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The aim of the work is to study a monotone finite-difference scheme of the second order of accuracy, created on the basis of a generalization of the one-dimensional Z-scheme. The study was carried out for model equations of the transfer of an incompressible medium. The paper describes a two-dimensional generalization of the Z-scheme with nonlinear correction, using instead of streams oblique differences containing values from different time layers. The monotonicity of the obtained nonlinear scheme is verified numerically for the limit functions of two types, both for smooth solutions and for nonsmooth solutions, and numerical estimates of the order of accuracy of the constructed scheme are obtained.

The constructed scheme is absolutely stable, but it loses the property of monotony when the Courant step is exceeded. A distinctive feature of the proposed finite-difference scheme is the minimality of its template. The constructed numerical scheme is intended for models of plasma instabilities of various scales in the low-latitude ionospheric plasma of the Earth. One of the real problems in the solution of which such equations arise is the numerical simulation of highly nonstationary medium-scale processes in the earth’s ionosphere under conditions of the appearance of the Rayleigh – Taylor instability and plasma structures with smaller scales, the generation mechanisms of which are instabilities of other types, which leads to the phenomenon F-scattering. Due to the fact that the transfer processes in the ionospheric plasma are controlled by the magnetic field, it is assumed that the plasma incompressibility condition is fulfilled in the direction transverse to the magnetic field.

The article considers the problem of the stability of the vertical position of a Maxwell pendulum during its periodic up-down movements. Two types of transition movements are considered: “stop” — occurs when the body of the pendulum in its highest position on the string (during its “standard” upward movement) stops for a moment; “two-link pendulum” — occurs when the entire thread from the body of the pendulum is selected (the lowest position of the body on the thread during its “standard” downward movement), and the body is forced to rotate relative to the thread around the point of its attachment to the body. It is shown that for any values of the pendulum parameters, this position is unstable in the sense that oscillations of the thread around the vertical of finite amplitude occur in the system for arbitrarily small initial deviations. In addition, it has been established that no shock phenomena occur during the movement of the Maxwell pendulum, and the model of this pendulum itself, with the values of its parameters often used in the literature, is incorrect according to Hadamard. In this work, it is shown that the vertical position of the pendulum threads during the indicated oscillatory movements of the body along the threads for any non-degenerate values of the parameters of the Maxwell pendulum is always unstable in the above sense. Moreover, this instability is caused precisely by transitional movements of the 2nd type. In this work, it is further shown that no jumps in speeds or accelerations (due to which shocks or “jerks” in the tension of the threads can occur) do not occur during the indicated movements of the Maxwell pendulum model under consideration. In our opinion, the “jerks” observed in the experiments are due to other reasons, for example, the technical imperfection of the instruments on which the experiments were carried out.

Execution or violation of the principle of competitive exclusion in communities is the subject of many studies. The principle of competitive exclusion means that coexistence of species in community is impossible if the number of species exceeds the number of controlling mutually independent factors. At that time there are many examples displaying the violations of this principle in the natural systems. The explanations for this paradox vary from inexact identification of the set of factors to various types of spatial and temporal heterogeneities. One of the factors breaking the principle of competitive exclusion is intraspecific competition. This study holds the model of community with two species and one influencing factor with density-dependent mortality and spatial heterogeneity. For such models possibility of the existence of stable equilibrium is proved in case of spatial homogeneity and negative effect of the species on the factor. Our purpose is analysis of possible variants of dynamics of the system with spatial heterogeneity under the various directions of the species effect on the influencing factor. Numerical analysis showed that there is stable coexistence of the species agreed with homogenous spatial distributions of the species if the species effects on the influencing factor are negative. Density-dependent mortality and spatial heterogeneity lead to violation of the principle of competitive exclusion when equilibriums are Turing unstable. In this case stable spatial heterogeneous patterns can arise. It is shown that Turing instability is possible if at least one of the species effects is positive. Model nonlinearity and spatial heterogeneity cause violation of the principle of competitive exclusion in terms of both stable spatial homogenous states and quasistable spatial heterogeneous patterns.

