## 3 System Properties for Constant Concentrations
### Dynamics on the Plane \(Q=0\)
Since there exist no periodic orbits and since \((P_{*},E_{*})\) is the only equilibrium point, it follows from Poincare-Bendixson theory that \(\Omega(\gamma)=\{(P_{*},E_{*})\}\), i.e., all trajectories starting in \(\mathbb{P}_{0}\) converge to \((P_{*},E_{*})\) as \(t\to\infty\). It is clear from Poincare-Bendixson theory that even if \(P_{\max,1}<P_{\max,2}\), the equilibrium point \((P_{*},E_{*})\) is globally asymptotically stable (in the sense that its region of attraction contains the set \(\mathbb{P}_{0}\), and only this region is relevant for the problem) as long as it is the only equilibrium point in \(\mathbb{P}_{0}\). This is shown in the phase portraits for the uncontrolled system in Figure 2; Figure 3 shows a case where \(P_{\max,1}>P_{\max,2}\). (The values of the parameters are given in Tables 1 and 2.) We also show the phase-portraits for the systems when one of the controls is set to be equal to \(1\) and all others are zero. The two sets of figures illustrate two different scenarios, one where the control parameters are such that the equilibrium can be effectively controlled by all the drugs (Figure 2), the other where it is essentially only the control \(u_{1}\) that is able to move the equilibrium point. However, this behavior depends on the fact that \(Q=0\). Figure 2: _Cont._  
Since the coefficient \(a_{1}\) is always positive, as \(a_{0}\) vanishes the Jacobian matrix has the eigenvalue \(0\) and the other eigenvalue is negative. At such a point _saddle-node bifurcations_ arise and two new equilibria, one stable, the other unstable, are born. **Proposition 2**. : _If \(P_{\max,2}>P_{\max,1}\), then multiple equilibria \((P_{*},E_{*})\) inside \(\mathbb{P}_{0}\) can exist. At points \((P_{*},E_{*})\) where_  
\[\frac{\delta_{P}}{r_{P}}\frac{\left(P_{\max,2}-P_{\max,1}\right)P_{*}}{\left(1 +P_{*}+P_{\max,1}P_{*}\right)^{2}}\frac{\delta_{E}}{s_{E}}\frac{E_{\max,2}E_{* }}{\left(1+E_{*}\right)^{2}}=1 \tag{14}\]  
Figure 3: Phase portraits of the reduced dynamics for \(Q=0\) and \(P_{\max,1}>P_{\max,2}\) for the uncontrolled system (**top**, **left**) and for constant controls \(u_{1}\equiv 1\) (**top**, **right**), \(u_{2}\equiv 1\) (**bottom**, **left**) and \(u_{3}\equiv 1\) (**bottom**, **right**). The numerical values for these phase portraits are given in Tables 1 and 2. Figure 2: Phase portraits of the reduced dynamics for \(Q=0\) and \(P_{\max,1}<P_{\max,2}\) for the uncontrolled system (**top**, **left**) and for constant controls \(u_{1}\equiv 1\) (**top**, **right**), \(u_{2}\equiv 1\) (**bottom**, **left**) and \(u_{3}\equiv 1\) (**bottom**, **right**). The numerical values for these phase portraits are given in Tables 1 and 2.  
saddle-node bifurcations occur in which a stable and an unstable equilibrium point merge._  
**Proof. ** The coefficient \(a_{0}\) vanishes if and only if  
\[r_{P}S_{E}\frac{1+P_{*}+P_{\max,1}P_{*}}{1+P_{*}}=\delta_{P}\frac{E_{\max,2}P_{*} }{\left(1+E_{*}\right)^{2}}\frac{\left(P_{\max,2}-P_{\max,1}\right)\delta_{E}E _{*}}{\left(1+P_{*}\right)\left(1+P_{*}+P_{\max,1}P_{*}\right)}.\]  
This condition is equivalent to (14). For the underlying biological problem it is natural that an inhibition effect would be smaller than a stimulation effect. Also, the denominators are quadratic in the respective variables \(E\) and \(P\), but these variables are scaled. In principle it is possible to satisfy (14), but we did not come across this in our simulations.