All issues

The paper considers an earlier proposed by the author discrete modification of the A. P. Mikhailov “power – society” model. The modification is based on a stochastic cellular automaton, it’s microdynamics being completely different from the c continuous model based on differential equations. However, the macrodynamics of the discrete modification is shown in previous works to be equivalent to one of the continuous model. This is important, but at the same time raises the question why use the discrete model. The answer lies in its flexibility, which allows adding a variety of factors, the consideration of which in a continuous model either leads to a significant increase in computational complexity or is simply impossible.

This paper considers several examples of such applicability expansion of the model, with the help of which a number of applied problems are solved.

One of the modifications of the model takes into account economic ties between regions and municipalities, which could not be studied in the basic model. Computational experiments confirmed the improvement of the socio-economic indicators of the system under the influence of the ties.

The second modification allows internal migration in the system. Using it we studied the socio-economic development of a more prosperous region that attracts migrants.

Next we studied the dynamics of the system while the number of regions and municipalities changes. The negative impact of this process on the socio-economic indicators of the system was shown and possible control was found to overcome this negative impact.

The results of this study, therefore, include both the solution of some applied problems and the demonstration of the broader applicability of the discrete model compared with the continuous one.

A mathematical model describing the dynamics of HIV-1 infection in a single lymph node during the initial period of infection development is presented. Within the framework of the model, the infection of an individual is set by a nonnegative finite function describing the rate of entry of the initial viral particles into the lymph node. The equations of the model are derived with consideration of two factors: 1) the interaction of viral particles with naive CD4+ T lymphocytes in various phases of the cell cycle; 2) contact interaction between multiplying naive CD4+ T lymphocytes and infected CD4+ T lymphocytes producing viral particles. The specific feature of intercellular contact interactions is the formation of complexes consisting of pairs of these cells. The duration of the complexes’ existence is determined by the distribution functions over finite time intervals. The model is presented as a high-dimensional system of nonlinear delay differential equations, including two equations with distributed delay, and is supplemented with non-negative initial data. In the absence of HIV-1 infection, the model is reduced to four delay differential equations describing the number of naive CD4+ T-lymphocytes in different phases of the cell cycle. The global solvability of the model (the existence and uniqueness of the solution on the semi-axis) is determined, and the non-negativity of the solution components is established. To carry out computational experiments with the model, an algorithm for numerically solving the used system of differential equations are developed based on the semi-implicit Euler scheme for the case of uniform distribution of durations of the complexes existence. The results of computational experiments aimed at approximation the numerical solution of the model to describing the kinetics of HIV-1 infection spread in its acute phase, including the eclipse phase, are presented. The variable used as the observable is the variable describing the number of viral particles per milliliter of blood on days 10–12 after the onset of acute infection. The dynamics of the observable variable is numerically studied depending on the variation of the model parameters reflecting the patterns of complex formation and the formation of cells producing viral particles. The possibility of attenuation of HIV-1 infection in the lymph node at certain values of some of the model parameters is shown.

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).

In the paper the model for a one-dimensional non-equilibrium riverbed process is proposed. The model takes into account the suspended and bed-load sediment transport. The bed-load transport is determined by using the original formula. This formula was derived from the thin bottom layer motion equation. The formula doesn’t contain new phenomenological parameters and takes into account the influence of bed slope, granulometric and physical mechanical parameters on the bed-load transport. A number of the model test problems are solved for the verification of the proposed mathematical model. The comparison of the calculation results with the established experimental data and the results of other authors is made. It was shown, that the obtained results have a good agreement with the experimental data in spite of the relative simplicity of the proposed mathematical model.

To study nonlinear effects of biological species interactions numerical-analytical approach is being developed. The approach is based on the cosymmetry theory accounting for the phenomenon of the emergence of a continuous family of solutions to differential equations where each solution can be obtained from the appropriate initial state. In problems of mathematical ecology the onset of cosymmetry is usually connected with a number of relationships between the parameters of the system. When the relationships collapse families vanish, we get a finite number of isolated solutions instead of a continuum of solutions and transient process can be long-term, dynamics taking place in a neighborhood of a family that has vanished due to cosymmetry collapse.

We consider a model for spatiotemporal competition of predators or prey with an account for directed migration, Holling type II functional response and nonlinear prey growth function permitting Alley effect. We found out the conditions on system parameters under which there is linear with respect to population densities cosymmetry. It is demonstated that cosymmetry exists for any resource function in case of heterogeneous habitat. Numerical experiment in MATLAB is applied to compute steady states and oscillatory regimes in case of spatial heterogeneity.

The dynamics of three population interactions (two predators and a prey, two prey and a predator) are considered. The onset of families of stationary distributions and limit cycle branching out of equlibria of a family that lose stability are investigated in case of homogeneous habitat. The study of the system for two prey and a predator gave a wonderful result of species coexistence. We have found out parameter regions where three families of stable solutions can be realized: coexistence of two prey in absence of a predator, stationary and oscillatory distributions of three coexisting species. Cosymmetry collapse is analyzed and long-term transient dynamics leading to solutions with the exclusion of one of prey or extinction of a predator is established in the numerical experiment.

