Mathematical Modeling of Biological Rehabilitation of the Taganrog Bay Considering Its Salinization

Abstract

Taganrog Bay is part of the Azov Sea, which has significant environmental value. However, in recent years, anthropogenic activity and climate change have increasingly impacted this coastal system. These factors have led to increased sea salinity. These factors also contribute to abundant blooms of potentially toxic cyanobacteria. One additional method for preventing the abundant growth of cyanobacteria may be the introduction of green algae into the bay. The aim of this study was to conduct a computational experiment on the biological rehabilitation of Taganrog Bay using mathematical modeling methods. For this purpose, the authors developed and analyzed a mathematical model of phytoplankton populations. A software model was developed based on modern mathematical modeling methods. The input data for the software module included grid points for advective transport velocities, salinity, and temperature, as well as phytoplankton population and nutrient concentrations. The software module outputs three-dimensional distributions of green algae and cyanobacteria concentrations. A computational experiment on biological rehabilitation of the Taganrog Bay by introducing a suspension of green algae was conducted. Green algae and cyanobacteria concentrations were obtained over 15 and 30-day time intervals. The concentration and volume of introduced suspension were empirically determined to prevent harmful cyanobacteria growth without leading to eutrophication of the bay by green algae.

1. Introduction

Taganrog Bay is the northeastern part of the Azov Sea, a unique and valuable natural resource. The Azov Sea is a shallow coastal estuarine system fed by freshwater rivers and connected to the Black Sea, so wide variations in salinity gradients and almost uniform warming in summer is typical for the sea. The significant volume of river runoff also brings large quantities of nutrients. The sea is significantly affected by climate change and human economic activity, which have led to changes in the hydrological regime. The salinity of sea water has only increased since 2007. The reasons for this are the long-term low water levels of the Don and Kuban rivers and high evaporation [1]. Changes in salinity lead to shifts in the habitats of aquatic organisms and the invasion of marine species from the Black Sea into the Azov Sea and Taganrog Bay.

The abundant bloom of phytoplankton populations in summer is one of the features of the Taganrog Bay. Potentially toxic cyanobacteria are of particular concern, as their blooms harm fisheries and the health of local populations. Cyanophytes are also known as cyanobacteria. This type of phytoplankton differs from other species in several ways. First of all, they are prokaryotes and have a very simple cellular structure, similar to that of bacteria, without a nucleus, mitochondria, Golgi apparatus, or endoplasmic reticulum. Individually, cyanophytes are small compared to other algae, but they often form massive colonies or filaments. Gas vacuoles provide buoyancy for cyanobacteria, which also allow them to change density, leading to vertical movement in the water column [2]. This mechanism provides better access to either light or nutrients [3]. This gives cyanobacteria an important advantage over other species. Nutrients also influence the ability of cyanoprokaryotes to move vertically in the water column [4]. Furthermore, turbulent diffusion and the velocity of the aquatic environment in the z-axis can influence the vertical migration of cyanobacteria [5]. In coastal systems, cyanobacteria are of interest due to their potential toxicity but also play an important role in the primary production of phytoplankton. Currently, many studies are being conducted aimed at studying the nature of toxins and the factors causing their release [6]. The most common species in the waters of Taganrog Bay are Aphanizomenon flos-aquae, Microcystis aeruginosa, and Anabaena spp. These species can secrete toxins classified as hepatotoxins and neurotoxins [7].

Many domestic and foreign studies are devoted to modeling the bloom of potentially harmful cyanobacteria. The role of phosphorus in stimulating the growth of cyanobacteria is investigated in [8]. The study [9] models the process of vertical movement of cyanobacteria of the Microcystis species, which allows them to occupy a dominant position during the bloom period in a reservoir. The influence of light radiation intensity and water mixing under the action of wind on development was modeled using the Ansys Fluent and MATLAB systems, which is described in [10]. In [11], autoregressive and multivariate versions of linear regression, random forest and Long Short-Term Memory (LSTM) neural networks were used to predict the development of cyanobacteria. The study [12] proposes a non-stationary three-component mathematical model for studying the competition between two types of phytoplankton, including toxic ones, and their absorption by zooplankton. The studies reviewed have a number of shortcomings, one of which is the lack of three-dimensional advection-diffusion-reaction models. Filling this gap in scientific knowledge will improve the accuracy of modeling.

The algolization is a method for preventing the abundant bloom of cyanobacteria in water systems. This is a method of biological purification and restoration of aquatic ecosystems by introducing a suspension of single-celled green algae Chlorella vulgaris, which was proposed in the work [13]. The idea of the method is that green algae are introduced into the aquatic system before the beginning of the cyanobacteria growing season, where they consume most of the nutrients. This inhibits the rapid growth of harmful cyanobacteria, which lack the resources for active reproduction. In turn, green algae are at the base of the reservoir food chain, providing food for higher-level aquatic organisms, including fish [14]. Green algae are also actively used in agriculture as fertilizers and feed for farm animals and are used to purify wastewater [15]. However, excessive introduction of green algae suspension should be avoided, as this may lead to eutrophication of the water system. In this article, the algolization method is of interest from the point of view of the problem statement and the implementation of a computational experiment.