The paper deals with a prey-predator model, which describes the spatiotemporal dynamics of plankton community and the nutrients. The system is described by reaction-diffusion-advection equations in a onedimensional vertical column of water in the surface layer. Advective term of the predator equation represents the vertical movements of zooplankton with velocity, which is assumed to be proportional to the gradient of phytoplankton density. This study aimed to determine the conditions under which these movements (taxis) lead to the spatially heterogeneous structures generated by the system. Assuming diffusion coefficients of all model components to be equal the instability of the system in the vicinity of stationary homogeneous state with respect to small inhomogeneous perturbations is analyzed.

Necessary conditions for the flow-induced instability were obtained through linear stability analysis. Depending on the local kinetics parameters, increasing the taxis rate leads to Turing or wave instability. This fact is in good agreement with conditions for the emergence of spatial and spatiotemporal patterns in a minimal phytoplankton–zooplankton model after flow-induced instabilities derived by other authors. This mechanism of generating patchiness is more general than the Turing mechanism, which depends on strong conditions on the diffusion coefficients.

While the taxis exceeding a certain critical value, the wave number corresponding to the fastest growing mode remains unchanged. This value determines the type of spatial structure. In support of obtained results, the paper presents the spatiotemporal dynamics of the model components demonstrating Turing-type pattern and standing wave pattern.

This work is devoted to the analysis of conformational changes in tubulin dimers and tetramers, in particular, the assessment of the bending of microtubule protofilaments. Three recently exploited approaches for estimating the bend of tubulin protofilaments are reviewed: (1) measurement of the angle between the vector passing through the H7 helices in $\alpha$ and $\beta$ tubulin monomers in the straight structure and the same vector in the curved structure of tubulin; (2) measurement of the angle between the vector, connecting the centers of mass of the subunit and the associated GTP nucleotide, and the vector, connecting the centers of mass of the same nucleotide and the adjacent tubulin subunit; (3) measurement of the three rotation angles of the bent tubulin subunit relative to the straight subunit. Quantitative estimates of the angles calculated at the intra- and inter-dimer interfaces of tubulin in published crystal structures, calculated in accordance with the three metrics, are presented. Intra-dimer angles of tubulin in one structure, measured by the method (3), as well as measurements by this method of the intra-dimer angles in different structures, were more similar, which indicates a lower sensitivity of the method to local changes in tubulin conformation and characterizes the method as more robust. Measuring the angle of curvature between H7-helices (method 1) produces somewhat underestimated values of the curvature per dimer. Method (2), while at first glance generating the bending angle values, consistent the with estimates of curved protofilaments from cryoelectron microscopy, significantly overestimates the angles in the straight structures. For the structures of tubulin tetramers in complex with the stathmin protein, the bending angles calculated with all three metrics varied quite significantly for the first and second dimers (up to 20% or more), which indicates the sensitivity of all metrics to slight variations in the conformation of tubulin dimers within these complexes. A detailed description of the procedures for measuring the bending of tubulin protofilaments, as well as identifying the advantages and disadvantages of various metrics, will increase the reproducibility and clarity of the analysis of tubulin structures in the future, as well as it will hopefully make it easier to compare the results obtained by various scientific groups.

The spatiotemporal dynamics of a three-component model for food web is considered. The model describes the interactions among resource, prey and predator that consumes both species. In a previous work, the author analyzed the model without taking into account spatial heterogeneity. This study continues the model study of the community considering the diffusion of individuals, as well as directed movements of the predator. It is assumed that the predator responds to the spatial change in the resource and prey density by occupying areas where species density is higher or avoiding them. Directed predator movement is described by the advection term, where velocity is proportional to the gradient of resource and prey density. The system is considered on a one-dimensional domain with zero-flux conditions as boundary ones. The spatiotemporal dynamics produced by model is determined by the system stability in the vicinity of stationary homogeneous state with respect to small inhomogeneous perturbations. The paper analyzes the possibility of wave instability leading to the emergence of autowaves and Turing instability, as a result of which stationary patterns are formed. Sufficient conditions for the existence of both types of instability are obtained. The influence of local kinetic parameters on the spatial structure formation was analyzed. It was shown that only Turing instability is possible when taxis on the resource is positive, but with a negative taxis, both types of instability are possible. The numerical solution of the system was found by using method of lines (MOL) with the numerical integration of ODE system by means of splitting techniques. The spatiotemporal dynamics of the system is presented in several variants, realizing one of the instability types. In the case of a positive taxis on the prey, both autowave and stationary structures are formed in smaller regions, with an increase in the region size, Turing structures are not formed. For negative taxis on the prey, stationary patterns is observed in both regions, while periodic structures appear only in larger areas.

