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The biohydrochemical portrait of the White Sea is constructed on the CNPSi-model calculations based on long-term mean annual observations (average monthly hydrometeorological, hydrochemical and hydrobiological parameters of the marine environment) as well as on updated information on the nutrient input to the sea with the runoff of the main river tributaries (Niva, Onega, Northern Dvina, Mezen, Kem, Keret). Parameters of the marine environment are temperature, light, transparency, and biogenic load. Ecological characteristics of the sea “portrait” were calculated for nine marine areas (Kandalaksha, Onega, Dvinsky, Mezensky Bays, Solovetsky Islands, Basin, Gorlot, Voronka, Chupa Bay), these are: the concentration changes of organic and mineral compounds of biogenic elements (C, N, P, Si), the biomass of organisms of the lower trophic level (heterotrophic bacteria, diatomic phytoplankton, herbivorous and predatory zooplankton) and other ones (rates of substance concentration and organism biomass changes, internal and external substance flows, balances of individual substances and nutrients as a whole). Parameters of the marine environment state (water temperature, ratio of mineral fractions N < P) and dominant diatom phytoplankton in the sea (abundance, production, biomass, chlorophyll content a) were calculated and compared with the results of individual surveys (for 1972–1991 and 2007–2012) of the White Sea water areas. The methods for estimating the values of these parameters from observations and calculations differ, however, the calculated values of the phytoplankton state are comparable with the measurements and are similar to the data given in the literature. Therefore, according to the literature data, the annual production of diatoms in the White Sea is estimated at 1.5–3 million tons C (at a vegetation period of 180 days), and according to calculations it is ~2 and 3.5 million tons C for vegetation period of 150 and 180 days respectively.

The article describes the modes with the exacerbation of social and biological history. The analysis of the possible causes of the sharp acceleration of biological and social processes in certain historical periods is carried out. Using mathematical modeling shows that hyperbolic trends in social and biological evolution may be the result of transitional processes in periods of expansion of ecological niches. Accelerating biological speciation due to the fact that its earlier life change inhabitancy, making it more diverse, saturating the organic, thus creating favourable conditions for the emergence of new species. In the social history of the expansion of ecological niches associated with technological revolutions, of which the most important were: Neolithic revolution — the transition from appropriating economy to producing economy (10 thousand years ago), “urban revolution” — a shift from the Neolithic epoch to the bronze epoch (5 thousand years ago), the “axial age” — transition to the development of iron tools (2.5 thousand years ago), the industrial revolution — the transition from manual labor to machine production (200 years ago). All of these technological revolutions have been accompanied by dramatic population growth, changes in social and political spheres. So, observed in the last century, hyperbolic nature of some demographic, economic growth and other indicators of world dynamics is a consequence of the transition process, which began as a result of the industrial revolution and to prepare for the transition of the society to a new stage of its development. Singularity point of hyperbolic trend shows the end of the initial phase of the process and marks the transition to the final stage. The mathematical model describing the demographic and economic changes in the era of change is proposed. It is shown that a direct analogue of the contemporary situation in this sense is the “axial age” (since 8 century BC to the beginning of our era). The existence of this analogy allows you to see into the future by studying the past.

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.

Mathematicity of physics is surprising, but it enables us to understand the laws of nature through the analysis of mathematical structures describing it. This concerns, however, only physics. The degree of the mathematization of biology is low, and attempts to mathematize it are limited to the application of mathematical methods used for the description of physical systems. When doing so, we are likely to commit an error of attributing to biological systems features that they do not have. Some argue that biology does need new mathematical methods conforming to its needs, and not known from physics. However, because of a specific complexity of biological systems, we should speak of their algorithmicity, rather than of their mathematicity. As an example of algorithmic approach one can indicate so called individual-based models used in ecology to describe population dynamics or fractal models applied to describe geometrical complexity of such biological structures as trees.

