Table 2.

Fixed points of the four regions of the phase diagram of Fig. (1b).

$d>q_0,d<\psi \alpha /\beta$
${ℱ}_1$
λ1 ∈ $ℝ$,λ1 < 0
λ2 ∈ $ℝ$,λ2 < 0
λ3 ∈ $ℝ$, λ3 < 0
λ4 ∈ $ℝ$, λ4 < 0
$d<q_0,d<\psi \alpha /\beta$
${ℱ}_1$	${ℱ}_2$
λ1 ∈ $ℝ$, λ1 < 0	λ1 ∈ $ℝ$, λ1 < 0
λ2 ∈ $ℝ$, λ2 < 0	λ2 ∈ $ℝ$, λ2 < 0
λ3 ∈ $ℝ$, λ3 > 0	${\lambda }_3\in ℂ,\Re \left[{\lambda }_3\right]<0$
λ4 ∈ $ℝ$, λ4 > 0	${\lambda }_4\in ℂ,\Re \left[{\lambda }_4\right]<0$
$d>q_0,d>\psi \alpha /\beta$
${ℱ}_1$	${ℱ}_3$
λ1 ∈ $ℝ$, λ1 < 0	λ1 ∈ $ℝ$, λ1 < 0
λ2 ∈ $ℝ$, λ2 < 0	λ2 ∈ $ℝ$, λ2 > 0
λ3 ∈ $ℝ$, λ3 < 0	${\lambda }_3\in \mathbb{C},\mathrm{\Re }[{\lambda }_3]\lessgtr 0$
λ4 ∈ $ℝ$, λ4 < 0	${\lambda }_4\in \mathbb{C},\mathrm{\Re }[{\lambda }_4]\lessgtr 00$
$d<q_0,d>\psi \alpha /\beta$
${ℱ}_1$	${ℱ}_2$	${ℱ}_3$
λ1 ∈ $ℝ$, λ1 < 0	λ1 ∈ $ℝ$, λ1 < 0	λ1 ∈ $ℝ$, λ1 < 0
λ2 ∈ $ℝ$, λ2 < 0	λ2 ∈ $ℝ$, λ2 < 0	λ2 ∈ $ℝ$, λ2 > 0
λ3 ∈ $ℝ$, λ3 > 0	${\lambda }_3\in ℂ,\Re \left[{\lambda }_3\right]<0$	${\lambda }_3\in \mathbb{C},\mathrm{\Re }[{\lambda }_3]\lessgtr 0$
λ4 ∈ $ℝ$, λ4 > 0	${\lambda }_4\in ℂ,\Re \left[{\lambda }_4\right]<0$	${\lambda }_4\in \mathbb{C},\mathrm{\Re }[{\lambda }_4]\lessgtr 0$

Every fixed point is classified with respect to the eigenvalues of the corresponding Jacobian matrix. Every eigenvalue is fully determined but for ${ℱ}_3$, the pair of complex conjugate eigenvalues may have either a positive or negative real part (in the table). This will not change the global stability, since an eigenvalue with positive real part (λ₂ in the Table) always exists as demonstrated in the Appendix C. In the blue region (d > q₀, d < ψα/β), only one fixed point exists with real and negative eigenvalues. Every initial concentration of cells and chemicals will then decay exponentially to ${ℱ}_1$, which indicates a complete tumor regression. In the orange region (d < q₀, d < ψα/β), the Jacobian near ${ℱ}_1$ has two positive and real eigenvalues, leading to a long time exponential relaxation for arbitrary initial concentration to ${ℱ}_2$. In the green region (d > q₀, d > ψα/β), every initial concentration in the basin of attraction of ${ℱ}_1$, will exponentially relax to full regression, whilst increasing if it lays outside the basin. A complete discussion about this scenario has been given in the main text (Section (Dynamics)). In the red region of Fig. 1, (d < q₀, d > ψα/β), the scenario is very rich, due to the existence of all fixed points. The Jacobian of ${ℱ}_1$ has two real and positive eigenvalues, making this point unstable. The tumor will never undergo regression. Instead, ${ℱ}_2$ is globally stable; an initial concentration of chemicals and cells in the vicinity of ${ℱ}_2$ will relax to a low population tumor. Differently from the orange region, the existence of ${ℱ}_3$ restricts the basin of attraction of ${ℱ}_2$, making the tumor less likely to relax to ${ℱ}_2$.