The paper covers the influence of petroleum hydrocarbons on the oxygen regime of a shallow water, as well as on zooplankton populations sensitive to lack of oxygen and oil pollution even at their low concentrations. A mathematical model of biological kinetics is constructed taking into account the number of abiotic and biotic factors that directly affect the water ecosystem: the movement of the water environment, microturbulent diffusion, temperature regime, illumination, wind stresses, salinity, concentration of oxygen dissolved in water. The model is implemented in a software focused on a high-performance computer system. Nonlinear dependences in the rate of growth of zooplankton concentrations are introduced, taking into account the influence of oil pollution. This makes it possible to determine their dynamics in conditions of insufficient oxygen supply with spatial and temporal variability of temperature and illumination. The problem discretization is based on the difference scheme that takes into account the partial occupancy of the computational cells. This made it possible to increase the simulation accuracy of hydrobiological processes and phenomena.

The paper considers a mathematical model of biological kinetics and geochemical cycles based on a system of convection-diffusion equations with nonlinear coefficients, supplemented by a spatially inhomogeneous three-dimensional mathematical model of wave hydrodynamics of a shallow reservoir, with a refined coefficient of turbulent vertical exchange. The task of monitoring the water surface in order to detect phytoplankton spots involves the creation and verification of effective methods for clustering these objects on the surface of reservoirs, in particular, restoring their boundaries based on remote sensing data. The article uses multispectral satellite images as sounding data. Based on the obtained images of plankton populations, the initial conditions for mathematical models of biogeochemical cycles can be determined, on the basis of which prognostic calculations are performed.

The model describes the following process. Two parties, called Left and Right, are involved in information warfare on two topics that play the role of battlefields. Each party has limited broadcasting resources for propaganda, which each allocates between these two topics. Each member of the population backs one of the parties for each topic. A situation is possible in which an individual backs different parties on different topics. In this case, the individual is considered a supporter of the party supported on a more salient topic. Party supporters participate in participatory propaganda, campaigning on the topic or two topics they support for their party. The saliency of a topic depends on the amount of media broadcasting and communication on it. The number of party supporters’ changes over time under the influence of media and party propaganda. The problem is to determine the parties' best strategies.

Each party apportion its broadcasting resource between two topics, thereby choosing its strategy. Therefore, a Blotto game appears. The Blotto game is a two-player zero-sum game in which the players distribute limited resources over several battlefields. In this matrix game, payoffs of the parties are the numbers of their supporters at the end of the propaganda battle. Numerical experiments were conducted in which these payoffs were calculated numerically and the obtained game was solved.

Typically, the best strategies are those where the resource is allocated between the topics very unevenly. Moreover, often the best strategy is spending all the resource on one topic.

The paper deals with the problem of preserving natural water systems, as well as maintaining their integrity, not only through the enterprise of organizational, engineering and technical solutions, but also through the development of highly effective mathematical modeling techniques that make it possible quickly and efficiently, based on interconnected high-precision models of hydrophysics and hydrobiology, predict the processes of pollution spreading and the occurrence of hazardous phenomena in coastal systems. The article considers algorithms for solving grid equations developed for high-performance cluster systems. A model of parallel computations is proposed, which makes it possible, when choosing the appropriate method for solving the problem of aquatic ecology, to estimate the cost of calculations, which is defined as the product of the time of parallel solution of the problem and the number of processors used. An estimate of the optimal amount of information packet for exchange between processors is obtained. The adaptive modified alternating-triangular method of minimum corrections is described, the results of numerical experiments for the parallel variant of this one are presented.

A hydrophysics mathematical model of a shallow water body is proposed, including the pollutants (PS) transport equation, taking into account their gravitational settling, numerically implemented in the software module form. The developed software module makes it possible to evaluate the pollutant concentrations distribution during their deposition from the aquatic environment surface. A model for predicting the distribution of pollutants along the vertical coordinate in a reservoir is proposed, taking into account the movement of the water environment.