## 3 System Properties for Constant Concentrations
### Dynamics on the Plane \(Q=0\)
The derivative \(\Phi^{\prime}\) of \(\Phi\) is given by  
\[\frac{dE}{dt} =\left[s_{E}\left(1+\frac{P_{\text{max},1}P}{1+P}\right)\ln\left( \frac{E_{\text{ss}}}{E}\right)-\delta_{E}\left(1+\frac{P_{\text{max},2}P}{1+P} \right)\right]E\]4  
and thus has the same sign as \(\Xi^{\prime}(P)\). Now  
\[\frac{dE}{dt} =\left[s_{E}\left(1+\frac{P_{\text{max},1}P}{1+P}\right)\ln\left( \frac{E_{\text{ss}}}{E}\right)-\delta_{E}\left(1+\frac{P_{\text{max},2}P}{1+P} \right)\right]E\]5  
Thus \(\Phi\) is strictly increasing for \(P_{\text{max},2}>P_{\text{max},1}\) and strictly decreasing for \(P_{\text{max},2}<P_{\text{max},1}\). If \(P_{\text{max},2}=P_{\text{max},1}\), then  
\[\frac{dE}{dt} =\left[s_{E}\left(1+\frac{P_{\text{max},1}P}{1+P}\right)\ln\left( \frac{E_{\text{ss}}}{E}\right)-\delta_{E}\left(1+\frac{P_{\text{max},2}P}{1+P} \right)\right]E\]6  
Equilibria are intersections of the graph of \(\Phi\) with the diagonal and thus there exists a unique equilibrium point \((P_{*},E_{*})\in\mathbb{P}_{0}\) if \(P_{\text{max},1}\geq P_{\text{max},2}\), but multiple solutions are possible if \(P_{\text{max},1}<P_{\text{max},2}\). We determine the stability of \((P_{*},E_{*})\) for the reduced system, i.e., within the invariant plane \(Q=0\). The Jacobian matrix at the equilibrium point is given by  
\[\left( \begin{array}{cc}-r_{P}&-\delta_{P}\frac{E_{\text{max},2}}{\left(1+E_ {*}\right)^{2}}P_{*}\\ \frac{s_{E}P_{\text{max},1}\ln\left(\frac{E_{\text{ss}}}{E_{*}}\right)-\delta_ {E}P_{\text{max},2}}{\left(1+P_{*}\right)^{2}}E_{*}&-s_{E}\left(1+\frac{P_{ \text{max},1}P_{*}}{1+P_{*}}\right)\end{array} \right).\]  
Using the equilibrium relations we can write the \((2,1)\)-term as  
\[\frac{\partial}{\partial P}_{\left|(P_{*},E_{*})\right.}\left( \frac{dE}{dt}\right) =\frac{\delta_{E}P_{\text{max},1}E_{*}}{\left(1+P_{*}\right)^{2}} \left\{\frac{s_{E}}{\delta_{E}}\ln\left(\frac{E_{\text{ss}}}{E_{*}}\right)- \frac{P_{\text{max},2}}{P_{\text{max},1}}\right\}\] \[=\frac{\delta_{E}P_{\text{max},1}E_{*}}{\left(1+P_{*}\right)^{2}} \left\{\frac{1+P_{*}+P_{\text{max},2}P_{*}}{1+P_{*}+P_{\text{max},1}P_{*}}- \frac{P_{\text{max},2}}{P_{\text{max},1}}\right\}\] \[=\delta_{E}E_{*}\frac{P_{\text{max},1}-P_{\text{max},2}}{\left(1+P _{*}\right)\left(1+P_{*}+P_{\text{max},1}P_{*}\right)}.\]The characteristic polynomial of this \(2\times 2\) matrix is given by  
\[\chi(t) =\left| \begin{array}{cc}t+r_{P}&\delta_{P}\frac{E_{\max,2}}{(1+E_{ *})^{2}}P_{*}\\ -\frac{(P_{\max,1}-P_{\max,2})\delta_{E}E_{*}}{(1+P_{*})(1+P_{*}+P_{\max,1}P_{*} )}&t+s_{E}\left(1+\frac{P_{\max,1}P_{*}}{1+P_{*}}\right)\end{array} \right|\] \[=t^{2}+\left(r_{P}+s_{E}\left(1+\frac{P_{\max,1}P_{*}}{1+P_{*}} \right)\right)t+r_{PS}E\left(1+\frac{P_{\max,1}P_{*}}{1+P_{*}}\right)\] \[\qquad+\delta_{E}\delta_{P}\frac{E_{\max,2}E_{*}}{(1+E_{*})^{2}} \frac{(P_{\max,1}-P_{\max,2})\,P_{*}}{(1+P_{*})\,(1+P_{*}+P_{\max,1}P_{*})}.\]  
If we write \(\chi(t)=t^{2}+a_{1}t+a_{0}\), then \(a_{1}\) is positive and thus the equilibrium point is locally asymptotically stable if \(a_{0}\) is positive while it is unstable if \(a_{0}\) is negative. A saddle node bifurcation occurs as \(a_{0}=0\). It immediately follows that \((P_{*},E_{*})\) is locally asymptotically stable if \(P_{\max,1}\geq P_{\max,2}\), i.e., if the stimulating effect of the tumor on the effector cells is larger than the inhibiting effect of the tumor on the effector cells. We have the following result:  
Proposition 1: If \(P_{\max,1}\geq P_{\max,2}\), then there exists a unique equilibrium point \((P_{*},E_{*})\) in \(\mathbb{P}_{0}\) and it is globally asymptotically stable in the sense that its region of attraction is the full rectangle \(\mathbb{P}_{0}\). Proof: The set \(\mathbb{P}_{0}\) is positively invariant and every trajectory \(\gamma\) starting in \(\mathbb{P}_{0}\) has a non-empty \(\omega\)-limit set \(\Omega(\gamma)\). Because of the stability properties of the equilibria in the boundary of \(\mathbb{P}_{0}\), this \(\omega\)-limit set \(\Omega(\gamma)\) lies in \(\mathbb{P}_{0}\).