$ conservation laws of the form $x_j=a_j x_1+b_j$ , where $a_j,b_j \in \mathbb{R}$ , for $j\in \{2,3,\dots, n\}$ . By substituting these $n-1$ relations into $f_1$ , we obtain a univariate polynomial in $x_1$ , which we denote by $h$ . If $h$ has multiple positive roots, then there is no ACR, which is a contradiction. If, on the other hand, $h$ does not have multiple positive roots, then $P_{x^*}$ does not contain multiple positive steady states (that is, $x^*$ is the unique positive steady state in $P_{x^*}$ ).

Case (b): $X_1$ is a catalyst-only species in all reactions of $G$ . In this case, $f_1=0$ , and $x_1=x_1^*$ is a conservation law of $G$ , and it is one of the defining equations of the compatibility class $P_{x^*}$ . By relabelling species $X_2,\dots, X_n$ , if needed, we may assume that $X_2$ is not a catalyst-only species (as $G$ is one-dimensional). Thus, we can “extend” the conservation law $x_1=T$ to a “basis” of $n-1$ conservation laws that define the compatibility class $P_{x^*}$ , by appending $n-2$ conservation laws of the form $x_j=a_jx_2+b_j$ , where $a_j,b_j \in \mathbb{R}$ , for $j\in \{3,4,\dots, n\}$ .

Next, we substitute these $n-2$ conservation relations into $f_2$ , which yields a polynomial in $x_1$ and $x_2$ , which we denote by $g$ . Consider the following set, which is the positive variety of $g$ in $\mathbb{R}^2_{\gt 0}$ (the values of $x_3,\dots, x_n$ are free, so we ignore them):

(17) \begin{align} \Sigma \,:\!=\, \left\{x\in \mathbb{R}^2_{\gt 0} \mid g(x_1,x_2)=0 \right\}. \end{align}

By construction and the fact that there is ACR in $X_1$ , the set $\Sigma$ is contained in the hyperplane (line) $x_1=x_1^*$ , and so is either one-dimensional or zero-dimensional. We consider these two subcases separately. First, assume that $\Sigma$ is one-dimensional. In this subcase, $\Sigma$ equals the subset of the hyperplane $x_1=x_1^*$ in the positive quadrant $\mathbb{R}^2_{\gt 0}$ , and so the compatibility class $P_{x^*}$ consists entirely of positive steady states. The Inverse Function Theorem now implies that every positive steady state of $P_{x^*}$ (in particular, $x^*$ ) is degenerate.

Consider the remaining subcase, in which $\Sigma$ is zero-dimensional (that is, $\Sigma$ consists of finitely many points). It follows that $g$ is either non-negative on $\mathbb{R}^2_{\gt 0}$ or non-positive on $\mathbb{R}^2_{\gt 0}$ , and so $f_2$ is either non-negative on $P_{x^*}$ or non-positive on $P_{x^*}$ . Consequently, as every $f_i$ is a scalar multiple of $f_2$ , the steady state $x^*$ is degenerate.

Proof. Let $g_i$ denote the nonzero polynomial obtained by evaluating $f_i$ at $x_j=a_j$ for all $j\neq i$ . Several properties of $g_i$ arise from the fact that $G$ is bimolecular: (1) $\text{deg}(g_i) \leq 2$ , (2) the coefficient of $x_i^2$ is non-positive, and (3) the constant coefficient is non-negative. Thus, $g_i$ has at most one sign change, and so Descartes’ rule of signs implies that $g_i$ has at most one positive root.

Proof. Let $\kappa ^*$ be a vector of positive rate constants for a bimolecular network $G$ with $n$ species, and let $f_i$ be the right-hand side of $(G,\kappa ^*)$ for the species $X_i$ . Let $\Sigma$ denote the set of reactions of $G$ in which $X_i$ is a non-catalyst-only species. Reactions not in $\Sigma$ do not contribute to $f_i$ , so we ignore them for the rest of the proof.

We claim that for all reactions in $\Sigma$ , the reactant complex is not one of the following five types: $0$ , $X_j$ , $X_j + X_{j^{\prime}}$ , $2X_j$ , $2X_i$ for any $j,j^{\prime} \in \{1,2,\dots, n \} \smallsetminus \{i\}$ . Indeed, any of the first four types of complexes would yield a constant term in $f_i$ (when viewed as a polynomial in $x_i$ ) consisting of a sum of monomials with positive coefficients; similarly, the last type ( $2X_i$ ) would yield a negative $x_i^2$ term (the fact that $G$ is bimolecular is used here). However, $f_i$ is zero, so the claim holds.