Thus, developing methods to model the dynamics, forecasting the growth of cyanobacteria depending on weather conditions, and developing measures to prevent their abundant blooms are the topical problems. Mathematical modeling methods are successfully used for this purpose due to their relative cheapness, ease of use, and accessibility compared to field studies. An analysis of current research on the mathematical modeling of cyanobacteria dynamics and the biological rehabilitation of water bodies revealed a gap in scientific knowledge regarding the comprehensiveness of the approach. Specifically, one- and two-dimensional mathematical models are used, not all factors significantly influencing the development of phytoplankton are considered, universal software packages do not simulate the movement of biomass with the flow of liquid accurately enough, non-deterministic models are used, which have less accuracy compared to models based on differential equations, etc. The authors of this study proposed an integrated approach that allows filling the gap in scientific knowledge. Specifically, a set of mathematical models of the phytoplankton populations growth and hydrodynamics will be used, considering advective and diffusive transfer, weather conditions, the geometry of the computational domain, limitation of microalgae growth by the presence of nutrients, salinity and temperature regimes [16]. To solve the problem, modern difference schemes and numerical methods will be used.

The objective of this study is to conduct a computational experiment on the biological rehabilitation of Taganrog Bay under salinization conditions through the introduction of green microalgae using mathematical modeling methods. To achieve this goal, the following problems will be solved: developing and analyzing a mathematical model of the dynamics of two phytoplankton populations competing for resources, discretizing the continuous model, and conducting a computational experiment. The model will be equipped with input data: a water flow vector obtained from a hydrodynamic model, salinity and temperature values obtained by processing cartographic information, and phytoplankton population and nutrient concentrations obtained from long-term observations.

2. Materials and Methods

Mathematical modeling methods are used in the study: a mathematical model of phytoplankton population dynamics and biological kinetics is described, difference equations approximating the initial model are presented, and a numerical method used to solve the resulting system of equations is mentioned. Input data for the model and a computational experiment on algolization of the Taganrog Bay are described.

2.1. Model of Phytoplankton Population Dynamics and Biogeochemical Cycles and Its Linearization

The most significant processes of biological kinetics were considered when constructing a mathematical model of the dynamics of phytoplankton populations: interspecies interactions, consumption and limitation of biogenic substances, the influence of salinity, temperature on the development of phytoplankton populations, and chemical transformations of substances that form the basis of plant nutrition in the aquatic environment—nitrogen and phosphorus—were also considered. The mathematical model of biological kinetics is based on the works of Sukhinov A.I., Yakushev E.V. [17,18]. A description of the transformation cycles of the modeled substances is given in

Table 1. Cycles of component $q_i$, $i=\overline{1,8}$ transformations.

No	Name	Designation	Description
1	green algae (Chlorella vulgaris)	F1	They are at the base of the food chain. Growth rate is determined by the availability of nutrients (phosphates, nitrates, nitrites, and ammonium), as well as optimal temperature, salinity, and light conditions. Biomass decreases due to excretion and mortality. They compete with cyanobacteria for resources.
2	cyanobacteria (Aphanizomenon flos-aquae)	F2	They are potentially toxic. Growth rate is determined by the availability of nutrients (phosphates, nitrates, nitrites, and ammonium), as well as optimal temperature, salinity, and light conditions. Biomass decreases due to excretion and mortality.
3	dissolved organic phosphorus	DOP	It is released by phytoplankton populations during excretion. Its concentration increases due to the autolysis of dissolved organic phosphorus and decreases during phosphatification.
4	particulate organic phosphorus	POP	It is a product of microalgae death. It is converted into a dissolved form through autolysis and, through phosphatification, into the mineral form of phosphorus.
5	Phosphates	PO4	Inorganic phosphorus compounds. Their concentration increases due to the conversion of organic forms of phosphorus into mineral forms during phosphatification. They are a biogen for phytoplankton populations.
6	Nitrates	NO3	It is the main form of nitrogen for consumption by microalgae.
7	Nitrites	NO2	in the presence of dissolved oxygen molecules, they are oxidized to nitrates. They are also consumed by phytoplankton populations.
8	Ammonia	NH4	In the presence of dissolved oxygen molecules, it is oxidized to nitrites. It is the least preferred form of nitrogen for microalgae to consume.

.

The mathematical model, nonlinear right-hand sides of the equations, and the formulation of the initial boundary value problem are described in detail in the work [19]. We will give a brief description of the mathematical model and its linearization.

This model is based on a system of unsteady advection-diffusion-reaction equations of parabolic type with nonlinear source functions and lower-order derivatives. The advective terms are presented in symmetric form, which guarantees the skew-symmetry of the transport operator and allows for a correct formulation of the problem. For each substance F_i, included in the model has the form:

$${\displaystyle \frac{\mathit{\partial }{q}_i}{\mathit{\partial }t}}+{\displaystyle \frac{1}{2}}\left(\nabla \cdot \left(V{q}_i\right)+\left(V\cdot \nabla \right)q_i\right)=\mathrm{div}\left(k\cdot \nabla q_i\right)+R_{q_i}$$

