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We analyze variants of considering the inhomogeneity of the environment in computer modeling of the dynamics of a predator and prey based on a system of reaction-diffusion–advection equations. The local interaction of species (reaction terms) is described by the logistic law for the prey and the Beddington –DeAngelis functional response, special cases of which are the Holling type II functional response and the Arditi – Ginzburg model. We consider a one-dimensional problem in space for a heterogeneous resource (carrying capacity) and three types of taxis (the prey to resource and from the predator, the predator to the prey). An analytical approach is used to study the stability of stationary solutions in the case of local interaction (diffusionless approach). We employ the method of lines to study diffusion and advective processes. A comparison of the critical values of the mortality parameter of predators is given. Analysis showed that at constant coefficients in the Beddington –DeAngelis model, critical values are variable along the spatial coordinate, while we do not observe this effect for the Arditi –Ginzburg model. We propose a modification of the reaction terms, which makes it possible to take into account the heterogeneity of the resource. Numerical results on the dynamics of species for large and small migration coefficients are presented, demonstrating a decrease in the influence of the species of local members on the emerging spatio-temporal distributions of populations. Bifurcation transitions are analyzed when changing the parameters of diffusion–advection and reaction terms.

This work is devoted to the numerical study of high-speed mixing layers of compressible flows. The problem under consideration has a wide range of applications in practical tasks and, despite its apparent simplicity, is quite complex in terms of modeling. Because in the mixing layer, as a result of the instability of the tangential discontinuity of velocities, the flow passes from laminar flow to turbulent mode. Therefore, the obtained numerical results of the considered problem strongly depend on the adequacy of the used turbulence models. In the presented work, this problem is studied based on the two-fluid approach to the problem of turbulence. This approach has arisen relatively recently and is developing quite rapidly. The main advantage of the two-fluid approach is that it leads to a closed system of equations, when, as is known, the long-standing Reynolds approach leads to an open system of equations. The paper presents the essence of the two-fluid approach for modeling a turbulent compressible medium and the methodology for numerical implementation of the proposed model. To obtain a stationary solution, the relaxation method and Prandtl boundary layer theory were applied, resulting in a simplified system of equations. In the considered problem, high-speed flows are mixed. Therefore, it is also necessary to model heat transfer, and the pressure cannot be considered constant, as is done for incompressible flows. In the numerical implementation, the convective terms in the hydrodynamic equations were approximated by the upwind scheme with the second order of accuracy in explicit form, and the diffusion terms in the right-hand sides of the equations were approximated by the central difference in implicit form. The sweep method was used to implement the obtained equations. The SIMPLE method was used to correct the velocity through the pressure. The paper investigates a two-liquid turbulence model with different initial flow turbulence intensities. The obtained numerical results showed that good agreement with the known experimental data is observed at the inlet turbulence intensity of $0.1 < I < 1 \%$. Data from known experiments, as well as the results of the $k − kL + J$ and LES models, are presented to demonstrate the effectiveness of the proposed turbulence model. It is demonstrated that the two-liquid model is as accurate as known modern models and more efficient in terms of computing resources.

Efficient method for numerical solving of the steady transport equation in x-y-z-geometry has been suggested. The equation is being solved on hexagonal mesh, reflecting real structure of the reactor active zone cross-section. Method of characteristics is used, that inherits all the outcomes from the two-dimensional r-z-geometry calculation. Two variants of the method of characteristics have been applied for solving the transport equation in a cell: method of short characteristics and its conservative modification. It has been confirmed that in three-dimensional geometry conservative method has advantage over pure characteristic and it produces highly accurate solution, especially for quasi-diffusion tensor components.

The results of the experimental and theoretical researches of kinetics of erythrosine penetration into collagenase molecules have represented in this paper. The case with introduction of the compound (fluorescent probe) which has dimers to enzyme solution as an injection has been considered. It was shown that monomers and dimers can penetrate into enzyme molecules with formation complexes monomer — enzyme, dimer- enzyme. Moreover, transformation of probe fluorescence spectra is at each time moment. Spectrum maximum shift, and its form change. At a time, the immobilized dye dimers greatly impact to formation of end fluorescence spectrum. Well correlation between experimental and theoretical results confirms reality of the obtained data.

Functional modeling of blood clotting and fibrin-polymer mesh formation is of a significant value for medical and biophysics applications. Despite the fact of some discrepancies present in simplified functional models their results are of the great interest for the experimental science as a handy tool of the analysis for research planning, data processing and verification. Under conditions of the good correspondence to the experiment functional models can be used as an element of the medical treatment methods and biophysical technologies. The aim of the paper in hand is a modeling of a point system of the fibrin-polymer formation as a multistage polymerization process with a sol-gel transition at the final stage. Complex-value Rosenbroke method of second order (CROS) used for computational experiments. The results of computational experiments are presented and discussed. It was shown that in the physiological range of the model coefficients there is a lag period of approximately 20 seconds between initiation of the reaction and fibrin gel appearance which fits well experimental observations of fibrin polymerization dynamics. The possibility of a number of the consequent $(n = 1–3)$ sol-gel transitions demonstrated as well. Such a specific behavior is a consequence of multistage nature of fibrin polymerization process. At the final stage the solution of fibrin oligomers of length 10 can reach a semidilute state, leading to an extremely fast gel formation controlled by oligomers’ rotational diffusion. Otherwise, if the semidilute state is not reached the gel formation is controlled by significantly slower process of translational diffusion. Such a duality in the sol-gel transition led authors to necessity of introduction of a switch-function in an equation for fibrin-polymer formation kinetics. Consequent polymerization events can correspond to experimental systems where fibrin mesh formed gets withdrawn from the volume by some physical process like precipitation. The sensitivity analysis of presented system shows that dependence on the first stage polymerization reaction constant is non-trivial.

