## 3 System Properties for Constant Concentrations
### Special Case: \(P_{\max,1}=P_{\max,2}\)  
We first discuss the dynamical behavior of the system for the case \(P_{\max,1}=P_{\max,2}\) which is quite different from the cases \(P_{\max,1}\
eq P_{\max,2}\). If these effective rates are equal, we have that  
\[\frac{dE}{dt}=\left(s_{E}\ln\left(\frac{E_{\rm ss}}{E}\right)-\delta_{E} \right)\left(1+\frac{P_{\max,1}P}{1+P}\right)E\]  
and it follows that \(E\) is strictly increasing for \(E<E_{*}=E_{\rm ss}\exp\left(-\frac{\delta_{E}}{s_{E}}\right)\) and strictly decreasing for \(E>E_{*}\). Therefore, as \(t\to\infty\), the \(E\)-dynamics approach \(E_{*}\), monotonically increasing if the initial condition is smaller, monotonically decreasing if it is higher. Consequently also the \(Q\)-dynamics approach the steady-state behavior  
\[\frac{dQ}{dt}=\left[r_{Q}-\delta_{Q}\left(1+\frac{E_{\max,1}E_{*}}{1+E_{*}} \right)\right]Q\]  
and \(Q\) will increase exponentially if  
\[r_{Q}>\delta_{Q}\left(1+\frac{E_{\max,1}E_{*}}{1+E_{*}}\right)\]  
and decrease exponentially if  
\[r_{Q}<\delta_{Q}\left(1+\frac{E_{\max,1}E_{*}}{1+E_{*}}\right).\]  
In the first case this also generates unbounded growth in \(P\) (Lemma 3) leading to behavior consistent with the _blast_ phase of the system. In the second case, \(Q\) decays exponentially to \(0\) and \(P\) converges to the unique and asymptotically stable equilibrium point \(P_{*}\) on \(Q=0\). Overall, and writing in the constant controls (the respective concentrations \(u_{i}\)) we have the following result:  
Proposition 3: _Suppose \(P_{\max,1}=P_{\max,2}\) and let_  
\[\hat{E}=E_{\rm ss}\exp\left(-\frac{\delta_{E}}{s_{E}}\right)\]  
and if \(P_{\max,1}>P_{\max,2}\), the solution is positive if and only if  
\[1>\frac{s_{E}}{\delta_{E}}\ln\left(\frac{E_{\rm ss}}{E_{*}}\right)>\frac{1+P_{ \max,2}}{1+P_{\max,1}}.\]  
If one of these inequalities is violated, no positive equilibrium solution \(P_{*}=P_{*}(E_{*})\) exists and the overall dynamics are determined either by exponential growth or decay of \(Q\). If \(P_{*}=P_{*}(E_{*})\) exists and is positive, then Equation (7) defines \(Q_{*}\) as  
\[k_{P}Q_{*}=\left[\delta_{P}\left(1+E_{\max,2}\frac{E_{*}}{1+E_{*}}\right)-r_{P }\ln\left(\frac{P_{\rm ss}}{P_{*}}\right)\right]P_{*}. \tag{19}\]  
Using the equilibrium relation for \(E_{*}\), this can equivalently be expressed in the form  
\[k_{P}Q_{*}=\left[1+\frac{E_{\max,2}}{E_{\max,1}}\left(\frac{r_{Q}}{\delta_{Q}} -1\right)-\frac{r_{P}}{\delta_{P}}\ln\left(\frac{P_{\rm ss}}{P_{*}}\right) \right]\delta_{P}P_{*}. \tag{20}\]  
Note that \(Q_{*}\) is positive if \(P_{*}\geq P_{\rm ss}\) while otherwise this becomes a requirement on the equilibrium value \(P_{*}=P_{*}(E_{*})\), namely  
\[P_{*}>P_{\rm ss}\exp\left(-\frac{\delta_{P}}{r_{P}}\left[1+\frac{E_{\max,2}}{E _{\max,1}}\left(\frac{r_{Q}}{\delta_{Q}}-1\right)\right]\right).\]  
If \(E_{\max,2}=E_{\max,1}\), then this simply becomes \(P_{*}>P_{\rm ss}\exp\left(-\frac{\delta_{P}}{r_{P}}\frac{r_{Q}}{\delta_{Q}}\right)\). In either case, there exists at most one positive equilibrium point given by Equations (16), (18) and (20). **Remark 1**. : _As \(P_{\max,1}\to P_{\max,2}\), condition (15) implies that along a positive solution \(P_{*}(E_{*})\) we must have_  
\[\frac{s_{E}}{\delta_{E}}\ln\left(\frac{E_{\rm ss}}{E_{*}}\right)\to 1\]  
_and thus the limit taken along these positive solutions only exists if \(E_{*}\to\hat{E}=E_{\rm ss}\exp\left(-\frac{\delta_{E}}{s_{E}}\right)\) and if_  
\[\frac{dQ}{dt}=\left[r_{Q}-\delta_{Q}\left(1+\frac{E_{\max,1}E_{*}}{1+E_{*}} \right)\right]Q\]0  
_In this degenerate case, the equilibrium conditions \(\dot{Q}=0\) and \(\dot{E}=0\) are automatically satisfied and there exists a one-dimensional equilibrium manifold, namely \(M=\left\{\left(Q_{*}(P),P,\hat{E}\right):P>0\right\}\) with the \(P\) value arbitrary and \(Q_{*}(P)\) given by_  
\[\frac{dQ}{dt}=\left[r_{Q}-\delta_{Q}\left(1+\frac{E_{\max,1}E_{*}}{1+E_{*}} \right)\right]Q\]1  
We now investigate the _stability of the positive equilibrium point_.