Study on stationary solutions to the problem of phytoplankton dynamics considering transformation of phosphorus, nitrogen and silicon compounds

Introduction. The solution to the problem of transformation of phosphorus, nitrogen and silicon forms is studied. This problem arises under modeling phytoplankton dynamics in shallow-water bodies including the Azov Sea. The phytoplankton dynamics model is formulated as a boundary value problem for the system of diffusion-convection-response equations and takes into account the absorption and release of nutrients by phytoplankton, as well as the transition of nutrients from one compound to another. To calculate the initial conditions and parameters of the equations under which the steady-state regime occurs, the software is developed, which is based on the model describing changes in phytoplankton concentrations without considering current effects. This model is represented by a system of inhomogeneous differential equations. Based on the developed software, the initial conditions and parameters of the phytoplankton dynamics model in the Azov Sea are calculated experimentally. Materials and Methods. A 3D model of phytoplankton dynamics is considered taking into account the transformation of phosphorus, nitrogen and silicon compounds based on the system of nutrient transport equations. The case of a spatially uniform distribution of substances is considered to specify the parameters of the model at which the stationary modes occur. Because of simplification, a system of ordinary differential equations solved through the Runge-Kutta method is obtained. Research Results. The software is developed to specify the initial conditions and parameters of the phytoplankton dynamics model considering the transformation of phosphorus, nitrogen and silicon compounds. Several numerical experiments are performed under the assumption that the development of phytoplankton is limited by Введение. Работа посвящена исследованию решения задачи трансформации форм фосфора, азота и кремния. Данная проблема возникает при моделировании динамики фитопланктона в мелководных водоемах, в том числе в Азовском море. Модель динамики фитопланктона сформулирована как краевая задача для системы уравнений диффузии-конвекции-реакции и учитывает поглощение и выделение питательных веществ фитопланктоном, а также переход питательных веществ из одного соединения в другое. Для расчета начальных условий и параметров уравнений, при которых наступает стационарный режим, разработано программное обеспечение, основой которого послужила модель, описывающая изменения концентраций фитопланктона без учета влияния течений. Данная модель представлена системой неоднородных обыкновенных дифференциальных уравнений. На основе разработанного программного обеспечения экспериментальным образом рассчитаны начальные условия и параметры модели динамики фитопланктона в Азовском море. Материалы и методы. Рассматривается трехмерная модель динамики фитопланктона с учетом трансформации соединений фосфора, азота и кремния, основанная на системе уравнений транспорта биогенных веществ. Для уточнения параметров модели, при которых наступают стационарные режимы, рассматривается случай пространственно-однородного распределения субстанций. В результате упрощения получена система обыкновенных дифференциальных уравнений, которая решена методом Рунге-Кутты. Результаты исследования. Разработано программное обеспечение для уточнения начальных условий и параметров модели динамики фитопланктона с учетом трансформации соединений фосфора, азота и кремния. Проведены несколько численных экспериментов в предположении, что развитие фитопланктона лимитируется единственным биогенным веществом. В результате вычисли-

Because of simplification, a system of ordinary differential equations solved through the Runge-Kutta method is obtained.
Research Results. The software is developed to specify the initial conditions and parameters of the phytoplankton dynamics model considering the transformation of phosphorus, nitrogen and silicon compounds.
Ключевые слова: фитопланктон, фосфор, азот, кремний, биоген, химико-биологический источник, уравнение конвекции-диффузии-реакции, задача Коши для системы обыкновенных дифференциальных уравнений, стационарный режим. Introduction. Because of the development of major cities on the coast of shallow water bodies and river systems that flow into these water bodies, eutrophication has become more frequent. The growth of algae in reservoirs is caused by an increase in the flow of nitrogen and phosphorus compounds from the adjacent land areas. Each water body is unique and requires a thorough study. Field investigations [1] and mathematical modeling are used to explore water bodies. Without downplaying the role of field experiments, we can say that mathematical modeling is less costly, and it allows us to predict the behavior of the ecosystem.
To study the Sea of Azov, a three-dimensional model of hydrodynamics [2,3] including the equations of motion in three spatial directions was developed. In [4], this model was made for the case of dynamic rebuilding of the computational domain geometry due to the tidal effects. The investigation of this model accuracy is given in [5]. In [6][7][8], the reconstruction of an ecological catastrophe that occurred in 2001 caused by an excessive concentration of algae in the eastern part of the Sea of Azov is given. In [9], methods of controlling the suffocation phenomena arising in the Sea of Azov were proposed. The [10][11][12] papers are devoted to studying the dynamics of phyto-and zooplankton.
The water condition in shallow water bodies is changing rapidly, and mathematical models need to be refined.
The parameters determination of the three-dimensional model of the phyto-and zooplankton dynamics is laborious; therefore, it is proposed to use a simplified model to calculate these parameters.
The work objective is to improve the parameters of the model of the phytoplankton dynamics considering the transformation of phosphorus, nitrogen and silicon compounds, under which stationary regimes occur with the assumption of a spatially uniform distribution of substances.

Vestnik of Don State Technical
, i{PO 4 , POP, DOP, NO 3 , NO 2 , NH 4 , Si}, where 1, 2, , i i   are sedimentation rates of algae and nutrients to the bottom.

Stationary Mode.
Consider the case of a spatially uniform distribution of substances (phytoplankton, forms of phosphorus, nitrogen and silica); then each of the equations (1) is simplified; and as a result, we get the following system of ordinary differential equations (ODE): (1 ) 3  3   2   2  3   1  3  3  2  4  23  1  3  2  4   ,  ,  1 , , We solve the system of ordinary differential equations by the Runge -Kutta method [15][16][17]. We will conduct several numerical experiments, assuming that the development of phytoplankton depends on a single limiting substance.
Research Results. For the ODE system (7) The obtained stationary modes of the ODE system (7) on the assumption that the development of phytoplankton is limited by a single nutrient (phosphorus, nitrogen or silica) are shown in Fig. 2-4, respectively.  Vol. 19, no. 1, pp. 4-12. ISSN 1992-5980 eISSN 1992-6006  Вестник Донского государственного технического университета. 2019. Т. 19, № 1. C. 4-12. ISSN 1992-5980 eISSN 1992-  The result of the computational experiment shows that with the above values of the initial concentrations and parameters of the equations, stationary modes occur for the ODE system (7), which describes the case of a spatially uniform distribution of substances. The obtained values will be used in further simulation of the spatially inhomogeneous distribution of substances, saltiness and temperature considering the movement of the aquatic environment [18].
Conclusion. A mathematical model of the transformation of forms of phosphorus, nitrogen and silica in the problem of phytoplankton dynamics is studied in the paper. The case of spatially uniform distribution of substances (phytoplankton, forms of phosphorus, nitrogen and silica) is considered. The system is divided into three systems of ordinary differential equations, each of which simulates the dependence of phytoplankton growth on a single nutrient. These systems are solved by the Runge-Kutta method (Fig. 2-4); stationary modes are obtained, for which the values of the system parameters and initial conditions are determined.
The results obtained will be used for the further simulation of the phytoplankton dynamics considering the transformation of phosphorus, nitrogen and silica compounds, taking into account diffusion-convection, saltiness, and temperature.