**5. Dynamics of Logistic Delay Equation with Large Coefficient of Spatially Distributed Control**

In this section, we study the logistic delay equation with large coefficient of spatially distributed control of the form

$$\dot{N} = r[1 - N(t - T, \mathbf{x})]N + \gamma \left[ \int\_{-\infty}^{\infty} F(s)N(t, \mathbf{x} + s)ds - N \right].\tag{79}$$

The dependence of the functions *N*(*t*, *x*) on the spatial variable *x* is assumed to be periodic:

$$N(t, \ge +2\pi) \equiv N(t, \ge). \tag{80}$$

Thus, we fix the space *C*[−*T*,0]×[0,2*π*] as a phase space of the boundary value problem (79), (80).

Function *F*(*s*) describing spatial interactions is defined by

$$F(s) = \frac{1}{\sqrt{\mu \tau t}} \exp[-\mu^{-1}(s+h)^2], \quad \mu > 0. \tag{81}$$

We note that <sup>∞</sup> <sup>−</sup><sup>∞</sup> *<sup>F</sup>*(*s*)*ds* <sup>=</sup> 1. Apparently, study of the problem (79) and (80) is of great interest, provided that parameter *μ* appearing in (81) is sufficiently small:

$$0 < \mu \ll 1. \tag{82}$$

This condition arises naturally in many applied problems (see, for example, [31]). The assumption that the coefficient *γ* is large enough, i.e.,

$$
\gamma \gg 1 \tag{83}
$$

allows us to apply special asymptotic methods. In the next section, we study the local dynamics of solutions to the boundary value problem (79) and (80) under the conditions (82) and (83). The corresponding constructions are based on the results from [28]. Section 5.2 considers the local dynamics (i.e., in a small equilibrium state neighborhood) of the problem (79), (80). Critical cases are distinguished in the stability problem. The distinctive

feature of these critical cases is that they have infinite dimension. As a basic result, nonlinear boundary value problems of parabolic type are constructed that do not contain small and large parameters. Their nonlocal dynamics determine the behavior of the solutions of problem (79), (80) from small equilibrium state neighborhood *N*<sup>0</sup> = 1. The corresponding investigation is based on the papers [26,28]. We immediately pay attention to one of the conclusions obtained in Sections 5.1 and 5.2. As it turns out, complicated bifurcation phenomena in the problem (79), (80) can appear even for sufficiently small values of the delay time *T*. Here, we use the results obtained in the paper [32].
