The article is devoted to the mathematical modelling of bioremediation processes of wastewater contaminated with nutrients. The model of a consumer-resource water system in which the resource is harmful impurities and the consumer is Euglena gracilis was built. The proposed system of diferential equations with predefined initial conditions in general describes the dynamics of changes in the concentrations of euglena and phosphorus impurities during their interaction quite qualitatively. In particular, the graphs of the functions of microalgae and impurities dependencies on time have a clearly expressed S-shape.
Today, many technologies are used to clean up environmental pollution and bioremediation processes,
as green technologies, are becoming one of the main environmental technologies that scientists use
in their research. Key research findings show that some strains of microalgae can be successfully
used for biological water treatment, providing fast and eficient removal of phosphorus and nitrogen,
which contribute to eutrophication of water bodies [
The efective management of natural resources and ecological systems has been a matter of concern in recent decades, as the problems of intense anthropogenic impact on the environment have put humanity in need of preserving natural systems and preventing their destruction. Achieving a reasonable and efective ecosystem management strategy is quite dificult, as there are many conflicting factors to consider, which must be balanced due to the complexity of the real-world problems. Moreover, various processes and activities are interconnected, leading to complex systems with interactive, dynamic, nonlinear, multi-purpose, multi-stage, multi-layered and uncertain features.
One of the prerequisites for this is the modelling and forecasting of anthropogenic processes in nature.
The most important socio-environmental problems that need to be addressed through modelling are
twofold. The first is the need to model the distribution of pollutants in the aquatic environment in
order to localise their impact and prevent negative social consequences. The second area, which is of
great social importance, is the preservation of water sources as an entire ecosystem, provided that its
parameters are stable within certain limits. It is a systematic approach to the study of a water body
that can ensure, on the basis of regulatory and search forecasting, compliance with environmental
requirements [
The use of modelling and forecasting methods determines the understanding of everything that is happening and can happen in water systems, atmosphere, soils, flora, the consequences of human intervention in them, pragmatises assessments and conclusions, and helps to find optimal technical, technological, organisational and environmental solutions. Mathematical models are recognised as efective tools that can help investigate the economic, environmental and ecological impact of alternative pollution control and resource conservation measures, and thus assist in decision-making in the formulation of environmentally and economically eficient management policies.
Therefore, the development of mathematical models of the interaction of pollutants, including nutrients contained in wastewater, with microalgae is an urgent task. Our task was to build and study a model of a consumer-resource water system in which the resource is harmful impurities and the consumer is Euglena gracilis. We will limit ourselves to the case when there is a certain amount of impurities in the reservoir and no new impurities enter the system.
3.1. Development of mathematical models for the interaction of microalgae with phosphates Consider the equation of the interaction of impurities in water with Euglena gracilis. Denote by and according to the concentration of algae and impurities. Let the concentrations of microalgae and impurities be equal, respectively, at the moment of time = 0:
(0) = 0 > 0, (0) = > 0.
The dynamics of changes in the concentration of impurities and microalgae, assuming that the trophic function is directly proportional to the amount of resource, i.e. the concentration of impurities, can be described by a system of diferential equations of the form: {︃ = 1 − = − 1, , where is the parameter, characterizes the process of increasing the concentration of algae with time consumption of impurities; is the parameter, characterizes the rate of decrease in the concentration of algae by account of natural processes (specific mortality); is the parameter, characterizes the process of reducing the concentration of impurities in water environment due to interaction with algae.
This system is a special case of the mathematical model of Lotka and Volterra, which they proposed
to describe the interaction of two species - a population of predators and a population of prey. In our
case, the microalgae Euglena gracilis act as predators, and impurities in water act as prey [
System (2) takes into account the following assumptions: (1) (2) • in the absence of microalgae, the concentration of impurities does not change according to the equation ; − , so = 0−
; • in the absence of impurities in the water, microalgae die of according to the equation = • the expression , taken with a certain coeficient, describes the efect of the microalgae population on the change in the amount of impurities in the water, i.e. the result of ‘eating’ impurities by the value , proportional to the concentration of microalgae. impurities by microalgae leads to a decrease in the rate of change in the concentration of In the following, we introduce the notation: 1 = 00 , 1 = 00 . Then the system (2) will {︃ = 0 · 0 − = − 0 · 0 .
,
Thus, it is necessary to solve the Cauchy problem - to find a solution to the system (3) that satisfies the initial conditions (1). For given values of the parameters, this Cauchy problem has a unique solution, and it is convenient to search for it approximately, using, for example, the fourth-order Runge-Kutta
Let’s find out the meaning of the parameters and in system (3) and propose an algorithm for determining their values depending on the initial concentrations 0 and 0.
About parameter values and Note that > 0, > 0, ≥ 0 . In addition, the following equations
Thus, the value of the parameter is equal to the sum of the initial growth rate of the microalgae concentration and the specific mortality rate multiplied by the initial value of the microalgae concentration, and the value of the parameter is equal to the absolute value of the rate of decrease in the concentration of impurities at the initial time = 0: = (0)
+ 0, = − (0) . take the form: method. hold where we get it from (3) (4) (6) (7) (0) = 0 0 · 0 0 − 0 = −
0, = (0) + 0, (0) = − 0 0 · 0 0 = −.
(5) If we assume that = 0 , then =
(0) . In the following, we will assign a certain value to the parameter , assuming that this value does not exceed 0.05, for example, 0.02.
