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We have investigated dynamics of synthetic genetic oscillators — repressilators — coupled through autoinducer diffusion. The model of the system with phase-repulsive coupling structure is under consideration. We have examined emergence of periodic regimes, stable inhomogeneous steady states depending on the main systems’ parameters: coupling strength and maximal transcription rate. It has been shown that autoinducer production module added to the isolated repressilator cause the limit cycle to disappear through infinite period bifurcation for sufficiently large transcription rate. We have found hysteresis of limit cycle and stable steady state the size of which is determined by ratio between mRNA and protein lifetimes. Two coupled oscillators system demonstrates stable anti-phase oscillations which can become a chaotic regime through invariant torus emergence or via Feigenbaum scenario.

In case of vessel wall damage or contact of blood plasma with a foreign surface, the chain of chemical reactions called coagulation cascade is launched that leading to the formation of a fibrin clot. A key enzyme of the coagulation cascade is thrombin, which catalyzes formation of fibrin from fibrinogen. The distribution of thrombin concentration in blood plasma determines spatio-temporal dynamics of clot formation. Contact pathway of blood coagulation triggers the production of thrombin in response to the contact with a negatively charged surface. If the concentration of thrombin generated at this stage is large enough, further production of thrombin takes place due to positive feedback loops of the coagulation cascade. As a result, thrombin propagates in plasma cleaving fibrinogen that results in the clot formation. The concentration profile and the speed of propagation of thrombin are constant and do not depend on the type of the initial activator.

Such behavior of the coagulation system is well described by the traveling wave solutions in a system of “reaction – diffusion” equations on the concentration of blood factors involved in the coagulation cascade. In this study, we carried out detailed analysis of the mathematical model describing the main reaction of the intrinsic pathway of coagulation cascade.We formulate necessary and sufficient conditions of the existence of the traveling wave solutions. For the considered model the existence of such solutions is equivalent to the existence of the wave solutions in the simplified one-equation model describing the dynamics of thrombin concentration derived under the quasi-stationary approximation.

Simplified model also allows us to obtain analytical estimate of the thrombin propagation rate in the considered model. The speed of the traveling wave for one equation is estimated using the narrow reaction zone method and piecewise linear approximation. The resulting formulas give a good approximation of the velocity of propagation of thrombin in the simplified, as well as in the original model.

The article reports the findings of an investigation into pollution migration processes at the municipal solid waste (MSW) landfill located in the water protection zone of Lake Seliger (Tver Region). The distribution of pollutants is investigated and migration parameters are determined in field and laboratory conditions at the landfill site. A mathematical model describing physical and chemical processes of substance migration in soil strata is constructed. Pollutant migration is found to be due to a variety of factors. The major ones, having a significant impact on the migration of MSW ingredients and taken into account mathematically, include convective transport, diffusion and sorption processes. A modified mathematical model differs from its conventional counterparts by considering a number of parameters reflecting the decrease in the concentration of ammonium and nitrate nitrogen ions in ground water (transpiration by plant roots, dilution with infiltration waters, etc.). An analytical solution to assess the pollutant spread from the landfill is presented. The mathematical model provides a set of simulation models helping to obtain a computational solution of specific problems, vertical and horizontal migration of substances in the underground flow. Numerical experiments, analytical solutions, as well as field and laboratory data was studied the dynamics of pollutant distribution in the object under study up to the lake. A long-term forecast for the spread of landfill pollution is made. Simulation experiments showed that some zones of clean groundwater interact with those of contaminated groundwater during the pollution migration from the landfill, each characterized by a different pollutant content. The data of a computational experiments and analytical calculations are consistent with the findings of field and laboratory investigations of the object and give grounds to recommend the proposed models for predicting pollution migration from a landfill. The analysis of the pollution migration simulation allows to substantiate the numerical estimates of the increase in $NH_4^+$ and $NO_3^-$ ion concentration with the landfill operation time. It is found that, after 100 years following the landfill opening, toxic filtrate components will fill the entire pore space from the landfill to the lake resulting in a significant deterioration of the ecosystem of Lake Seliger.

High-purity hydrogen is required for clean energy and a variety of chemical technology processes. A considerable part of hydrogen is to be obtained by methane conversion. Different alloys, which may be wellsuited for use in gas-separation plants, were investigated by measuring specific hydrogen permeability. One had to estimate the parameters of diffusion and sorption to numerically model the different scenarios and experimental conditions of the material usage (including extreme ones), and identify the limiting factors. This paper presents a nonlinear model of hydrogen permeability in accordance with the specifics of the experiment, the numerical method for solving the boundary-value problem, and the results of parametric identification for the alloy V₈₅Ni₁₅.