(1)

where $q_i$ is the concentration of the i-th component, [mg/L], i ∈ M, M = {F₁, F₂, DOP, POP, PO₄, NO₃, NO₂, NH₄}; $V=\left\{u,v,w\right\}$ is the water flow velocity vector, [m/s]; $k=\left(k_h,k_h,k_v\right)$ are the turbulent exchange coefficients, [m²/s]; $\nabla$ is the gradient operator notation, $\left(x,y,z\right)\in G$, $0<t\le T$, $R_{q_i}$ is the function-source of biogenic substances, [mg/(L∙s)], $q_{F_1}$ means that the concentration of green algae is being considered, $q_{F_2}$ is the cyanobacteria concentration. The following are the biogenic components: $q_{DOP}$ means that the component belongs to dissolved phosphorus, $q_{POP}$ is the particulate organic phosphorus, $q_{P{O}_4}$ are the phosphates, $q_{N{O}_3}$ are the nitrates, $q_{N{O}_2}$ are the nitrites, $q_{N{H}_4}$ is the ammonia (ammonium nitrogen).

Biochemical interactions between the components of the system (1), functions of the right parts $R_{q_i}=R_{q_i}\left(x,y,z,t\right)$, in the general case, nonlinear dependencies that may depend on the temperature of the aquatic environment and its salinity, have the form:

$$R_{F_i}=C_{F_i}(1-K_{F_{i}R})q_{F_i}-K_{F_{i}D}{q}_{F_i}-K_{F_{i}E}{q}_{F_i}$$

(2)

$$R_{DOP}={\displaystyle \sum _{i=1}^3{s}_P{K}_{F_{i}E}{q}_{F_i}}+K_{PD}{q}_{POP}-K_{DN}{q}_{DOP}$$

$$R_{POP}={\displaystyle \sum _{i=1}^3{s}_P{K}_{F_{i}D}{q}_{F_i}}-K_{PD}{q}_{POP}-K_{PN}{q}_{POP}$$

$$R_{P{O}_4}={\displaystyle \sum _{i=1}^3{s}_P{C}_{F_i}\left(K_{F_{i}R}-1\right)q_{F_i}}+K_{PN}{q}_{POP}+K_{DN}{q}_{DOP}$$

$$R_{N{O}_3}={\displaystyle \sum _{i=1}^3{s}_N{C}_{F_i}\left(K_{F_{i}R}-1\right)\frac{f_N^{\left(1\right)}\left(q_{N{O}_3},q_{N{O}_2},q_{N{H}_4}\right)}{f_N\left(q_{N{O}_3},q_{N{O}_2},q_{N{H}_4}\right)}\cdot \frac{q_{N{O}_3}}{q_{N{O}_2}+q_{N{O}_3}}{q}_{F_i}}+K_{23}{q}_{N{O}_2}$$

$$R_{N{O}_2}={\displaystyle \sum _{i=1}^3{s}_N{C}_{F_i}(K_{F_{i}R}-1)\frac{f_N^{\left(1\right)}\left(q_{N{O}_3},q_{N{O}_2},q_{N{H}_4}\right)}{f_N\left(q_{N{O}_3},q_{N{O}_2},q_{N{H}_4}\right)}\cdot \frac{q_{N{O}_2}}{q_{N{O}_2}+q_{N{O}_3}}{q}_{F_i}}+K_{42}{q}_{N{H}_4}-K_{23}{q}_{N{O}_2}$$

$$R_{N{H}_4}={\displaystyle \sum _{i=1}^3{s}_N{C}_{F_i}\left(K_{F_{i}R}-1\right)\frac{f_N^{\left(2\right)}\left(q_{N{H}_4}\right)}{f_N\left(q_{N{O}_3},q_{N{O}_2},q_{N{H}_4}\right)}{q}_{F_i}}+{\displaystyle \sum _{i=1}^3{s}_N\left(K_{F_{i}D}+K_{F_{i}E}\right)q_{F_i}}-K_{42}{q}_{N{H}_4}$$

where $K_{F_{i}R}$ is the specific respiration rate of phytoplankton; $K_{F_{i}D}$ is the specific rate of phytoplankton dying; $K_{F_{i}E}$ is the specific rate of phytoplankton excretion; $K_{PD}$ is the specific speed of autolysis POP; $K_{PN}$ is the phosphatification coefficient POP; $K_{DN}$ is the phosphatification coefficient DOP; $K_{42}$ is the specific rate of oxidation of ammonium to nitrites in the process of nitrification; $K_{23}$ is the specific rate of oxidation of nitrites to nitrates in the process of nitrification, $s_P$, $s_N$ are the normalization coefficients between the content of N, P in organic matter. The diagram of the model components interactions is presented in Figure 1.

The growth rate of phytoplankton populations is expressed as a function of dependence on salinity S, temperature T:

$$C_{F_{1,2}}=K_{N{F}_{1,2}}{f}_T\left(T\right)f_S\left(S\right)f_I\left(I\right)min\left\{f_P\left(q_{P{O}_4}\right),f_N\left(q_{N{O}_3},q_{N{O}_2},q_{N{H}_4}\right)\right\}$$

where $K_{NF}$ is the maximum specific growth rate of phytoplankton. Microalgae growth also depends on the concentration of essential nutrients—nitrogen compounds (nitrates, nitrites, ammonia) and phosphorus (phosphates, dissolved organic phosphorus, suspended organic phosphorus), the functional relationships for which are written in the Michaelis-Menten form. All of these factors are limiting, and their influence is reflected by Liebig’s law.