The article covered results of three-dimensional modeling of ecologic situation of shallow water on the example of the Azov Sea with using schemes of increased order of accuracy on multiprocessor computer system of Southern Federal University. Discrete analogs of convective and diffusive transfer operators of the fourth order of accuracy in the case of partial occupancy of cells were constructed and studied. The developed scheme of the high (fourth) order of accuracy were used for solving problems of aquatic ecology and modeling spatial distribution of polluting nutrients, which caused growth of phytoplankton, many species of which are toxic and harmful. The use of schemes of the high order of accuracy are improved the quality of input data and decreased the error in solutions of model tasks of aquatic ecology. Numerical experiments were conducted for the problem of transportation of substances on the basis of the schemes of the second and fourth orders of accuracy. They’re showed that the accuracy was increased in 48.7 times for diffusion-convection problem. The mathematical algorithm was proposed and numerically implemented, which designed to restore the bottom topography of shallow water on the basis of hydrographic data (water depth at individual points or contour level). The map of bottom relief of the Azov Sea was generated with using this algorithm. It’s used to build fields of currents calculated on the basis of hydrodynamic model. The fields of water flow currents were used as input data of the aquatic ecology models. The library of double-layered iterative methods was developed for solving of nine-diagonal difference equations. It occurs in discretization of model tasks of challenges of pollutants concentration, plankton and fish on multiprocessor computer system. It improved the precision of the calculated data and gave the possibility to obtain operational forecasts of changes in ecologic situation of shallow water in short time intervals.

This article presents an integrated dynamic model of eco-economic system of the Republic of Armenia (RA). This model is constructed using system dynamics methods, which allow to consider the major feedback related to key characteristics of eco-economic system. Such model is a two-objective optimization problem where as target functions the level of air pollution and gross profit of national economy are considered. The air pollution is minimized due to modernization of stationary and mobile sources of pollution at simultaneous maximization of gross profit of national economy. At the same time considered eco-economic system is characterized by the presence of internal constraints that must be accounted at acceptance of strategic decisions. As a result, we proposed a systematic approach that allows forming sustainable solutions for the development of the production sector of RA while minimizing the impact on the environment. With the proposed approach, in particular, we can form a plan for optimal enterprise modernization and predict long-term dynamics of harmful emissions into the atmosphere.

The paper presents a comparative analysis of the enterprises of the Arctic Zone of the Russian Federation (AZ RF) on economic indicators in accordance with the rating of the Polar index. This study includes numerical data of 193 enterprises located in the AZ RF. Machine learning methods are applied, both standard, from open source, and own original methods — the method of Optimally Reliable Partitions (ORP), the method of Statistically Weighted Syndromes (SWS). Held split, indicating the maximum value of the functional quality, this study used the simplest family of different one-dimensional partition with a single boundary point, as well as a collection of different two-dimensional partition with one boundary point on each of the two combining variables. Permutation tests allow not only to evaluate the reliability of the data of the revealed regularities, but also to exclude partitions with excessive complexity from the set of the revealed regularities. Patterns connected the class number and economic indicators are revealed using the SDT method on one-dimensional indicators. The regularities which are revealed within the framework of the simplest one-dimensional model with one boundary point and with significance not worse than p < 0.001 are also presented in the given study. The so-called sliding control method was used for reliable evaluation of such diagnostic ability. As a result of these studies, a set of methods that had sufficient effectiveness was identified. The collective method based on the results of several machine learning methods showed the high importance of economic indicators for the division of enterprises in accordance with the rating of the Polar index. Our study proved and showed that those companies that entered the top Rating of the Polar index are generally recognized by financial indicators among all companies in the Arctic Zone. However it would be useful to supplement the list of indicators with ecological and social criteria.