The study tested correctness of three interatomic potentials available in the scientific literature in reproducing a vacancy diffusion in concentrated Fe–Cr alloys by molecular dynamic simulations. It was necessary for further detailed study of vacancy diffusion mechanism in these alloys with Cr content 5–25 at.% at temperatures in the range of 600–1000 K. The analysis of the potentials was performed on alloys models with Cr content 10, 20, 50 at.%. The consideration of the model with chromium content 50 at.% was necessary for further study of diffusion processes in chromium-rich precipitates in these alloys. The formation energies and the atomic mobilities of iron and chromium atoms were calculated and analyzed in the alloys via an artificially created vacancy for all used potentials. A time dependence of mean squared displacement of atoms was chosen as а main characteristic for the analysis of atomic mobilities. The simulation of vacancy formation energies didn’t show qualitative differences between the investigated potentials. The study of atomic mobilities showed a poor reproduction of vacancy diffusion in the simulated alloys by the concentration-dependent model (CDM), which strongly underestimated the mobility of chromium atoms via vacancy in the investigated range of temperature and chromium content. Also it was established, that the two-band model (2BM) of potentials in its original and modified version doesn’t have such drawbacks. This allows one to use these potentials in simulations of vacancy diffusion mechanism in Fe–Cr alloys. Both potentials show a significant dependence of the ratio of chromium and iron atomic mobilities on temperature and Cr content in simulated alloys. The quantitative data of the diffusion coefficients of atoms obtained by these potentials also differ significantly.

In this paper, a mathematical model of cellular tissue dynamics is considered. The first part gives the conclusion of the model, the main provisions and the formulation of the problem. In the second part, the final system is investigated numerically and the simulation results are presented. It is postulated that cellular tissue is a three-phase medium that consists of a solid skeleton (which is an extracellular matrix), cells and extracellular fluid. In addition, the presence of nutrients in the tissue is taken into account. The model is based on the equations of conservation of mass, taking into account mass exchange, the equations of conservation of momentum for each phase, as well as the diffusion equation for nutrients. The equation describing the cellular phase also takes into account the term describing the chemical effect on the tissue, which is called chemotaxis — the movement of cells caused by a gradient in the concentration of chemicals. The initial system of equations is reduced to a system of three equations for finding porosity, cell saturation and nutrient concentration. These equations are supplemented by initial and boundary conditions. In the one-dimensional case, the distribution of porosity, concentration of the cell phase and nutrients is set at the initial moment of time. A constant concentration of nutrients is set on the left border, which corresponds, for example, to the supply of oxygen from the vessel, as well as the flow of cell concentration on it is zero. Two types of conditions are considered at the right boundary: the first is the condition of impermeability of the right boundary, the second is the condition of constant concentration of the cell phase and zero flow of nutrient concentration. In both cases, the conditions for the matrix and extracellular fluid are the same, it is assumed that there is a source of nutrients (blood vessel) on the left border of the modeling area. As a result of modeling, it was revealed that chemotaxis has a significant effect on tissue growth. In the absence of chemotaxis, the compaction zone extends to the entire modeling area, but with an increase in the effect of chemotaxis on the tissue, a degradation area is formed in which the concentration of cells becomes lower than the initial one.

We developed a model of honeybee colony foraging based on reaction – diffusion equations. Employed bees transmit information about their food sources using dance, and job seekers in the hive can choose any dance they like and thus join the exploitation of the corresponding source. We consider two strategies of dance selection: a targeted one, when bees analyze information on the dance floor and choose the most energetic and longest dance corresponding to the most profitable source, and a simple random choice of the first dance they encounter. Modelling showed that the greatest profit (food influx into the hive) is provided by the random choice of dance, as paradoxical as it may seem at first glance. Optimization of profit by each agent for itself (targeted choice of dances) is rather a disadvantage for the colony, and “non-optimality” in dance choice can be the result of useful evolutionary adaptation.

A correct model of lipid molecule (distearoylphosphatidylcholine, DSPC) and lipid membrane in water was constructed. Model lipid membrane is stable and has a reliable energy distribution among degrees of freedom. Also after equilibration model system has spatial parameters very similar to those of real DSPC membrane in liquid-crystalline phase. This model was used for studying of lipid membrane permeability to oxygen and water molecules and sodium ion. We obtained the values for transmembrane mobility and diffusion coefficients profiles, which we used for effective permeability coefficients calculation. We found lipid membranes to have significant diffusional resistance to penetration not only by charged particles, such as ions, but also by nonpolar molecules, such as oxygen molecule. We propose theoretical approach for calculation of particle flow across a membrane, as well as methods for estimation of distribution coefficients between bilayer and water phase.

The possibility to simplify the integration of Langevin equation using the variation of correlation between augmentation was researched. The analytical expression for a set of numerical schemes is presented. It’s shown that asymptotic limits for squared velocity depend on step size. The region of convergence and the convergence orders were estimated. It turned out that the incorrect correlation between increments decrease the accuracy down to the level of first-order methods for schemes based on precise solution.