It follows that, for every reaction in $\Sigma$ , the reactant complex either is $X_i$ or has the form $X_i+X_j$ for some $j \in \{1,2,\dots, n \} \smallsetminus \{i\}$ . It is straightforward to check that all possible such reactions (in which $X_i$ is a non-catalyst-only species) are listed in the proposition. Next, reactions of type (1) in the proposition contribute positively to $f_i$ , while those of type (2) contribute negatively, as follows:

(19) \begin{align} f_i = \left ( \kappa ^*_{1,0} - \sum _\ell \kappa ^*_{2, 0, \ell } \right ) x_i \,+ \sum _{j\in \{1,2,\dots, n \} \smallsetminus \{i\}} \left ( \kappa ^*_{1,j} - \sum _\ell \kappa ^*_{2,j,\ell } \right ) x_i x_j. \end{align}

As $f_i=0$ , the coefficient of $x_i$ and the coefficient of each $x_{ij}$ in (19) must be $0$ , which yields the desired equalities (18).

Proof. Assume that $f_i$ is nonzero, but $f_i|_{x_1=\alpha }$ is zero. Using properties of polynomial rings over a field, it follows that $(x_1- \alpha )$ divides $f_i$ . From the fact that $G$ is bimolecular, we conclude that:

(22) \begin{align} \notag f_i &= (x_1 - \alpha ) \left ( \beta x_i + \gamma + \sum _{j \in [n] \smallsetminus \{i\}} \delta _j x_j \right ) \\[5pt] &=\, \beta x_1 x_i + \gamma x_1 + \left ( \sum _{j \in [n] \smallsetminus \{i\}} \delta _j x_1 x_j \right ) -\alpha \beta x_i -\alpha \gamma - \left ( \sum _{j \in [n] \smallsetminus \{i\}} \alpha \delta _j x_j \right ), \end{align}

for some real numbers $\beta, \gamma, \delta _j$ , at least one of which is nonzero.

In the right-hand side of (22), the variable $x_i$ does not appear in any of the following monomials (here the hypothesis $i\neq 1$ is used):

\begin{equation*} \gamma x_1, \quad -\alpha \gamma, \quad \delta _j x_1 x_j, \quad -\alpha \delta _j x_j, \end{equation*}

for $ j \in \{1,2,\dots,n\} \smallsetminus \{i\}$ , so Lemma 2.3 implies that the coefficients of these monomials must be non-negative. Since $\alpha \gt 0$ , we conclude that $\gamma =0$ and $\delta _j=0$ (for all $j \in \{1,2,\dots,n\} \smallsetminus \{i\}$ ).

Thus, using (22), we have $f_i= \beta x_1 x_i - \alpha \beta x_i$ , for some $\beta \in \mathbb{R} \smallsetminus \{0\}$ . Next, we investigate which reactions contribute to the two monomials in $f_i$ . For $ \beta x_1 x_i$ , the contributing reactions have the form $X_1+X_i\to 2X_i$ and $X_1 + X_i \to \star$ , where $\star$ does not involve $X_i$ . The first reaction contributes positively, while the second type contributes negatively. Let $\kappa ^*_{1,1}$ be the reaction rate constant for $X_1+X_i\to 2X_i$ (as in the statement of the lemma) and $\kappa ^*_{2,1,\ell }$ be the rate constant for reactions of type $X_1 + X_i \to \star$ , where $\ell$ is an index for all the reactions of this type. We conclude that

(23) \begin{align} \kappa ^*_{1,1} - \sum _\ell \kappa ^*_{2,1,\ell } \,= \, \beta. \end{align}

Similarly, the monomial $-\alpha \beta x_i$ in $f_i$ comes from reactions of the form $X_i \to 2X_i$ , which contributes positively, and $X_i \to \star$ , which contributes negatively, where $\star$ is a complex that does not involve $X_i$ . Hence,

(24) \begin{align} \kappa ^*_{1,0} - \sum _\ell \kappa ^*_{2,0,\ell } = - \alpha \beta. \end{align}

Now the equations (23) and (24) together imply the desired equality (20).