The paper investigates the dynamics of a finite-dimensional model describing the interaction of three populations: prey $x(t)$, its consuming predator $y(t)$, and a superpredator $z(t)$ that feeds on both species. Mathematically, the problem is formulated as a system of nonlinear first-order differential equations with the following right-hand side: $[x(1-x)-(y+z)g;\,\eta_1^{}yg-d_1^{}f-\mu_1^{}y;\,\eta_2^{}zg+d_2^{}f-\mu_2^{}z]$, where $\eta_j^{}$, $d_j^{}$, $\mu_j^{}$ ($j=1,\,2$) are positive coefficients. The considered model belongs to the class of cosymmetric dynamical systems under the Lotka\,--\,Volterra functional response $g=x$, $f=yz$, and two parameter constraints: $\mu_2^{}=d_2^{}\left(1+\frac{\mu_1^{}}{d_1^{}}\right)$, $\eta_2^{}=d_2^{}\left(1+\frac{\eta_1^{}}{d_1^{}}\right)$. In this case, a family of equilibria is being of a straight line in phase space. We have analyzed the stability of the equilibria from the family and isolated equilibria. Maps of stationary solutions and limit cycles have been constructed. The breakdown of the family is studied by violating the cosymmetry conditions and using the Holling model $g(x)=\frac x{1+b_1^{}x}$ and the Beddington–DeAngelis model $f(y,\,z)=\frac{yz}{1+b_2^{}y+b_3^{}z}$. To achieve this, the apparatus of Yudovich's theory of cosymmetry is applied, including the computation of cosymmetric defects and selective functions. Through numerical experimentation, invasive scenarios have been analyzed, encompassing the introduction of a superpredator into the predator-prey system, the elimination of the predator, or the superpredator.

The article dwells upon investigating the issue of the most adequate tools developing and justifying to forecast the interregional migration flows value and structure. Migration processes have a significant impact on the size and demographic structure of the population of territories, the state and balance of regional and local labor markets.

To analyze the migration processes and to assess their impact an economic-mathematical tool is required which would be instrumental in modelling the migration processes and flows for different areas with the desired precision. The current methods and approaches to the migration processes modelling, including the analysis of their advantages and disadvantages, were considered. It is noted that to implement many of these methods mass aggregated statistical data is required which is not always available and doesn’t characterize the migrants behavior at the local level where the decision to move to a new dwelling place is made. This has a significant impact on the ability to apply appropriate migration processes modelling techniques and on the projection accuracy of the migration flows magnitude and structure.

The cellular automata model for interregional migration flows modelling, implementing the integration of the households migration behavior model under the conditions of the Bounded Rationality into the general model of the area migration flow was developed and tested based on the Primorye Territory data. To implement the households migration behavior model under the conditions of the Bounded Rationality the integral attractiveness index of the regions with economic, social and ecological components was proposed in the work.

To evaluate the prognostic capacity of the developed model, it was compared with the available cellular automata models used to predict interregional migration flows. The out of sample prediction method which showed statistically significant superiority of the proposed model was applied for this purpose. The model allows obtaining the forecasts and quantitative characteristics of the areas migration flows based on the households real migration behaviour at the local level taking into consideration their living conditions and behavioural motives.

This paper studies a model of a population with non-overlapping generations and density-dependent regulation of birth rate. The population breeds seasonally, and its reproductive potential is determined genetically. The model proposed combines an ecological dynamic model of a limited population with non-overlapping generations and microevolutionary model of its genetic structure dynamics for the case when adaptive trait of birth rate controlled by a single diallelic autosomal locus with allelomorphs A and a. The study showed the genetic composition of the population, namely, will it be polymorphic or monomorphic, is mainly determined by the values of the reproductive potentials of heterozygote and homozygotes. Moreover, the average reproductive potential of mature individuals and intensity of self-regulation processes determine population dynamics. In particularly, increasing the average value of the reproductive potential leads to destabilization of the dynamics of age group sizes. The intensity of self-regulation processes determines the nature of emerging oscillations, since scenario of stability loss of fixed points depends on the values of this parameter. It is shown that patterns of occurrence and evolution of cyclic dynamics regimes are mainly determined by the features of life cycle of individuals in population. The life cycle leading to existence of non-overlapping generation gives isolated subpopulations in different years, which results in the possibility of independent microevolution of these subpopulations and, as a result, the complex dynamics emergence of both stage structure and genetic one. Fixing various adaptive mutations will gradually lead to genetic (and possibly morphological) differentiation and to differences in the average reproductive potentials of subpopulations that give different values of equilibrium subpopulation sizes. Further evolutionary growth of reproductive potentials of limited subpopulations leads to their number fluctuations which can differ in both amplitude and phase.