Functional dependencies on abiogenic factors:

$$f_T\left(T\right)=\exp \left(-a_i{\left\{\left(T-T_{opt}\right)/T_{opt}\right\}}^2\right),\hspace{1em}i=1,2;\hspace{1em}{f}_S\left(S\right)=\exp \left(-b_2{\left\{\left(S-S_{opt}\right)/S_{opt}\right\}}^2\right)$$

$$f_S\left(S\right)=\left\{\begin{array}{l}{k}_s,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for} \hspace{0.17em}\hspace{0.17em}S\le S_{opt},\\ \exp \left(-b_1{\left\{\left(S-S_{opt}\right)/S_{opt}\right\}}^2\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for} \hspace{0.17em}\hspace{0.17em}S>S_{opt},\end{array}\right.$$

where $k_s=1$; $T_{opt}$, $S_{opt}$ are the temperature and salinity optimal for a given type of aquatic organisms; $a_i>0$, $b_i>0$, $i=1,2$ are the coefficients of the width of the range of aquatic organisms tolerance to temperature and salinity, respectively.

Functional dependence of phytoplankton growth rate on illumination:

$$f_I\left(I\right)=I/I_{opt}\cdot \exp \left(1-I/I_{opt}\right)$$

where $I\left(h\right)=I_0\exp \left(-\theta h\right)$, $I_0$ is the total solar radiation on the water surface, $\theta =k_0+k_p{q}_F$ is the attenuation coefficient due to self-shading at a phytoplankton concentration equal to $q_F$, $h$ is the depth of the reservoir, $I_{opt}$ is the optimal illumination.

The initial boundary value problem is set for system (1), and the corresponding initial and boundary conditions are added. The initial conditions for (1) are:

$$q_i\left(x,y,z,0\right)=q_{0i}\left(x,y,z\right), i\in M, t=0, \left(x,y,z\right)\in \overline{G}$$

(3)

$$V\left(x,y,z,0\right)=V_0\left(x,y,z\right), T\left(x,y,z,0\right)=T_0\left(x,y,z\right), S\left(x,y,z,0\right)=S_0\left(x,y,z\right)$$

where G—the computational domain of a closed reservoir, limited by the lateral surface (cylindrical domain) σ, by the bottom $\mathit{\partial }{\Sigma }_H=\mathit{\partial }{\Sigma }_H\left(x,y\right)$ and by the undisturbed free surface of the reservoir Σ₀. $\Sigma$ is the piecewise smooth boundary G, given for $0<t\le T$ at $\Sigma ={\Sigma }_0\cup {\Sigma }_H\cup \sigma$.

Considering the introduced notations, the boundary conditions for Equation (1) are formulated:

$$\mathrm{on} \sigma : q_i=0, \mathrm{if} u_n<0; {\displaystyle \frac{\mathit{\partial }{q}_i}{\mathit{\partial }n}}=0, \mathrm{if} u_n\ge 0$$

(4)

$${\displaystyle \frac{\mathit{\partial }{q}_i}{\mathit{\partial }z}}=0, \mathrm{on} {\Sigma }_0; {\displaystyle \frac{\mathit{\partial }{q}_i}{\mathit{\partial }z}}=-{\epsilon }_i{q}_i \mathrm{on\; the\; bottom} {\Sigma }_H$$

where ${\epsilon }_i$ are the non-negative constants; i ∈ M; ${\epsilon }_i$ consider the sinking of algae to the bottom and their flooding for i ∈ {F₁, F₂} and consider the absorption of nutrients by bottom sediments. In this study, the simplification is adopted that the coefficients ${\epsilon }_i$ are the same. The values of all kinetic coefficients in the model (1)–(4) are given in [18].

On a uniform time grid ${\omega }_{\tau }=\left\{t_n=n\tau ,\hspace{0.33em}n=0,1,\dots ,N;\hspace{0.33em}N\tau =T\right\}$ on a time interval $0<t\le T$ linearization of the nonlinear with respect to the functions of the right-hand sides of the system of initial-boundary value problems (1)–(9) was carried out for a continuous model. Solutions of the linearized problem will be designated as functions of ${\tilde{q}}_i^n$, $n=1,2,\dots ,N$ considering the initial and boundary conditions. Linearization involves specifying the concentration functions of substances included in the right-hand sides of the equations on the previous time layer $t_{n-1}$. Known initial conditions (3) are used if n = 1.

The non-linearized (initial) system (1) is formulated as a chain of connected initial-boundary value problems:

$${\displaystyle \frac{\mathit{\partial }{q}_i^n}{\mathit{\partial }t}}+{\displaystyle \frac{1}{2}}\left(\mathrm{div}\left(V\cdot q_i^n\right)+q_i^n\cdot \mathrm{div}V\right)=\mathrm{div}\left(k\cdot \hspace{0.17em}\mathrm{grad}\hspace{0.17em}{q}_i^n\right)+R_{q_i}^n$$

(5)

where $i\in M$, $\left(x,y,z\right)\in G$, $n=1,2,\dots ,N$, $t_{n-1}<t\le t_n$, $t\in {\omega }_{\tau }=\left\{t_n=n\tau ,n=1,2,\dots ,N\right\}$ with initial and boundary conditions considered on the interval $t_{n-1}<t\le t_n$ for each of the equations.

