**5. Conclusions**

The dynamics of the logistic equation with delays and with bounding modified nonlinearity are studied.

First, oscillations close to harmonic are studied. Using the methods of bifurcation analysis, we singled out the critical case in the problem of stability of stationary state and constructed the normal form. Its nonlocal dynamics determine the local behavior of the initial equation solutions in the neighborhood of the equilibrium state.

The solutions of the relaxation (step-like) type are studied for sufficiently large values of the parameter *λ*. We stress that, from a computational point of view, Equations (4) and (8) are difficult, since the relaxation solution approaches *A* and 0 very closely, so even a small error in the calculations will take us out of the class of solutions under consideration. Therefore, asymptotic methods play a special role. They not only allow one to find an approximation of the solution, but also reduce the problem of the dynamics of the original infinite-dimensional problem to the problem of the dynamics of the constructed finitedimensional mapping (this object is much more simpler than the initial equation). Asymptotic formulas that couple the solutions of(4) and (8) with such mappings trajectories are obtained. The resulting formulas are suitable for engineering calculations. It is important to note that this method is applicable to equations with different types of nonlinearities [30,31] and to systems of two, three, and more singular perturbed equations with delay [30,32,33].

The use of the modified nonlinearity leads to the fact that the oscillations turn out to be 'safer' in comparison with Equation (1): their largest value *A* is significantly less than the value of exp(*λT*) for Equation (1), and the smallest value of exp(−*λconst*) is significantly greater than the value of exp(− exp *λT*).

As a generalization of the obtained results, we note that they do not change if the function (*A* − *u*) in (8) is replaced by a more general function *F*0(*u*) for which the conditions *F*0(*u*) > 0 as 0 < *u* < *A*, *F*0(0) = *F*0(*A*) = 0 and *F* <sup>0</sup>(0) = 0, *F* <sup>0</sup>(*A*) = 0 are satisfied. We also note that the proposed method allows extension to the case of more than two delays in Equation (8).

**Author Contributions:** Conceptualization, A.K. and S.K.; Methodology, A.K. and S.K.; Validation, A.K. and S.K.; Formal analysis, A.K. and S.K.; Investigation, A.K. and S.K.; Writing—original draft, A.K. and S.K.; Writing—review and editing, A.K. and S.K.; Visualization, A.K. and S.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Russian Science Foundation (project No. 21-71-30011), https://rscf.ru/en/project/21-71-30011/, accessed on 28 February 2023.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.