A vertically distributed three-component model of marine ecosystem is considered. State of the plankton community with nutrients is analyzed under the active movement of zooplankton in a vertical column of water. The necessary conditions of the Turing instability in the vicinity of the spatially homogeneous equilibrium are obtained. Stability of the spatially homogeneous equilibrium, the Turing instability and the oscillatory instability are examined depending on the biological characteristics of zooplankton and spatial movement of plankton. It is shown that at low values of zooplankton grazing rate and intratrophic interaction rate the system is Turing instable when the taxis rate is low. Stabilization occurs either through increased decline of zooplankton either by increasing the phytoplankton diffusion. With the increasing rate of consumption of phytoplankton range of parameters that determine the stability is reduced. A type of instability depends on the phytoplankton diffusion. For large values of diffusion oscillatory instability is observed, with a decrease in the phytoplankton diffusion zone of Turing instability is increases. In general, if zooplankton grazing rate is faster than phytoplankton growth rate the spatially homogeneous equilibrium is Turing instable or oscillatory instable. Stability is observed only at high speeds of zooplankton departure or its active movements. With the increase in zooplankton search activity spatial distribution of populations becomes more uniform, increasing the rate of diffusion leads to non-uniform spatial distribution. However, under diffusion the total number of the population is stabilized when the zooplankton grazing rate above the rate of phytoplankton growth. In general, at low rate of phytoplankton consumption the spatial structures formation is possible at low rates of zooplankton decline and diffusion of all the plankton community. With the increase in phytoplankton predation rate the phytoplankton diffusion and zooplankton spatial movement has essential effect on the spatial instability.

Theoretical and experimental investigations of water vapor interaction with porous materials are carried out both at the macro level and at the micro level. At the macro level, the influence of the arrangement structure of individual pores on the processes of water vapor interaction with porous material as a continuous medium is studied. At the micro level, it is very interesting to investigate the dependence of the characteristics of the water vapor interaction with porous media on the geometry and dimensions of the individual pore.

In this paper, a study was carried out by means of mathematical modelling of the processes of water vapor interaction with suffering pore of the cylindrical type. The calculations were performed using a model of a hybrid type combining a molecular-dynamic and a macro-diffusion approach for describing water vapor interaction with an individual pore. The processes of evolution to the state of thermodynamic equilibrium of macroscopic characteristics of the system such as temperature, density, and pressure, depending on external conditions with respect to pore, were explored. The dependence of the evolution parameters on the distribution of the diffusion coefficient in the pore, obtained as a result of molecular dynamics modelling, is examined. The relevance of these studies is due to the fact that all methods and programs used for the modelling of the moisture and heat conductivity are based on the use of transport equations in a porous material as a continuous medium with known values of the transport coefficients, which are usually obtained experimentally.

The background social strain of a society can be quantitatively estimated using various statistical indicators. Mathematical models, allowing to forecast the dynamics of social strain, are successful in describing various social processes. If the number of interacting groups is small, the dynamics of the corresponding indicators can be modelled with a system of ordinary differential equations. The increase in the number of interacting components leads to the growth of complexity, which makes the analysis of such models a challenging task. A continuous social stratification model can be considered as a result of the transition from a discrete number of interacting social groups to their continuous distribution in some finite interval. In such a model, social strain naturally spreads locally between neighbouring groups, while in reality, the social elite influences the whole society via news media, and the Internet allows non-local interaction between social groups. These factors, however, can be taken into account to some extent using the term of the model, describing negative external influence on the society. In this paper, we develop a continuous social stratification model, describing the dynamics of two societies connected through migration. We assume that people migrate from the social group of donor society with the highest strain level to poorer social layers of the acceptor society, transferring the social strain at the same time. We assume that all model parameters are constants, which is a realistic assumption for small societies only. By using the finite volume method, we construct the spatial discretization for the problem, capable of reproducing finite propagation speed of social strain. We verify the discretization by comparing the results of numerical simulations with the exact solutions of the auxiliary non-linear diffusion equation. We perform the numerical analysis of the proposed model for different values of model parameters, study the impact of migration intensity on the stability of acceptor society, and find the destabilization conditions. The results, obtained in this work, can be used in further analysis of the model in the more realistic case of inhomogeneous coefficients.

We consider a model of populations that are closely related and share a common areal. System of nonlinear parabolic equations is formulated that incorporates nonlinear diffusion and migration flows induced by nonuniform densities of population and carrying capacity. We employ the method of lines and study the impact of migration on scenarios of local competition and coexistence of species. Conditions on system parameters are determined when a nontrivial family of steady states is formed.

Population dynamics is considered in a modified chemostat model including diffusion, chemotaxis, and nonlocal competitive losses. To account for influence of the external environment on the population of the ecosystem, a random parameter is included into the model equations. Computer simulations reveal three dynamic modes depending on system parameters: the transition from initial state to a spatially homogeneous steady state, to a spatially inhomogeneous distribution of population density, and elimination of population density.

A model, describing the influence of external factors on temporal evolution of phytoplankton distribution in a horizontally-homogenous water layer, is presented. This model is based upon the reactiondiffusion equation and takes into account the main factors of influence: mineral nutrients, insolation and temperature. The mineral nutrients and insolation act oppositely on spatial phytoplankton distribution. The results of numerical modeling are presented and the prospect of applying this model to reconstruction of phytoplankton distribution from sea-surface satellite data is discussed. The model was used to estimate the chlorophyll content of the Peter the Great Bay (Sea of Japan).