## Appendix A  
Figure 2 shows the exemplary solution of Equation (6), which is a function of \(P_{H}(P_{C})\) or \(P_{C}(P_{H})\) (which creates the curve in \(P_{H}-P_{C}\) phase space). Each point on the mentioned curve has the constant quantity:  
\[C=k_{H}P_{C}+\varepsilon_{H}\ln P_{C}+k_{C}P_{H}+k_{C}P_{H}-\varepsilon_{C} \ln P_{H} \tag{11}\]  
The function \(C\) has a local extremum and the critical point for \((P_{H},P_{C})=\left(\frac{\varepsilon_{C}}{k_{C}},-\frac{\varepsilon_{H}}{k_ {H}}\right)\). In the situation when \(\frac{\varepsilon_{C}}{k_{C}}\approx-\frac{\varepsilon_{H}}{k_{H}}\), the phase space is a circle (thus, Equation (1) is maintained). The exemplary phase space with a circle solution is presented in the mentioned Figure 2. For the ideal condition of \(P_{H}=P_{C}\), the phase space is a single point only.  
But in the more probable situation, the fluctuation around the critical point denoted as \((P_{H},P_{C})=\left(\frac{\varepsilon_{C}}{k_{C}},-\frac{\varepsilon_{H}}{k_ {H}}\right)\), which can be visualized as in Figure 1, is analogous to the situation of the jostling process between detrimental and beneficial factors in the mentioned system. Thus, the Jacobian matrix for that case can be written as  
\[J(P_{C},P_{H})=\begin{bmatrix}\varepsilon_{C}-k_{C}P_{H}&-k_{C}P_{C}\\ k_{H}P_{H}&\varepsilon_{H}+k_{H}P_{C}\end{bmatrix}=\begin{bmatrix}0&k_{C}\frac {\varepsilon_{H}}{k_{H}}\\ k_{H}\frac{\varepsilon_{C}}{k_{C}}&0\end{bmatrix} \tag{12}\]  
with its eigenvalues equal to \(\lambda=\pm\sqrt{\varepsilon_{C}\varepsilon_{H}}\). So, the oscillations of solutions around the mentioned critical points are given with the frequency \(\omega=i\sqrt{\varepsilon_{C}\varepsilon_{H}}\).  
Next, assuming that the \(C\) function is a constant of motion, it would be easier to use  
\[K=e^{-C}=P_{C}^{-\varepsilon_{H}}\;P_{H}{}^{\varepsilon_{C}}\;e^{-k_{H}P_{C} -k_{C}P_{H}} \tag{13}\]  
Which is also constant for each closed curve (orbit). For higher \(K\), the radius of the circle (orbit) decreases, and for  
\[K_{max}=\left(\frac{\varepsilon_{C}}{e\,k_{C}}\right)^{\varepsilon_{C}}\left( \frac{-\varepsilon_{H}}{e\,k_{H}}\right)^{-\varepsilon_{H}} \tag{14}\]  
the solution is a single (stable) point only, as mentioned earlier.  
Now, in the phase-space plot of \(P_{C}\) and \(P_{H}\), one can calculate the function \(K(P_{C},P_{H})\) given by Equation (13). It reaches a maximum value for the mentioned point \((P_{H},P_{C})=\left(\frac{\varepsilon_{C}}{k_{C}},-\frac{\varepsilon_{H}}{k_ {H}}\right)\), which is equal to \(K_{max}\) (see Figure 11). However, when one takes the \(K\) values going through the diagonal axis (first for \(P_{C}=P_{H}\) and second, perpendicular to the first one, for  
The conclusion of this deliberation is that the homeostatic balance state given by Equation (1) is preserved for the conditions around the \(K_{max}\), which is the stable jostling between beneficial and detrimental factors. In the close vicinity around the \(K_{max}\), the shape of the \(K\) distribution is quasisymmetrical (quasi-Gaussian). The slight disturbance of \(P_{H}\) or \(P_{C}\) creates the oscillations (with the frequency \(\omega=i\sqrt{\epsilon_{C}\epsilon_{H}}\)). However, the strong disturbance would break the symmetry in the \(K\) function and can create the situation for potential cancer transformation.  
The \(K\) function, which joins both \(P_{C}\) and \(P_{H}\), can be treated as a form of so-called radiosensitivity, which is usually described by quasi-Gaussian distributions with one nonsymmetrical long tail corresponding to the hyper-radiosensitivity [22; 26]. However, the practical representation of the \(K\) function will not be discussed here.