Linearization involves setting the concentration functions of substances included in the right-hand sides of the equations on the previous, relative to the current, time layer:

$${\displaystyle \frac{\partial {\tilde{q}}_i^n}{\partial t}}+{\displaystyle \frac{1}{2}}\left(\mathrm{div}\left(V\cdot {\tilde{q}}_i^n\right)+{\tilde{q}}_i^n\cdot \mathrm{div}V\right)=\mathrm{div}\left(k\cdot \mathrm{grad}{\tilde{q}}_i^n\right)+{\tilde{R}}_{q_i}^{n-1}$$

(6)

$${\tilde{R}}_{q_i}^{n-1}=R_{q_i}\left(x,y,z,t_{n-1},{\tilde{q}}_i^{n-1}\right), i\in M$$

The linearization error is denoted as $z_i^n\left(x,y,z,t\right)\equiv {\tilde{q}}_i^n\left(x,y,z,t\right)-q_i^n\left(x,y,z,t\right)$, $i=1,\dots ,8$ ($i\in M$), $t_{n-1}<t\le t_n$, $\left(x,y,z\right)\in G$, $n=1,2,\dots ,N$. Equation (6) is subtracted from Equation (5). The resulting problems have a form characteristic of a linearized problem, where the functions of the right-hand side are replaced by the difference (error) in the specification of the right-hand sides:

$${\displaystyle \frac{\mathit{\partial }{z}_i^n}{\mathit{\partial }t}}+{\displaystyle \frac{1}{2}}\left(\mathrm{div}\left(V\cdot z_i^n\right)+z_i^n\cdot \mathrm{div}V\right)=\mathrm{div}\left(k\cdot \mathrm{grad}{z}_i^n\right)+{\tilde{R}}_{q_i}^{n-1}-R_{q_i}^n$$

for $n=1,\dots ,N$, $\left(x,y,z\right)\in G$, $t_{n-1}<t\le t_n$, with appropriate initial and boundary conditions.

The idea of studying the error function in a Hilbert space $L_2\left(G\right)$ with a scalar product $\left(\xi ,\eta \right)$, where the norm is determined by the expression ${‖\xi ‖}_{L_2(x,y,z)}\equiv {\left(\xi ,\xi \right)}^{1/2}\equiv {\left({\displaystyle \underset{G}{\iiint }{\xi }^2\left(x,y,z\right)dxdydz}\right)}^{1/2}$, is briefly outlined below.

Both sides of each equation are multiplied by the corresponding $z_i^n$. The error norm ${‖z_i^n‖}_{L_2(G)}$ is proven to approach zero for any n and i under conditions based on hydrophysical and biogeochemical constraints. The proof involved rather cumbersome transformations, using the Ostrogradsky-Gauss theorem, Green’s formula, and Poincaré inequalities. Inequalities are obtained that guarantee the closeness of the solutions to the linearized and nonlinear problems for each of the substances $q_{F_i}$ in $L_2\left(G\right)$ on a sequence of grids ${\omega }_{\tau }$ for $\tau \to 0$:

$$\underset{C_1\equiv const>0}{{‖z_1^n\left(x,y,z,t_n\right)‖}_{L_2\left(G\right)}}\le \underset{n=1,2,\dots ,N}{C_1\tau }$$

2.2. Specifying the Model Input Data

In 2022–2024, the researchers of the Azov-Black Sea branch of the Russian Federal Research Institute of Fisheries and Oceanography (“AzNIIRKH”) studied the hydrobiological characteristics of the Azov Sea, in particular the salinity and temperature of the waters. Salinity values at points on the hydrobiological survey grid are presented in the paper [20]. The data from field measurements are consistent with the assumption of the authors of this article about an increase in the salinity of the Azov Sea in the Taganrog Bay by 30% of the normal values for the water system, which is reflected in the paper [19]. A forecast was also made for the growth of the main types of phytoplankton populations in summer under various scenarios of salinization of the Azov Sea. As a result of the Azov Sea salinization, the range of cyanobacteria replaced to the eastern part of the Taganrog Bay. They are almost absent in the main part of the sea, which is confirmed by the data of “AzNIIRKH” [21]. Considering the above, it can be assumed that the obtained ranges of phytoplankton populations at salinity values increased by 30% from normal can be used as initial distributions of phytoplankton population concentrations for conducting a computational experiment on the algolization of the water system. Also, the forecast of the geographic location of phytoplankton populations presented in Figure 2 reflects the ratio of green algae and cyanobacteria, the biomass of which in Taganrog Bay constitutes 60–70% of the total phytoplankton biomass [22]. According to research [22], green algae predominate in the eastern part of the bay and prefer fresh water, although they can also live in brackish water.

