# Vol 24, No 4 (2018)

**Year:**2018**Articles:**8**URL:**https://journals.ssau.ru/est/issue/view/317

## Full Issue

## Articles

### PROBLEM WITH AN INTEGRAL CONDITION FOR ONE-DIMENSIONAL HYPERBOLIC EQUATION

#### Abstract

In this paper, we study a nonlocal problem with an integral condition for a one-dimensional hyperbolic equation arising in the study of vibrations of the rod. The conditions for the input data providing unambiguous solvability of the problem are obtained, the proof of the existence and uniqueness of the solution of the problem is carried out.

**Vestnik of Samara University. Natural Science Series**. 2018;24(4):7-12

### MACKAY FUNCTIONS IN SPACES OF HIGHER LEVELS

#### Abstract

In the article we prove structure theorems for spaces of cusps forms with the levels that are divisible by the minimal levels for MakKay functions. There are 28 eta–products with multiplicative Fourier coeffi- cients. They are called MacKay functions. Let f(z) be such function. It belongs to the space Sl(Γ0(N), χ) for a minimal level N. In each space of the level N there is the exact cutting by the function f(z). Also the function f(z) is a cusp form for multiple levels. In this case the exact cutting doesn’t take place and the additional spaces exist. In this article we find the conditions for the divisor of functions that are divisible by f(z) and we study the structure of additional spaces. Dimensions of the spaces are calculated by the Cohen — Oesterle formula, the orders in cusps are calculated by the Biagioli formula.

**Vestnik of Samara University. Natural Science Series**. 2018;24(4):13-18

### NONLOCAL PROBLEMS FOR ONE-DIMENSIONAL HYPERBOLIC EQUATION

#### Abstract

This article discusses some nonlocal problems and methods of proving solvability of them. We show that choosing of effective method depends on the form of nonlocal conditions. One of these methods is based on reducing nonlocal problem to a boundary-value problem for a loaded equation and allows us to use many well-known methods of justification solvability. In the article, we consider the problem with nonlocal integral conditions for a one-dimensional hyperbolic equation and prove the equivalence to a problem with classical boundary conditions for a loaded equation.

**Vestnik of Samara University. Natural Science Series**. 2018;24(4):19-23

### ABOUT ONE TASK WITH A NONLOCAL CONDITION ON TIME VARIABLE FOR THE HYPERBOLIC EQUATION

#### Abstract

In this article, boundary value problem for hyperbolic equation with nonlocal initial data in integral form is considered. The main result is that the nonlocal problem is equivalent to the classical boundary value problem for a loaded equation. This fact helps to prove the uniqueness of a solution to the problem.

**Vestnik of Samara University. Natural Science Series**. 2018;24(4):24-28

### CALCULATION OF THE NUMBER OF PALINDROMS IN A BINARY SYSTEM

#### Abstract

The work deals with symmetric numbers in the binary number system, called palindromes. The aim of the work is to derive the dependence of the number of palindromes on their digit. The dependences of the number of palindromes for even and odd digits are obtained separately.

**Vestnik of Samara University. Natural Science Series**. 2018;24(4):29-32

### PARAMETRIZATION OF INVARIANT MANIFOLDS OF SLOW MOTIONS

#### Abstract

The method of integral manifolds is used to study the multidimensional systems of differential equations. This approach allows to solve an important problem of order reduction of differential systems. If a slow invariant manifold cannot be described explicitly then its parametrization is used for the system order reduction. In this case, either a part of the fast variables, or all fast variables, supplemented by a certain number of slow variables, can play a role of the parameters.

**Vestnik of Samara University. Natural Science Series**. 2018;24(4):33-40

### WHITE NOISE EFFECT IN THE DYNAMIC MODEL OF THE ELECTROCHEMICAL REACTION

#### Abstract

The work is devoted to the problem of the Gaussian white noise influence on a canard cycle in a dynamic model of an electrochemical reaction. This study is conducted on the example of an electrochemical reaction of the Cooper-Slyter type. An analysis of noise-induced transitions was performed, the effect of external disturbances on the limit cycle is investigated, the sensitivity of the cycle to the noise is found. A critical noise intensity, at which the small-amplitude oscillations are transformed into mixedmode oscillations, is obtained. It is shown that an increase in the intensity of random perturbations can lead to significant deformations of the modes in the model up to their destruction.

**Vestnik of Samara University. Natural Science Series**. 2018;24(4):41-47

### SIMULATION OF FLUCTUATIONS OF AGGRESSIVE ALIEN SPECIES IN CONTINUOUS MODELS WITH INDEPENDENT REGULATION

#### Abstract

Traditional models in biology do not describe modern extraordinary situations when the whole species composition is mixed. The article deals with oscillatory equations and non-dissipative population dynamics for specific environmental situations that are associated with alien species in ecosystems. During the invasion of new species, the resistance of the biotic environment may be completely absent for a considerable time. Under such conditions, with a high specific fecundity, nonstationary regimes of change in numbers arise. A pest outbreak is realized with an explosive growth phase. All outbreaks of fish and insects are some brief extreme episodes that end with a new state of the environment and the aggressive new species. Completion options are varied even in the example of one malicious species of comb jelly Mnemiopsis leidyi in Azov and Caspian Sea. Transition to relaxation oscillations after a comb jelly outbreak is possible. A new species may become a small group or even disappear in case minN(t; r) = 0. The paper proposes a model based on the lagging regulation for actual scenarios of population behaviour in a new environment. In computational experiments we have shown the conditions for stabilization after an outbreak in an extremely small group of individuals or complete disappearance after collapse. A separate scenario describes the complete depletion of environmental resources during fluctuations with a significant amplitude. The most relevant is the model scenario of stabilization at minimum values after a rapid change in the phases of the outbreak and a transition to the depression of the pest number in the modification of the differential equation of Bazykin and Hutchinson model.

**Vestnik of Samara University. Natural Science Series**. 2018;24(4):48-58