Next, let $\Sigma$ denote the set of reactions of $G$ in which $X_i$ is a non-catalyst-only species. We showed above that $\Sigma$ contains a (nonempty) subset of reactions with rate constants labelled by $\kappa ^*_{1,0}$ , $\kappa ^*_{1,1}$ , $\kappa ^*_{2,0,\ell }$ , $\kappa ^*_{2,1,\ell }$ . Let $\Sigma ^{\prime} \subseteq \Sigma$ denote the remaining reactions, and let $G^{\prime}$ denote the subnetwork defined by the reactions in $\Sigma ^{\prime}$ . Let $\kappa ^{\prime}$ be obtained from $\kappa ^*$ by restricting to coordinates corresponding to reactions in $\Sigma ^{\prime}$ . By construction, the mass-action ODE of $(G^{\prime}, \kappa ^{\prime})$ for species $X_i$ has the right-hand side equal to $0$ . So, Proposition 3.14 applies (where reactions arising from $j=0,1$ in that proposition are absent from $G^{\prime}$ by construction), and yields two conclusions. First, $\Sigma ^{\prime}$ is a subset of the reactions listed in Proposition 3.15 (specifically, with $j \not = 0,1$ ), and so $\Sigma$ is a subset of the full list (including $j = 0,1$ ). Second, the equations (18) hold (for $j \not = 0,1$ ), which are the desired equalities (21).

Remark 3.16. The reactions listed in Propositions 3.14 and 3.15 (the lists are the same) are not reversible. Hence, if $G$ is a reversible network satisfying the hypotheses of either proposition, then $X_i$ is a catalyst-only species in every reaction of $G$ .

Remark 3.17. The formula for the ACR value, in (20), is related to the concept of “robust ratio” introduced by Johnston and Tonello [Reference Tonello and Johnston32].

Proof. Let $G$ be a one-dimensional, bimolecular reaction network. Up to relabelling species, the one-dimensional stoichiometric subspace is spanned by one of the following seven vectors:

(25) \begin{equation} (1,0,0,\dots, 0), (1,-1,0,0,\dots, 0), \end{equation}

(26) \begin{align} & (1,1,0,0,\dots, 0), (1,-2, 0,0,\dots, 0), (1,1,-1,0,0,\dots, 0), (1,1,-2, 0,0,\dots, 0),\nonumber\\& (1,1,-1,-1, 0,0,\dots, 0). \end{align}

We first consider the case when the stoichiometric subspace is spanned by one of the five vectors listed in (26). The network $G$ is then a subnetwork of one of the following networks (where we use $A,B,C,D$ in place of $X_1,X_2,X_3,X_4$ for ease of notation);

\begin{align*} \{0 \leftrightarrows A+B\},\, \{A \leftrightarrows 2B\},\, \{A+B \leftrightarrows C\},\, \{A+B \leftrightarrows 2C\},\, \{A+B \leftrightarrows C+D\}. \end{align*}

A direct calculation shows that the deficiency of $G$ is 0, so the deficiency-zero theorem (Lemma 2.5) implies that $G$ is not multistationary. In particular, $\textrm{cap}_{\text{pos}}(G) \lt \infty$ .

Having shown that the case of (26) is consistent with Lemma 3.19, we now consider the remaining two cases, from (25), separately. First, assume the stoichiometric subspace of $G$ is spanned by $(1,0,0,\dots, 0)$ . It follows that the reactions of $G$ form a subset of the following $2n+4$ reactions:

\begin{align*} &0 \mathop{\leftrightarrows }_{k_0}^{m_0} X_1 \mathop{\leftrightarrows }_{m_1}^{\ell _0} 2X_1 \quad \quad 0 \mathop{\leftrightarrows }_{k_1}^{\ell _1} 2X_1 \\[5pt] & X_i \mathop{\leftrightarrows }_{k_i}^{m_i} X_1 + X_i \quad \text{for } i=2,3,\dots, n. \end{align*}

The ODEs for species $X_2, X_3, \dots, X_n$ are $\frac{dx_i}{dt}=0$ so, $x_i=T_i$ (with $T_i\gt 0$ ) for $i=2,3,\dots,n$ are the corresponding conservation laws. We substitute these conservation laws into the ODE for $X_1$ :