At the beginning of the growing season, nutrients are abundant. They flow abundantly into Taganrog Bay with the Don River runoff during the winter. At the beginning of the experiment, the distribution of the main nutrients is set to be uniform. This simplification is adopted in the model, since it is difficult to obtain field data on the spatial distribution of biogenic substances. The phosphate concentration is 0.04 mg/L, and nitrate concentration is 0.204 mg/L. According to “AzNIIRKH” [21], the average concentration of phytoplankton biomass in Taganrog Bay is 1 mg/L, with cyanobacteria accounting for 70%. The initial distribution areas of phytoplankton populations are shown in Figure 2; the maximum concentration of green algae was 0.1 mg/L, and that of cyanobacteria was 0.7 mg/L. The optimal temperature during the experiment was 25 °C for green algae and 28 °C for cyanobacteria. The distributions of salinity and temperature values fed to the input of the software module for modeling algolization of the Taganrog Bay are shown in Figure 3.

When solving the linearized problem (1)–(4), the input data are the values of the components of the water flow vector at the nodes of the hydrodynamic computational grid, which is calculated based on a 3D hydrodynamic model implemented in the software package [23], salinity values $S_0$, temperatures $T_0$ and calculated concentrations $q_{0i}$ at the time $t_0$. The methods of discretization and grid equation solution used in the software module are described below.

Data Sources and Validation

The initial values of phytoplankton population concentrations [21,22] and salinity [20] were obtained from literature sources, using the work of researchers of the “AzNIIRKH”, an organization that regularly conducts field research. The values of nutrients and temperature in the spring were obtained from data from the ESIMO geoinformation system [24]. Depth values obtained from processing pilot charts were used to determine the boundaries of the calculation area [25]. The values of salinity, temperature, and depth were obtained using the value retrieval method described in the work [26]. This method allows us to obtain maps of depths, salinity, and temperature distributions that are sufficiently smooth at the gluing points.

2.3. Construction of a Discrete Model and Its Numerical Implementation

The problem numerical solution involves constructing a discrete model (difference scheme) using input data and applying the method of numerically solving grid equations. The simulation area is assumed to be inscribed in a three-dimensional stepped region and is covered by a computational grid ${\omega }_{\tau }\times {\omega }_h$, which is uniform in time and three spatial directions:

$${\omega }_{\tau }=\left\{t_n=n\tau ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=0,1,\dots ,N,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}N\tau =T\right\}$$

$${\overline{\omega }}_h=\left\{x_j=j\cdot h_x,\hspace{0.17em}{y}_k=k\cdot h_y,\hspace{0.17em}{z}_l=l\cdot h_z;\hspace{0.17em}j=0,1,\dots ,N_x,\hspace{0.17em}k=0,1,\dots ,N_y,\hspace{0.17em}l=0,1,\dots ,N_z\right\}$$

where $\tau$ is the time step, $0\le t\le T$ is the time interval, $h_x,h_y,h_z$ are the steps in spatial directions Ox, Oy and Oz, respectively, $N_x,N_y,N_z$ are the maximum number of grid nodes in each spatial direction. The linearization considered above allows us to obtain a system of linear grid equations.

Discretization of problem (1), based on the system of advection-diffusion-reaction equations, is carried out using implicit schemes constructed on hydrodynamic grids. For each of the substances of number i included in the linearized formulation of problem (1)–(4), we have a system of difference equations:

$$\begin{gathered} {\displaystyle \frac{{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_l\right)-{\tilde{q}}_i^n\left(x_j,y_k,z_l\right)}{\tau }}+ \\ +{\displaystyle \frac{1}{2}}\left[u^n\left(x_{j+1/2},y_k,z_l\right){\displaystyle \frac{{\tilde{q}}_i^{n+1}\left(x_{j+1},y_k,z_l\right)-{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_l\right)}{h_x}}+u^n\left(x_{j-1/2},y_k,z_l\right){\displaystyle \frac{{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_l\right)-{\tilde{q}}_i^{n+1}\left(x_{j-1},y_k,z_l\right)}{h_x}}\right]+ \\ +{\displaystyle \frac{1}{2}}\left[v^n\left(x_j,y_{k+1/2},z_l\right){\displaystyle \frac{{\tilde{q}}_i^{n+1}\left(x_j,y_{k+1},z_l\right)-{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_l\right)}{h_y}}+v^n\left(x_j,y_{k-1/2},z_l\right){\displaystyle \frac{{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_l\right)-{\tilde{q}}_i^{n+1}\left(x_j,y_{k-1},z_l\right)}{h_y}}\right]+ \\ +{\displaystyle \frac{1}{2}}\left[w^n\left(x_j,y_k,z_{l+1/2}\right){\displaystyle \frac{{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_{l+1}\right)-{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_l\right)}{h_z}}+w^n\left(x_j,y_k,z_{l-1/2}\right){\displaystyle \frac{{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_l\right)-{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_{l-1}\right)}{h_z}}\right]= \\ ={\displaystyle \frac{1}{h_x}}\left[{\mu }_x\left(x_{j+1/2},y_k,z_l\right){\displaystyle \frac{{\tilde{q}}_i^{n+1}\left(x_{j+1},y_k,z_l\right)-{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_l\right)}{h_x}}+{\mu }_x\left(x_{j-1/2},y_k,z_l\right){\displaystyle \frac{{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_l\right)-{\tilde{q}}_i^{n+1}\left(x_{j-1},y_k,z_l\right)}{h_x}}\right]+ \\ +{\displaystyle \frac{1}{h_y}}\left[{\mu }_y\left(x_j,y_{k+1/2},z_l\right){\displaystyle \frac{{\tilde{q}}_i^{n+1}\left(x_j,y_{k+1},z_l\right)-{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_l\right)}{h_y}}+{\mu }_y\left(x_j,y_{k-1/2},z_l\right){\displaystyle \frac{{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_l\right)-{\tilde{q}}_i^{n+1}\left(x_j,y_{k-1},z_l\right)}{h_y}}\right]+ \\ +{\displaystyle \frac{1}{h_z}}\left[{\mu }_y\left(x_j,y_k,z_{l+1/2}\right){\displaystyle \frac{{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_{l+1}\right)-{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_l\right)}{h_z}}+{\mu }_y\left(x_j,y_k,z_{l-1/2}\right){\displaystyle \frac{{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_l\right)-{\tilde{q}}_i^{n+1}\left(x_j,y_k,z_{l-1}\right)}{h_z}}\right]+{\tilde{R}}_i^n \end{gathered}$$

