## 3 Results

Table 2: Testing results of the evolutionary approach to the identification of dynamical systems in the form of an ODE. Figure 5: Initial sample fit with the output of the obtained model of the change in the rate of a chemical reaction on temperature. The tasks presented in Table 3 are applied to test the evolutionary approach of EvolDES. The systems of ODEs of various types and orders represented these tasks. The analysis of the error of the obtained models was conducted in a similar way to the previous testing scheme. This depended on the amount of data and the presence of noise (Table 4). Let us consider the application of this approach in solving a practical task from the field of biology. Let values describing the change in the population size of predators and prey over time be given in some ecosystem. The sample size is 50 points. It is necessary to calculate the change in the number of prey and predator populations over time. As a result, an expression was obtained represented by a system of differential equations. \[\left\{ \begin{array}{l}\frac{dx}{dt}=0.998x-\frac{xy}{1.023+2x-x}-0.124x^{2} +0.009x^{2}\\ \frac{dy}{dt}=\frac{(0.255-0.101\frac{x}{x}+0.08)y^{2}}{y}\\ \frac{dy}{dt}=\frac{dy}{dt}-\frac{dy}{dt}+e^{-t}+\cos t\\ \frac{dy}{dt^{2}}=\frac{dy_{1}}{dt}-\frac{dy_{2}}{dt}-2e^{t}-\sin t\end{array}  \right. \tag{8}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**No** & **System of Differential Equations** & **Initial Sample Point** \\ \hline
1 & \(\left\{\begin{array}{l}\frac{dy_{1}}{dt}=2y_{1}-5y_{2}+3\\ \frac{dy_{2}}{dt}=5y_{1}-6y_{2}-1\end{array}\right.\) & \(y_{1}(0)=6\) \\ \hline
2 & \(\left\{\begin{array}{l}\frac{dy_{1}}{dt}=2y_{1}+y_{2}\\ \frac{dy_{2}}{dt}=3y_{1}+te^{t}\end{array}\right.\) & \(y_{1}(1)=1\) \\ \hline
3 & \(\left\{\begin{array}{l}\frac{dy_{1}}{dt}=y_{1}+2y_{2}+e^{-2t}\\ \frac{dy_{2}}{dt}=4y_{1}-y_{2}\end{array}\right.\) & \(y_{1}(0.1)=5\) \\ \hline
4 & \(\left\{\begin{array}{l}\frac{dy_{1}}{dt}=y_{2}-y_{1}^{2}-y_{1}\\ \frac{dy_{2}}{dt}=3y_{1}-y_{2}^{2}-y_{1}\end{array}\right.\) & \(y_{1}(0)=0\) \\ \hline
5 & \(\left\{\begin{array}{l}\frac{dy_{1}}{dt}=-y_{2}\\ \frac{dy_{2}}{dt}=y_{1}\end{array}\right.\) & \(y_{2}(0)=1\) \\ \hline
6 & \(\left\{\begin{array}{l}\frac{dy_{1}}{dt}=2y_{1}-y_{2}-y_{3}\\ \frac{dy_{2}}{dt}=3y_{1}-2y_{2}-3y_{3}+2t\\ \frac{dy_{3}}{dt}=2y_{3}-y_{1}-y_{2}-t^{2}\end{array}\right.\) & \(y_{1}(0)=1\) \\ \hline
7 & \(\left\{\begin{array}{l}\frac{dy_{1}}{dt}=-10y_{1}+10y_{2}\\ \frac{dy_{2}}{dt}=25y_{1}-y_{2}-y_{1}y_{2}\\ \frac{dy_{3}}{dt}=y_{1}y_{2}-10y_{3}\end{array}\right.\) & \(y_{2}(0)=0\) \\ \hline
8 & \(\left\{\begin{array}{l}\frac{dy_{1}}{dt}=3y_{1}+2y_{2}\\ \frac{dy_{2}^{2}}{dt^{2}}=-10y_{1}-y_{2}\end{array}\right.\) & \(y_{1}(0)=-1\) \\ \hline
9 & \(\left\{\begin{array}{l}\frac{d^{2}y_{1}}{dt^{2}}=-ty_{2}\\ \frac{dy_{2}^{2}}{dt}=y_{2}-2\frac{dy_{1}}{dt}\end{array}\right.\) & \(y_{1}(1)=4\) \\ \hline
10 & \(\left\{\begin{array}{l}\frac{dy_{1}}{dt^{2}}=\frac{dy_{1}}{dt}-\frac{dy_{2} }{dt}+e^{-t}+\cos t\\ \frac{dy_{2}}{dt^{2}}=\frac{dy_{1}}{dt}-\frac{dy_{2}}{dt}-2e^{t}-\sin t\end{array}\right.\) & \(y_{1}(0)=2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Test tasks for the evolutionary approach to the identification of dynamical systems in the form of an ODE system. The presented system corresponds to the best solution found among 50 runs. It corresponds to an error of 0.0019. The best solution was found 24 times from 50 runs. The averaged error over 50 runs was 0.0031. The presented system of differential equations corresponds to the Holding-Tanner model. Generally, it is represented as follows [27]:  
\[\left\{ \begin{array}{rl}\frac{dx}{dt}=\left(a-\frac{cq}{\dot{S}+x}-ex\right)x \\ \frac{dy}{dt}=\left(b-f\frac{x}{x}\right)y\end{array} \right. \tag{9}\]  
where \(x\) is the size of the predator population, \(y\) is the size of the prey population, \(\frac{cy}{\dot{S}+x}\) is the predation coefficient, and \(a,b,f\) are positive parameters.