Despite the widespread use of cancer chemotherapy drugs, the molecular mechanisms of action of many of them remain unclear. Some of these drugs, such as taxol, are known to affect the dynamics of microtubule assembly and stop the process of cell division in prophase-prometaphase. Recently, new spatial structures of microtubules and individual tubulin oligomers have emerged associated with various regulatory proteins and cancer chemotherapy drugs. However, knowledge of the spatial structure in itself does not provide information about the mechanism of action of drugs.

In this work, we applied the molecular dynamics method to study the behavior of taxol-bound tubulin oligomers and used our previously developed method for analyzing the conformation of tubulin protofilaments, based on the calculation of modified Euler angles. Recent structures of microtubule fragments have demonstrated that tubulin protofilaments bend not in the radial direction, as many researchers assume, but at an angle of approximately 45◦ from the radial direction. However, in the presence of taxol, the bending direction shifts closer to the radial direction. There was no significant difference between the mean bending and torsion angles of the studied tubulin structures when bound to the various natural regulatory ligands, guanosine triphosphate and guanosine diphosphate. The intra-dimer bending angle was found to be greater than the interdimer bending angle in all analyzed trajectories. This indicates that the bulk of the deformation energy is stored within the dimeric tubulin subunits and not between them. Analysis of the structures of the latest generation of tubulins indicated that the presence of taxol in the tubulin beta subunit pocket allosterically reduces the torsional rigidity of the tubulin oligomer, which could explain the underlying mechanism of taxol’s effect on microtubule dynamics. Indeed, a decrease in torsional rigidity makes it possible to maintain lateral connections between protofilaments, and therefore should lead to the stabilization of microtubules, which is what is observed in experiments. The results of the work shed light on the phenomenon of dynamic instability of microtubules and allow to come closer to understanding the molecular mechanisms of cell division.

An approach to computation of open-loop optimal Nash–Cournot strategies in dynamical games which is based on the Z-transform method and factorization is proposed. The main advantage of the proposed approach is that it permits to overcome the problems of instability of economic indicators of oligopolies arising when generalized Riccati equations are used.

The paper presents the results of applying a scheme of very high accuracy and resolution to obtain numerical solutions of the Navier – Stokes equations of a compressible gas describing the occurrence and development of instability of a two-dimensional laminar boundary layer on a flat plate. The peculiarity of the conducted studies is the absence of commonly used artificial exciters of instability in the implementation of direct numerical modeling. The multioperator scheme used made it possible to observe the subtle effects of the birth of unstable modes and the complex nature of their development caused presumably by its small approximation errors. A brief description of the scheme design and its main properties is given. The formulation of the problem and the method of obtaining initial data are described, which makes it possible to observe the established non-stationary regime fairly quickly. A technique is given that allows detecting flow fluctuations with amplitudes many orders of magnitude smaller than its average values. A time-dependent picture of the appearance of packets of Tollmien – Schlichting waves with varying intensity in the vicinity of the leading edge of the plate and their downstream propagation is presented. The presented amplitude spectra with expanding peak values in the downstream regions indicate the excitation of new unstable modes other than those occurring in the vicinity of the leading edge. The analysis of the evolution of instability waves in time and space showed agreement with the main conclusions of the linear theory. The numerical solutions obtained seem to describe for the first time the complete scenario of the possible development of Tollmien – Schlichting instability, which often plays an essential role at the initial stage of the laminar-turbulent transition. They open up the possibilities of full-scale numerical modeling of this process, which is extremely important for practice, with a similar study of the spatial boundary layer.