Vol 24, No 4 (2018)

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.

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.

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.

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.