(27) \begin{align} \frac{d x_1}{dt}|_{x_2=T_2,\dots, x_n=T_n} = & (k_0 + 2 k_1) + (k_2T_2 + \dots + k_nT_n) +m_1 x_1\\[5pt] \notag & - (m_0+ m_2T_2 + \dots + m_nT_n) x_1 - (\ell _0 + 2 \ell _1) x_1^2. \end{align}

When at least one $k_i$ is positive and all other $k_j$ ’s are non-negative, the right-hand side of (27), viewed as a polynomial in $x_1$ , has a nonzero constant term and, hence, is not the zero polynomial. Similarly, if $\ell _0$ or $\ell _1$ is positive and $\ell _0,\ell _1 \geq 0$ , then the right-hand side of (27) has a nonzero coefficient of $x_1^2$ and is again a nonzero polynomial. We conclude that if $G$ contains at least one of the reactions labelled by $k_i$ or $\ell _i$ , then $\textrm{cap}_{\text{pos}}(G) \lt \infty$ , which is consistent with Lemma 3.19.

We now consider the case when $G$ contains no reactions labelled by $k_i$ or $\ell _i$ , that is, every reaction of $G$ is one of the following $n+1$ reactions:

\begin{align*} &0 \stackrel{m_0}{\leftarrow } X_1 \stackrel{m_1}{\to } 2X_1 \\[5pt] & X_i \stackrel{m_i}{\leftarrow } X_1 + X_i \quad \text{for } i=2,3,\dots, n. \end{align*}

The right-hand side of the ODE for $X_1$ , as in (27), becomes $x_1(m_1 - m_0 - m_2T_2 - \dots - m_nT_n)$ . In order for this polynomial in $x_1$ to become the zero polynomial for some choice of positive rate constants of $G$ (equivalently, $\textrm{cap}_{\text{pos}}(G) = \infty$ ), we must have $m_1\gt 0$ and $m_j\gt 0$ for at least one of $j=0,2,3,\dots,n$ . This gives exactly the reactions listed in Lemma 3.19(2)(b). In this case, given $m_j\gt 0$ for the reactions appearing in the network, we can always choose $T_j\gt 0$ , such that the right-hand side of the ODE for $X_1$ vanishes (i.e. $\textrm{cap}_{\text{pos}}(G) = \infty$ ). Moreover, when this right-hand side vanishes is the only situation in which there are positive steady states, and an easy calculation shows that all such positive steady states are degenerate. This concludes our analysis of networks with stoichiometric subspace spanned by the vector $(1,0,0,\dots, 0)$ .

Our final case is when the stoichiometric subspace is spanned by the vector $(1,-1,0,\dots, 0)$ . In this case, the reactions of $G$ form a subset of the following $2n+4$ reactions:

\begin{align*} & X_1 \mathop{\leftrightarrows }_{k_1}^{\ell _1} X_2 \quad \quad 2 X_1 \mathop{\leftrightarrows }_{k_2}^{\ell _2} 2X_2 \quad \quad 2 X_1 \mathop{\leftrightarrows }_{k_3}^{m_1} X_1 + X_2 \mathop{\leftrightarrows }_{m_2}^{\ell _3} 2 X_2 \\[5pt] & X_1 + X_i\mathop{\leftrightarrows }_{k_{i+1}}^{\ell _{i+1}} X_2 + X_i\quad \text{for } i=3,4,\dots, n. \end{align*}

The conservation laws are $x_1+x_2=T_2$ and $x_i=T_i$ for $i=3,4,\dots, n$ . The ODE for species $X_1$ is:

(28) \begin{align} \frac{dx_1}{dt} = & -(2k_2+k_3) x_1^2 -k_1x_1 -(k_4 x_3 + \dots + k_{n+1}x_n) x_1 + (m_1-m_2) x_1 x_2\\[5pt] & \notag \quad + \left(\ell _1 x_2 + 2 \ell _2 x_2^2 + \ell _3 x_2^2\right) + (\ell _4 x_3 + \dots + \ell _{n+1} x_n) x_2. \end{align}

Consider the subcase when at least one of the $\ell _i$ is positive and all other $\ell _j$ ’s are non-negative. After substituting the expressions arising from the conservation laws (namely, $x_2= T_2-x_1$ and $x_i=T_i$ for $i=3,4,\dots, n$ ) into the right-hand side of the ODE (28), we obtain a polynomial in $x_1$ that has a positive constant term (see the second line of the right-hand side of (28)). Hence, if $G$ contains at least one of the reactions labelled by $\ell _i$ , then $\textrm{cap}_{\text{pos}}(G) \lt \infty$ .

