Том 242, № 1 (2019)

The population size has far-reaching effects on the fitness of a population that in turn influences the population extinction or persistence. Understanding the density- and age-dependent factors will facilitate more accurate predictions about the population dynamics and its asymptotic behavior. In this paper, we develop a rigorous mathematical analysis of the study of positive and negative effects of the increased population density in the classical nonlinear age-structured population model. One of the main results expresses the global stability of the system in terms of the newborn function only. We establish the existence of a threshold population size implying the population extinction, which is well known in the population dynamics as the Allee effect.

Necessary and sufficient conditions for the validity of the central limit theorem for densities are considered with respect to the norms in Orlicz spaces. The obtained characterization unites several results due to Gnedenko and Kolmogorov (uniform local limit theorem), Prokhorov (convergence in total variation) and Barron (entropic central limit theorem).

We consider the system of boundary layer equations governing a viscous medium subject to the nonlinear rheological law in the sense of Ladyzhenskaya. Owing to the use of the Crocco transformation for reducing the system to a single quasilinear equation, it becomes possible to study both stationary and nonstationary boundary layers. We obtain asymptotic estimates for the solution.

For the space of (not necessarily polynomial) Hermite type splines we develop algorithms for constructing the spline-wavelet decomposition provided that an arbitrary coarsening of a nonuniform spline-grid is a priori given. The construction is based on approximate relations guaranteeing the asymptotically optimal (with respect to the N-diameter of standard compact sets) approximate properties of this decomposition. We study the structure of restriction and extension matrices and prove that each of these matrices is the one-sided inverse of the transposed other. We propose the decomposition and reconstruction algorithms consisting of a small number of arithmetical actions.

We discuss homogenization of the singular perturbation problem

\( {\Delta}_p{u}_{\delta}^{\varepsilon }={f}^{\varepsilon }{\beta}_{\delta}\left({u}_{\delta}^{\varepsilon}\right)\kern1em in\kern0.5em {\mathrm{\mathbb{R}}}^n\backslash \overline{B_1} \)

with a constant boundary value on the ball. Here, Δ_p is the usual p-Laplacian operator. It is generally understood that the two parameters δ and ε are in competition and two different behaviors may be exhibited, depending on which parameter tends to zero faster. We consider one scenario where we assume that ε, the homogenization parameter, tends to zero faster than δ, the singular perturbation parameter. We show that there is a universal speed for which the limit solves a standard Bernoulli free boundary problem.