## 3 Damping Processes and Homeostatic Balance Equation  
The jostling process between detrimental and beneficial factors described in the previous section is stable over time. However, in a real situation, many disturbances of the system exist. After such a disturbance and creation of a detrimental pulse, the system naturally goes back to stable conditions due to additional dumping mechanism(s). This situation was recently simulated by the damped harmonic oscillator [13] related to the adaptive response phenomenon, which is a good analogy to the mentioned situation. Figure 2: The phase space of \(P_{H}\) and \(P_{C}\) functions, given by Equation (4) for exemplary data from Figure 1. One can observe the regular oscillation of \(P_{H}\) and \(P_{C}\) functions, which is another representation of periodic functions in Figure 1.  
In the case of Lotka-Volterra equations, one needs to implement damping terms into Equation (4), which creates their new forms as  
\[ \begin{cases}\frac{dP_{C}}{dt}=\left(\varepsilon_{C}-k_{C}P_{H}\right)P_{C}-\mu_ {C}P_{C}^{2}\\ \frac{dP_{H}}{dt}=\left(\varepsilon_{H}+k_{H}P_{C}\right)P_{H}-\mu_{H}P_{H}^{2} \end{cases}  \tag{7}\]  
with damping parameters (factors) \(\mu_{C}\) and \(\mu_{H}\) (which shall be generally lower than \(\{k\}\) to keep the damping process not too strong). In this situation, the shape of functions given by Equation (7) and originally presented in Figure 1 (without damping) became the ones in Figure 3. Similarly, the perfect circle from Figure 2 became spiral (see Figure 4), which goes to the critical point. This reasoning can allow us to make analogous calculations like Equations (5)-(9), which are much more complicated, but one can use a simpler solution and find that Equation (7) stabilizes for \(t\to\infty\), where \(\frac{dP_{C}}{dt}=\frac{dP_{H}}{dt}=0\). Therefore, after a long time, one obtains stable values of  
\[P_{H}=\frac{\mu_{C}\varepsilon_{H}+k_{H}\varepsilon_{C}}{k_{H}k_{C}+\mu_{C} \mu_{H}} \tag{8}\]  
and  
\[P_{C}=\frac{\mu_{H}\varepsilon_{C}-k_{C}\varepsilon_{H}}{k_{H}k_{C}+\mu_{C} \mu_{H}} \tag{9}\]  
The proposed damping factors, \(\mu_{C}\) and \(\mu_{H}\), were selected as the simplest constant values. This is rather a simplification of the reality, where such processes, which are analogous to the damping, can be described by some function. Indeed, both \(\mu\) factors are dependent on many variables (like age), which makes the damping process more complicated. For simplicity, to show a more qualitative approach, both \(\mu\) factors are assumed to be constant values. One has to note that the exact conditions determine which probability factor, \(P_{C}\) or \(P_{H}\), will dominate after damped fluctuations (Figure 3). Let us focus on the special condition of homeostatic balance (equilibrium, namely, \(P_{C}=P_{H}\), see Equation (1)), which generates the equation of  
\[\frac{\varepsilon_{C}}{\varepsilon_{H}}=\frac{\mu_{C}+k_{C}}{\mu_{H}-k_{H}} \tag{10}\]  
This equation can be rewritten in a different form, as  
\[\theta=\frac{\mu+k_{C}}{\mu-k_{H}} \tag{11}\]  
which joins both dynamic factors (\(k_{C}\) is responsible for an decrease in detrimental processes due to beneficial processes, and \(k_{H}\) is responsible for an increase in beneficial processes due to detrimental processes, see Equation (4)) and damping factors, which can be assumed to be equal both for detrimental and beneficial processes, i.e., \(\mu=\mu_{C}=\mu_{H}\). The coefficient \(\theta\) is negative for low doses (where \(\varepsilon_{H}<0\) and \(\mu<k_{H}\)), which was presented in Figure 5 for \(\mu=0.1\). One can call the Equation (11) a homeostatic balance equation with a damping factor [12]. Please note that the plateau in Figure 5 represents the quasistable situation where detrimental and beneficial factors fluctuate around a stable point (which is finally reached).