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Рассмотрено важное для приложений логистическое уравнение с запаздыванием и диффузией. Предполагается, что все его коэффициенты, а также коэффициенты в граничных условиях являются быстро осциллирующими функциями времени. Построено усредненное уравнение и рассмотрен вопрос о связи между его решениями и решениями исходного уравнения. Сформулирован результат об устойчивости решений и изучена задача о локальной динамике в критическом случае. Предложен алгоритм построения асимптотики решений и алгоритм исследования устойчивости. Важно отметить, что соответствующий алгоритм содержит как регулярные, так и погранслойные составляющие. Приведены содержательные примеры. Библиография: 23 названия.

A system of two logistic equations with delay coupled by delayed control has been considered. It has been shown that, in the case of a fairly large delay control coefficient, the problem of the dynamics of the initial systems has been reduced to investigating the nonlocal dynamics of special families of partial differential equations that do not contain small and large parameters. New interesting dynamic phenomena are discovered based on the results of numerical analysis. Systems of three logistic delay equations with two types of diffusion relations have been considered. Special families of partial differential equations that do not contain small and large parameters have also been constructed for each of these systems. The research results for the dynamic properties of the original equations have been presented. It has been shown that the difference in the dynamics of the considered systems of three equations may be of a fundamental nature.

A new asymptotic method for investigating complex relaxation oscillations in a system with delay is offered. By applying it, we can reduce the problem of predator–prey system dynamics to that of one-dimensional maps analysis. Some conclusions of a biological nature based on asymptotic analysis are made.

Asymptotic solutions of parabolic boundary value problems are studied in a neighborhood of both an equilibrium state and a cycle in near-critical cases which can be considered as infinite-dimensional due to small values of the diffusion coefficients. Algorithms are developed to construct normalized equations in such situations. Principle difference between bifurcations in two-dimensional and one-dimensional spatial systems is demonstrated.

In this paper the local dynamics of systems of nonlinear PDEs with small diffusion is studied. The main feature of these systems lies in the fact that the dimension of a critical case in the stability problem for an equilibrium state is equal to infinity. Algorithms that reduce the initial problem to the analysis of nonlocal dynamics of special evolution equations playing the role of normal forms are developed.

We consider the problem of density wave propagation in a logistic equation with delay and diffusion (Fisher-Kolmogorov equation with delay). A Ginzburg-Landau equation was constructed in order to study the qualitative behavior of the solution near the equilibrium state. The numerical analysis of wave propagation shows that for a sufficiently small delay this equation has a solution similar to the solution of a classical Fisher-Kolmogorov equation. The delay increasing leads to existence of the oscillatory component in spatial distribution of solutions. A further increase of delay leads to destruction of the traveling wave. That is expressed in the fact that undamped spatio-temporal fluctuations exist in a neighborhood of the initial perturbation. These fluctuations are close to the solution of the corresponding boundary value problem with periodic boundary conditions. Finally, when the delay is sufficiently large we observe intensive spatio-temporal fluctuations in the whole area of wave propagation.

This paper considers the behavior of solutions from the neighborhood of an equilibrium state of nonlinear two-component parabolic problems with diffusion matrixes of one or two eigenvalues to zero. It has been shown that problems related to stability have infinite dimension. Reported here is the development of an algorithm that constructs universal families of nonlinear boundary-value problems which do not contain small parameters and whose nonlocal dynamics describes local dynamics of original boundary-value problems. In addition, an exhaustive set of universal systems for two-component parabolic equations is presented. It is concluded that a hyper multistability phenomenon is one characteristic of these systems.