where $\left(x,y,z\right)\in {\omega }_h$, $n=0,1,\dots ,N-1$.

Also, the boundary conditions are approximated with the second order of accuracy with respect to the spatial grid steps, which we will not dwell on for brevity [27]. For the constructed difference scheme, the conditions of its monotonicity and positive definiteness of the grid operator of the problem on the upper time layer are satisfied, provided that the Courant–Friedrichs–Lewy constraint on the time step is satisfied and the grid Péclet number is bounded, not exceeding 1 [27]. Estimates of the permissible time step are 10–20 s when using steps in the horizontal directions of 300–500 m, and in the vertical direction—0.2–0.5 m. For the numerical solution of the obtained system of difference equations, an iterative alternating-triangular method of variational type is used for grid equations with a non-self-adjoint operator [28]. The general flowchart of the modeling process is shown in Figure 4.

2.4. Description of the Computational Experiment

The algolization experiment consists of introducing a suspension of green algae at the beginning of their growing season, approximately March–April. By the beginning of the cyanobacteria growing season (in May–June), the green algae have consumed most of the nutrients, and there are insufficient nutrients for abundant cyanobacteria bloom. The chlorella suspension is best introduced into areas of the water system where advection is greatest, such as riverbeds, spits, and other areas. Water flow velocity values were obtained using the “Azov3D” software package (developed by Sukhinov A.I. and Chistyakov A.E. [23]), which implements a three-dimensional non-stationary mathematical model of hydrodynamics. In the Azov Sea, easterly and northeasterly winds prevail from October to April. Such directions are formed under the influence of the spur of the Siberian anticyclone [29]. Therefore, the current structure obtained with an easterly wind direction was chosen as the input data for a computational experiment on the algolization of the Taganrog Bay under conditions of increased salinity. The current pattern in the Azov Sea with an easterly wind speed of 5 m/s is shown in Figure 5. The red dots mark the places where the suspension was introduced. When selecting the points, the speed of the currents and the fact that Chlorella vulgaris is a freshwater alga, as well as the accessibility for introducing the suspension from the shore, were considered. The concentration of Chlorella vulgaris in the suspension was 1167 mg/L, the release rate was 5 L/s, a total of 25 tons were released, 2.5 tons at each of the 10 release points.

3. Results

The authors of this article conducted a computational experiment on the biological rehabilitation of the Taganrog Bay under conditions of its salinization based on the introduction of green microalgae. Distributions of concentrations of green algae and cyanobacteria were obtained over time intervals of 15 days (Figure 6) and 30 days (Figure 7) for the concentration of Chlorella vulgaris in suspension—1167 mg/L and volume 25 tons as a result of the modeling.

Figure 8 shows the distribution of concentrations of green algae and cyanobacteria over a 30-day time interval for a Chlorella vulgaris concentration in suspension of 2333 mg/L and a volume of 25 tons.

The figures show the concentration values of two microalgae types on the Taganrog Bay surface.

4. Discussion

The obtained modeling results for the distribution of green algae and cyanobacteria concentrations indicate the success of the experiment on the biological rehabilitation of Taganrog Bay at given values of concentration and volume of the introduced suspension. The introduction of a suspension of Chlorella vulgaris phytoplankton into the Taganrog Bay in the spring before the onset of the growing season of potentially toxic cyanobacteria Aphanizomenon flos-aquae was modeled. The introduction points were selected in the desalinated zone (salinity values up to 7–8 ‰), which allowed the freshwater green algae to survive and successfully vegetate. The green microalgae consumed phosphates ($P{O}_4$) and nitrates ($N{H}_4$), so by the beginning of the growing season of cyanobacteria, nutrients were deficient. The concentration of cyanobacteria exceeded that of green algae at the beginning of the experiment (0.7 mg/L and 0.1 mg/L, respectively). After 15 days, the concentration of cyanobacteria was 131 times less than the concentration of green algae (0.034 mg/L and 4.462 mg/L, respectively). After 30 days, the difference in concentrations increased even more (1.349∙10⁻³ mg/L and 1.475 mg/L, respectively). It is also shown (Figure 6) that if the concentration of the introduced green alga is doubled (to 2333 mg/L) at the same volume, the Chlorella vulgaris concentration after 30 days (9.267 mg/L) becomes potentially dangerous. This concentration of green algae, combined with other phytoplankton species, can lead to eutrophication of the marine system and fish kills. Increasing the amount of green algae introduced is also expensive and, therefore, economically unviable. The 30-day modeling period was chosen based on the fact that after a decrease in the concentration of cyanobacteria by 100 or more times compared to the unregulated number without the introduction of Chlorella vulgaris, the change in the number of these cyanobacteria cannot increase, based on the properties of the biological model used, since a practically equilibrium state of these two competing populations is established on the 25–30th day of this process, which is confirmed by the data of numerical modeling.