By symmetry, if $G$ has at least one of the reactions labelled by $k_i$ , then again $\textrm{cap}_{\text{pos}}(G) \lt \infty$ . Hence, if $G$ contains a reaction labelled by $\ell _i$ or $k_i$ , then this subcase is consistent with the lemma.

Consider the remaining subcase, when $G$ is a subnetwork of $\left\{ 2 X_1 \stackrel{m_1}{\leftarrow } X_1 + X_2 \stackrel{m_2}{\to } 2 X_2 \right\}$ , and so consists of only one or two reactions. If $G$ has only one reaction, then Proposition 3.5 implies that $\textrm{cap}_{\text{pos}}(G) = 0 \lt \infty$ (which is consistent with the lemma).

Now assume that $G$ has two reactions, that is, $G=\left\{ 2 X_1 \stackrel{m_1}{\leftarrow } X_1 + X_2 \stackrel{m_2}{\to } 2 X_2 \right\}$ . If $m_1 \neq m_2$ , then the ODE for $X_1$ is $\frac{dx_1}{dt}= (m_1 - m_2) x_1 x_2$ and so there are no positive steady states. When $m_1=m_2$ , the ODE for $X_1$ becomes $\frac{dx_1}{dt}=0$ and it follows that $\textrm{cap}_{\text{pos}}(G) = \infty$ . Moreover, a simple computation shows that all the positive steady states are degenerate. This concludes the proof.

Proof. Assume that $G$ is a one-dimensional network that is nondegenerately multistationary. We must show that $G$ is not bimolecular. We claim that $\textrm{cap}_{\text{pos}}(G)$ is finite. Indeed, if $\textrm{cap}_{\text{pos}}(G) = \infty$ , then Lemma 3.19 implies that all positive steady states are degenerate and so $G$ is not nondegenerately multistationary, which is a contradiction. Hence, $\textrm{cap}_{\text{pos}}(G) \lt \infty$ .

The hypotheses of part (1) of Lemma 3.18 are satisfied, that is, $\textrm{cap}_{\text{pos}}(G) \lt \infty$ , and $G$ is one-dimensional and multistationary. Therefore, $G$ has an embedded one-species network with arrow diagram $({\leftarrow}, \to)$ . Such an embedded network (e.g. $\{0 \leftarrow A,\, 2A \to 3A\}$ ) involves at least one complex that is not bimolecular, and so $G$ is also not bimolecular.

Proof. Assume that $G$ is bimolecular and consists of one or two pairs of reversible reactions. Let $p$ denote the number of pairs of reversible reactions (so, $p=1$ or $p=2$ ), and $\ell$ the number of linkage classes. Let $s$ be the dimension of the stoichiometric subspace (so, $s=1$ or $s=2$ ).

Case 1: $p=1$ . The deficiency of $G$ is $\delta = 2-1-1=0$ and $G$ is weakly reversible. Hence, by the deficiency-zero theorem (Lemma 2.5) the network is not multistationary.

Case 2: $p=s=2$ . If $\ell =1$ , then the deficiency is $\delta = 3-1-2=0$ . If $\ell =2$ , then the deficiency is $\delta =4-2-2=0$ . Therefore, for either value of $\ell$ , the deficiency-zero theorem (Lemma 2.5) implies that the network is not multistationary.

Case 3: $p=2$ and $s=1$ . $G$ is one-dimensional, bimolecular, and reversible. So, Lemma 3.19 implies that $\textrm{cap}_{\text{pos}}(G) \lt \infty$ . Now Lemma 3.18(2) yields $\textrm{cap}_{\text{pos}}(G) = \textrm{cap}_{\text{nondeg}}(G)$ , and Lemma 3.20 implies that $\textrm{cap}_{\text{nondeg}}(G) \,\leq \, 1$ . Thus, $\textrm{cap}_{\text{pos}}(G) \leq 1$ , or, equivalently, $G$ is non-multistationary.