## 2 Methodology: Living System's Disturbance and Lotka-Volterra Equations  
In a normal (healthy) organism, the jostling process between detrimental and beneficial processes keep the system stable (balance or homeostasis). In the special case of ionizing radiation as a detrimental factor, one can rewrite the simple homeostatic balance equation as  
\[P_{C}\approx P_{H} \tag{1}\]  
where \(P_{C}\) corresponds to the probability function of detrimental (cancerous) effects' influence, while \(P_{H}\) means beneficial (hormetic) effects' influence. Both probability functions can change over time:  
\[ \begin{cases}\frac{dP_{C}}{dt}=\varepsilon_{C}\;P_{C}\\ \frac{dP_{H}}{dt}=\varepsilon_{H}\;P_{H}\end{cases}  \tag{2}\]  
This is, however, not the final set of equation of the analyzing problem. The crucial importance should be put on \(\varepsilon_{C}\) and \(\varepsilon_{H}\) parameters, which can be presented in their simplest form as:  
\[ \begin{cases}\varepsilon_{C}=\alpha_{C}\;D\\ \varepsilon_{H}=-\alpha_{H}\;D+\beta_{H}\;D^{2}\end{cases}  \tag{3}\]  
where \(\varepsilon_{C}>0\) corresponds to the linear growth with the analyzed substance dose (e.g., ionizing radiation), while \(\varepsilon_{H}<0\) corresponds to low doses only (for \(D<\alpha_{H}/\beta_{H}\)), which is consistent with the hormetic effect. In general, the hormetic relationship can be J-shaped (as mentioned) or U-shaped (for \(\varepsilon_{H}=\alpha_{H}-\beta_{H}\;D+\gamma_{H}\;D^{2}\)). Of course, those relationships are valid only for \(D>0\). In other words: both relationships in Equation (3) are responsible for the proper signal dynamics: \(\varepsilon_{C}\) for the detrimental signal change with dose \(P_{C}\) and for the beneficial signal change with dose \(P_{H}\). The overall response of the organism is a joint effect of both beneficial and detrimental processes, and will be discussed later. An increase in cancerogenic processes, \(P_{C}\), however, causes an increase in beneficial ones (\(P_{H}\)) to protect the system. And the opposite: a large quantity of \(P_{H}\) causes a reduction in \(P_{C}\). This is analogical to the prey-predator phenomena, which can be described by the Lotka-Volterra equations as [12]  
\[ \begin{cases}\frac{dP_{C}}{dt}=(\varepsilon_{C}-k_{C}P_{H})\ P_{C}\\ \frac{dP_{H}}{dt}=(\varepsilon_{H}+k_{H}P_{C})\ P_{H}\end{cases}  \tag{4}\]  
where predators consume preys, and an increase in the number of preys (\(P_{C}\)) causes an increase in the number of predators (\(P_{H}\)). Additionally, the \(\varepsilon_{C}\) and \(\varepsilon_{H}\) parameters correspond to the dose-response model of detrimental and beneficial factors, respectively. This approach was described in the previous study in the context of the ionizing radiation's influence on the evolution of species [12]. The presented approach, here given by Equation (4), seems to be a good approximation of the system's behavior, where the competition between detrimental and beneficial factors keeps the system in a meta-stable state. In that situation, the homeostatic balance equation, given here by Equation (1), is maintained; however, regular fluctuations are observed when one factor dominates over another one, and vice versa (see Figure 1). Equation (4) can be rewritten in a time-independent form as  
\[\frac{dP_{C}}{dP_{H}}=\frac{P_{C}}{P_{H}}\ \frac{\varepsilon_{C}-k_{C}P_{H}}{ \varepsilon_{H}+k_{H}P_{C}} \tag{5}\]  
which results in the closed curves characteristic to the Lotka-Volterra equations (phase space). Next, the Equation (5) can be rewritten in a splitting variables form:  
\[\frac{k_{C}P_{H}-\varepsilon_{C}}{P_{H}}\ dP_{H}+\frac{k_{H}P_{C}+\varepsilon_{ H}}{P_{C}}\ dP_{C}=0 \tag{6}\]  
Figure 1: The time-related solution of Lotka–Volterra Equation (4) for exemplary in-put values of \(\varepsilon_{C}=2.1\), \(\varepsilon_{H}=-2.1\) and \(k_{C}=k_{H}=1\). The solution of Equation (6) is a function \(P_{H}(P_{C})\) or \(P_{C}(P_{H})\), which creates the curve in \(P_{H}-P_{C}\) phase space.