As a result of the computational experiment, the optimal concentration and volume of the introduced Chlorella vulgaris suspension were empirically determined. It should be noted that the results of the computational experiment were obtained using reliable data on salinity, temperature, and distribution of the simulated substances, confirmed by field studies and long-term observations. A situation where the concentration of green algae exceeds that cyanobacteria (Figure 7) while remaining within the maximum permissible limits is economically favorable. Green algae are the basis of the food chain and provide food for zooplankton, which in turn provides food for many species of commercial fish. A situation where the concentration of cyanobacteria exceeds that of green algae (Figure 1), as typically occurs in summer, is unfavorable, as cyanobacteria have low nutritional value, are potentially toxic, and their abundant blooms lead to fish kills, which negatively impacts the regional economy and the health of local populations. Conducting an experiment to control algae growth in the Taganrog Bay involves repeated application of a Chlorella vulgaris suspension at the beginning of active cyanobacterial growth—in early June, when water temperatures reach optimal levels for cyanobacteria. A third application is possible during the summer. Otherwise, cyanobacteria annually predominate over green algae in the Taganrog Bay, as shown in Figure 2. This predominance has been confirmed by field measurements [21]. The approximate cost of one such injection is approximately 5000 USD. 2–3 injections are necessary per year. These costs appear modest compared to other methods of predicting algae blooms in Taganrog Bay, such as cleaning riverbeds to increase river flow, improving wastewater treatment, and others. Some factors, such as weather and climate conditions, agricultural activities, and others, are not controlled.

Some considerations regarding the time interval for predictive modeling using model (1)–(4) are given below. To increase the prediction period to 1–2 years, retrospective data on the combined influence of the main factors on salinity changes are used. Changes in the runoff of the Don and Kuban rivers, which are the main sources of fresh water for the Azov Sea, as shown by long-term observations, are cyclical with a period that can be approximately estimated as 12–15 years. In the first decade of the 21st century, some increase in freshwater runoff was observed. In the last 7–8 years, a significant decrease in this runoff has been observed with a simultaneous increase in average annual temperatures and evaporation, especially in summer. According to data from our colleagues, researchers at the Southern Scientific Center of the Russian Academy of Sciences, in the spring of 2025, the salinity of water in the Taganrog Bay increased twofold compared to 2007; the reasons for this are discussed in [1]. In 2026–2027, the increase in salinity should slow due to the expected cyclical increase in freshwater runoff in the Don and Kuban rivers. The increase may amount to 1–2 g per liter over these two years in Taganrog Bay. Calculations based on a previously published study by the authors [19] indicate a shift in the entire desalinated water zone by 10–12 km toward the mouth of the Don River in a northeasterly direction. Therefore, a high probability of correctness should be expected for the obtained predictions of changes in the abundance of cyanobacteria with the introduction of green algae in the eastern part of Taganrog Bay, however, over a somewhat smaller area. Therefore, considering these clarifications, the proposed forecast remains valid through 2027.

The authors envision several directions for further improvement of the model: adding modeling of zooplankton populations and considering the consumption of phytoplankton populations by zooplankton (at this stage, grazing is factored into the phytoplankton mortality rate); considering the influence of illumination on the vertical distribution of phytoplankton; considering the buoyancy of cyanobacteria; model sensitivity analysis; validating predictions against long-term field data; exploring uncertainties through sensitivity analysis and probabilistic modeling; and integrating machine learning to handle complex, nonlinear interactions.

5. Conclusions

The modeling results were obtained using modern, high-precision mathematical modeling methods. To achieve this goal, the authors developed a mathematical model of phytoplankton population dynamics and biological kinetics, considering the most significant factors influencing microalgae growth: salinity, temperature, and nutrient availability. The input data for this model included salinity and temperature distributions and depth values obtained by processing cartographic information. Current velocity values obtained from a hydrodynamic model were also input. Discretization of the continuous model was performed using implicit schemes built on hydrodynamic grids. An iterative alternating-triangular variational method was used to solve the grid equations. The software module was developed based on the aforementioned mathematical modeling methods and integrated into the “Azov3D” forecasting and research software. The study results demonstrate the advantages of using an integrated approach to mathematical modeling of processes occurring in complex natural systems. They can be successfully used to simulate various scenarios for the development and rehabilitation of marine systems. The mathematical model and numerical methods described in this study can be replicated to other shallow coastal water systems. Practical application of the algolization method also requires preliminary field studies, such as expeditions.

Despite the obtained results, the use of chlorella cannot be considered the primary method for improving the ecological state of a water system. There are both positive and negative experiences with the use of chlorella in water system rehabilitation. Based on the computational experiment conducted, it can be concluded that this method could be an additional and effective tool for water system rehabilitation.