## 4 Model Dynamics  
Now, we study the mathematical properties and dynamics of our two models. For Model 1, we shall state the results without providing proofs as they are routine. The detailed mathematical analysis for Model 2 will be presented. Proposition 1 summarizes the mathematical dynamics of Model 1. Since \(P\) is decoupled from the system, we shall refer only to the dynamics of Equations (2)-(4). This proposition reveals that there is no cure for cancer. Since ADT is non-curative, this property is biologically reasonable. **Proposition 1**. : _Solutions of the system Equations (2)-(4) are positive and bounded. The system Equations (2)-(4) has a cancer free steady state \(E_{0}=(0,0,\frac{TQ_{m}+\mu q}{\mu+\gamma})\) that is unstable, and a steady state \(E_{1}=(\frac{pY}{\delta}\frac{Q_{m}-\mu q}{YQ^{m}+\mu q},0,\frac{TQ_{m}+\mu q} {\mu+\gamma})\) that is globally stable._  
Next, we do a thorough mathematical analysis of Model 2. First, we study boundedness and positivity of the system. Followed by the number and existence of steady states. Finally, we analyze the local stability of the steady states. Observe that \(P\) is also decoupled from Equations (2)-(4) and we do not include it in the analysis. **Proposition 2**. : _Assume \(q_{2}\leq q_{1}<Q_{m}\) and \(\delta_{1}\geq\delta_{2}\). Then, solutions of Equations (7)-(9) with initial conditions \(x_{1}(0)>0\), \(x_{2}(0)>0\), and \(q_{2}\leq Q(0)\leq Q_{m}\) stay in the region \(\{(x_{1},x_{2},Q):x_{1}\geq 0,x_{2}\geq 0,x_{1}+x_{2}\leq\frac{G_{2}(Q_{m})-D_{m}(q _{2})}{\delta_{2}},q_{2}\leq Q\leq Q_{m}\}\), where \(D_{m}=\min\{D_{1}(q_{2}),D_{2}(q_{2})\}\)._  
Proof. : We note that in Equation (7), \(x_{1}\) appears in every term ensuring its positivity. Since \(x_{2}\) appears in the first two terms of (8) and \(x_{1}\) appears in the last term, the positivity of \(x_{2}\) is also guaranteed. In addition, \(q_{2}\leq q_{1}<Q_{m}\), and  
\[Q^{\prime}=\gamma(Q_{m}-Q)-\frac{\mu(Q-q_{1})x_{1}+\mu(Q-q_{2})x_{2}}{x_{1}+x_ {2}}.\]  
We see that \(Q^{\prime}(q_{2})>0\) and \(Q^{\prime}(Q_{m})<0\). It is thus easy to see that \(q_{2}\leq Q(t)\leq Q_{m}\) for \(t>0\) with initial conditions \(q_{2}\leq Q(0)\leq Q_{m}\). For boundedness of \(x_{1}\) and \(x_{2}\), we let \(N=x_{1}+x_{2}\). Since we have that \(\delta_{1}\geq\delta_{2}\), and the growth rate \(G_{i}(Q)\), \(i=1,2\) are increasing functions of \(Q\), we have  
\[N^{\prime} \leq (G_{2}(Q)-D_{m})N-\delta_{2}N^{2}, \tag{16}\] \[\leq (G_{2}(Q_{m})-D_{m})N-\delta_{2}N^{2}, \tag{17}\]  
which implies that \(\limsup_{t\to\infty}N(t)\leq\frac{G_{2}(Q_{m})-D_{m}}{\delta_{2}}\). Now, we study the steady states of Model 2. We seek to understand the conditions under which one population will overtake the other, and the circumstances under which they may coexist. **Proposition 3**. : _Assume \(q_{2}\leq q_{1}<Q_{m}\) and \(\delta_{1}\geq\delta_{2}\). The system Equations (7)-(9) have a CR cell only steady state \(E_{1}=(0,\frac{G_{2}(Q^{\prime})-D_{1}(Q^{\prime})}{\delta_{2}},Q^{1})\), and a coexistence steady state \(E_{2}=(\frac{G_{1}(Q^{\prime})-D_{1}(Q^{\prime})-D_{1}(Q^{\prime})}{\delta_{1}},x_{2}^{*},Q^{*})\), where \(Q^{1}=\frac{YQ_{m}+\mu q_{2}}{\mu+\gamma}\) and \(Q^{*}>Q^{1}\)._  
**Proof. ** Let \(E=(x_{1}^{*},x_{2}^{*},Q^{*})\) be a steady state of the system Equations (7)-(9). We have two mutually exclusive cases: \(x_{1}^{*}=0\) and \(x_{1}^{*}>0\). If \(x_{1}^{*}=0\), then we have two possibilities: (i) \(x_{2}^{*}=0\) or (ii) \(x_{2}^{*}>0\). In the case of (i), we see that \(E=E_{0}.\) In the case of (ii), we see that \(E=E_{1}.\)  
If \(x_{1}^{*}>0\), we see that \(x_{2}^{*}>0\) from the equation of \(dx_{2}/dt.\) In this case, \(E=E_{2}.\) In addition, we have the following:  
\[0 = \gamma(Q_{m}-Q^{*})-\frac{\mu(Q^{*}-q_{1})x_{1}^{*}+\mu(Q^{*}-q_{2 })x_{2}^{*}}{x_{1}^{*}+x_{2}^{*}}\] \[\geq \gamma(Q_{m}-Q^{*})-\mu(Q^{*}-q_{2})\] \[Q^{*} \geq \frac{\gamma Q_{m}+\mu q_{2}}{\gamma+\mu}=Q^{1}.\]  
This proves the proposition.