**6. Conclusions**

It has been shown that the considered critical cases in the stability problem of the distributed chain of logistic equations with delay have an infinite dimension. This leads to the fact that description of their local dynamics is reduced to the study of the nonlocal behavior of boundary value problems of Ginzburg–Landau type solutions. It is known (see, for example, [30]) that the dynamics of such objects can be complicated, and they are characterized by irregular oscillations, multistability phenomena, etc. The dynamic effects essentially depend on the choice of couplings. It has been shown that in a number of cases the solutions are rapidly and slowly oscillating with respect to spatial variable components. The basic results define the structure of problems that are asymptotic with respect to residual solutions to the initial boundary value. The problem of existence, stability, and more complicated asymptotic expansions of exact solutions close to those constructed can be solved, for example, for the case of periodic solutions of normalized equations.

We considered separately the role of the above parameter *θ* = *θ*(*ε*) ∈ [0, 1). We recalled that the dynamic properties of the initial system are determined by the QNF (49), (50), which includes the parameter *θ*. The dynamics of (49), (50) and hence of the boundary value problem (9), (10) may change for different values of this parameter. This is shown in detail in [31]. This implies that an infinite process of forward and reverse bifurcations can occur as *ε* → 0.

Below, we formulate one more conclusion of the general plan. It was shown above that the quasinormal forms that determine the dynamics of the initial boundary value problem are equations of Ginzburg–Landau type. We note that parabolic boundary value problems with one and two spatial variables can act as quasinormal forms depending on the coefficient *σ* of the function *F*(*s*,*ε*) ((12), (13)). The stability of the simplest solutions of these equations is studied in [33]. In particular, it has been established that their stability properties are determined to a large extent by the imaginary components of the diffusion coefficients and of the Lyapunov quantity (coefficients *g* and *q* in (49) and (50)). Numerical analysis of the corresponding criterion makes it possible to formulate the conclusion about the instability of all the simplest solutions of the form · exp(*iωt* + *ikx*). Thus, solution synchronization is a rather rare phenomenon in the considered chains.

It has been demonstrated that the study of the dynamics of logistic equations with delay is reduced to nonlinear dynamics analysis of special families of the parabolic and degenerately parabolic boundary value problems for large values of the coefficient of spatially distributed control. In particular, the phenomenon of hypermultistability is described.

In the study of local dynamics, bifurcation phenomena can be realized in the equilibrium state neighborhood even for asymptotically small delays. Here, the critical case has an infinite dimension in the stability problem. Analogues of the normal form, so-called quasinormal forms, are constructed in this situation, which are universal nonlinear boundary value problems of the parabolic type. Their nonlocal dynamics determine the local behavior of the solutions of the initial boundary value problem.

**Funding:** This work was supported by the Russian Science Foundation (project no. 21-71-30011).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The author declares no conflict of interest. The funder 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.