Bioenergetic regularities determining the maximal biomass yield in aerobic microbial growth on various substrates have been considered. The approach is based on the method of mass-energy balance and application of GenMetPath computer program package. An equation system describing the balances of quantities of 1) metabolite reductivity and 2) high-energy bonds formed and expended has been formulated. In order to formulate the system, the whole metabolism is subdivided into constructive and energetic partial metabolisms. The constructive metabolism is, in turn, subdivided into two parts: forward and standard. The latter subdivision is based on the choice of nodal metabolites. The forward constructive metabolism is substantially dependent on growth substrate: it converts the substrate into the standard set of nodal metabolites. The latter is, then, converted into biomass macromolecules by the standard constructive metabolism which is the same on various substrates. Variations of flows via nodal metabolites are shown to exert minor effects on the standard constructive metabolism. As a separate case, the growth on substrates requiring the participation of oxygenases and/or oxidase is considered. The bioenergetic characteristics of the standard constructive metabolism are found from a large amount of data for the growth of various organisms on glucose. The described approach can be used for prediction of biomass growth yield on substrates with known reactions of their primary metabolization. As an example, the growth of a yeast culture on ethanol has been considered. The value of maximal growth yield predicted by the method described here showed very good consistency with the value found experimentally.

The biomass growth yield is the ratio of the newly synthesized substance of growing cells to the amount of the consumed substrate, the source of matter and energy for cell growth. The yield is a characteristic of the efficiency of substrate conversion to cell biomass. The conversion is carried out by the cell metabolism, which is a complete aggregate of biochemical reactions occurring in the cells.

This work newly considers the problem of maximal cell growth yield prediction basing on balances of the whole living cell metabolism and its fragments called as partial metabolisms (PM). The following PM’s are used for the present consideration. During growth on any substrate we consider i) the standard constructive metabolism (SCM) which consists of identical pathways during growth of various organisms on any substrate. SCM starts from several standard compounds (nodal metabolites): glucose, acetyl-CoA 2-oxoglutarate, erythrose-4-phosphate, oxaloacetate, ribose-5- phosphate, 3-phosphoglycerate, phosphoenolpyruvate, and pyruvate, and ii) the full forward metabolism (FM) — the remaining part of the whole metabolism. The first one consumes high-energy bonds (HEB) formed by the second one. In this work we examine a generalized variant of the FM, when the possible presence of extracellular products, as well as the possibilities of both aerobic and anaerobic growth are taken into account. Instead of separate balances of each nodal metabolite formation as it was made in our previous work, this work deals at once with the whole aggregate of these metabolites. This makes the problem solution more compact and requiring a smaller number of biochemical quantities and substantially less computational time. An equation expressing the maximal biomass yield via specific amounts of HEB formed and consumed by the partial metabolisms has been derived. It includes the specific HEB consumption by SCM which is a universal biochemical parameter applicable to the wide range of organisms and growth substrates. To correctly determine this parameter, the full constructive metabolism and its forward part are considered for the growth of cells on glucose as the mostly studied substrate. We used here the found earlier properties of the elemental composition of lipid and lipid-free fractions of cell biomass. Numerical study of the effect of various interrelations between flows via different nodal metabolites has been made. It showed that the requirements of the SCM in high-energy bonds and NAD(P)H are practically constants. The found HEB-to-formed-biomass coefficient is an efficient tool for finding estimates of maximal biomass yield from substrates for which the primary metabolism is known. Calculation of ATP-to-substrate ratio necessary for the yield estimation has been made using the special computer program package, GenMetPath.

The traditional approach to computing optimal game strategies of firms on oligopolistic markets and of indicators of such markets consists in studying linear dynamical games with quadratic criteria and solving generalized matrix Riccati equations.

The other approach proposed by the author is based on methods of operational calculus (in particular, Z-transform). This approach makes it possible to achieve economic meaningful decisions under wider field of parameter values. It characterizes by simplicity of computations and by necessary for economic analysis visibility. One of its advantages is that in many cases important for economic practice, it, in contrast to the traditional approach, provides the ability to make calculations using widespread spreadsheets, which allows to study the prospects for the development of oligopolistic markets to a wide range of professionals and consumers.

The article deals with the practical aspects of determining the optimal Nash–Cournot strategies of participants in oligopolistic markets on the basis of operational calculus, in particular the technique of computing the optimal Nash–Cournot strategies in Excel. As an illustration of the opportinities of the proposed methods of calculation, examples close to the practical problems of forecasting indicators of the markets of high-tech products are studied.

The results of calculations obtained by the author for numerous examples and real economic systems, both using the obtained relations on the basis of spreadsheets and using extended Riccati equations, are very close. In most of the considered practical problems, the deviation of the indicators calculated in accordance with the two approaches, as a rule, does not exceed 1.5–2%. The highest value of relative deviations (up to 3–5%) is observed at the beginning of the forecasting period. In typical cases, the period of relatively noticeable deviations is 3–5 moments of time. After the transition period, there is almost complete agreement of the values of the required indicators using both approaches.