The values of the parameters and are also not subject to precise determination. However, using the concept of the first-order diferential of the functions () and (), we can estimate these values. 3.2. Improvement of mathematical models of microalgae-phosphate interaction on the example of euglena gracilis strain Let us formulate the basic requirements for a mathematical model of the interaction of impurities with microalgae. The concentrations of impurities and microalgae, respectively, and the curves describing the dynamics of their changes, must meet certain requirements: number of days during which microalgae interact with impurities. Most of the time, the concentration of ( ) is close to zero, and as a rule, ∈ [4; 7];
2) the graph of the function = () must have a pronounced S-shape. In particular, the function = () at the initial point in time = 0 should decrease at a rate ′(0), determined by formula (3), so that the value of (1) is approximately equal to the calculated value ((1))* , determined by formula (3). The fastest decay of the function () should be reached at the point = * and the ordinate of the point (* ; (* )) – the curve inflection point = (), should satisfy the inequality (* ) > 0 . The steepest function graph = () should be mostly in the interval ∈ (1; 3) 2 (during the second or third day). In the interval (0; * ) the curve () must be convex and concave in the interval (* ; );
3) the function = () should increase in the interval ∈ [0; 1], where 1 ≤ . If 1 = , then the function = () should increase during all days of interaction between microalgae and impurities; if 1 < , then the function would increase in the interval ∈ [0; 1], acquire values from 0 = (0) to (), and decrease in the interval ∈ [1; ]. The rate of growth of the function () should initially increase from ′ (0) to (′) ′* and then the rate should monotonically decrease to a certain small value or be equal to zero at the point = 1. The graph of the function = () on the interval [0; 1] should also have a pronounced S-shape. In the interval (0; * ), the curve () must be concave, and in the interval (* ; ) – convex. The point (* ; (* )) of the curve () must be an inflection point, and the following conditions must be met: * approximately (* ) > 23 (), i.e. the curve () should change from concavity to convexity essentially at the final stage (in the last one or several days, depending on the initial values of the > 2 , concentrations of impurities and microalgae) of the interaction of microalgae with impurities.
The achievement of the corresponding efect depends on the perfection of the choice of trophic
functions that determine the qualitative properties of the mathematical model of the systems
‘predatorprey’, ‘consumer-resource’, etc. Many works [
It should be noted that today, most scientists who study these issues do not have a unanimous opinion
on how to solve the problem of choosing trophic functions [
The above requirements for a mathematical model of the interaction of impurities with microalgae are generally satisfied by the system of diferential equations: the multiplier () = 1 − ︁( 1 than 20. The multiplier () has the following properties: 1) at the initial time = 0 : (0) = 1; The main diference between them is the presence in the numerator of the function 1(, ) of )︁ , where the indicator n takes on a certain value, mostly greater ⎧ ⎨ ⎪⎪ = · · ⎪ ⎪⎪⎩ ⎪ = − · · − 1 (︁ )︁ 1 0 0 0 · 1 − 1+ 1(︁ 0
1 )︁ 1 · 1+ 2(︁ )︁ 2 −
, 1 1 0 · 0 · 1+ 3(︁ )︁ 3 · 1+ 4(︁ )︁ 4 0 0
)︃ ︃( 1 1 1 − ︁( 1 2) () is an increasing function, which, when the argument is continuously changing from 0 to 0, takes continuous values from 1 to 0, and in the interval [; 0], where = 0.150, the double inequality 0.99 < () ≤ 1 is satisfied. A significant change in the value of the function () from 0 to 0.99 is observed in the interval [0; ]. For example, if we take [0 = 14, = 30, = 2, then on 1; the interval [[0; ] the graph of the function [ () = 1 − ︁( 1 )︀ 30 on the interval [0;2].
︁( 1 3) at small values of [ concentration, the approximate equality [ () = 1 − is true, i.e., this function approaches the linear one.
From the above properties of the function (), its role in the trophic function 1 (, ) follows, namely, this function is intended primarily to prevent a rapid increase in the concentration of () of microalgae during the first few days of their interaction with impurities.
In order to reproduce the dynamics close to the results of the experiments, regulating factors that appear in the denominators of trophic functions are introduced into the mathematical model. In particular, based on the results of the research, the following values of the indicators are proposed 1 − 4
: time = 0:
1 = 4.5, 2 = 0.89, 3 = 5, 4 = 2.2.
To determine the values of the coeficients 1 −
4, the parameter q is introduced, which has a certain relationship with these coeficients. Let us illustrate it. To do this, consider the system (8) at the initial ⎧ (0) = · · ⎨ ⎩ (0) = − · ·
1 (1+ 1)(1+ 2) −
1 (1+ 3)(1+ 4) 0,
To preserve the meaning of the parameters and we require that at the initial time = 0 the following equations are satisfied: {︃ = · · − = − · · 1+ 1 · 1+1 2 , 1 1
1 1+ 3 · 1+ 4 research, we propose calculation formulas for determining the values of the coeficients which means that (1 + 1) (1 + 2) = and (1 + 3) (1 + 4) = . Given that the coeficients 1 − 4 take on positive values, let us assume, for example, that = 4, then 1 ∈ (0; 3) - a certain constant,
1 = (1 + 4.928(00)0.058), 3 = (1 + 19.96(00)−0.037 ).
For the case of 0 = 120, 0 = 14, the values of the coeficients 1 − 4 to the nearest thousandth are as follows: 1 = 2.150, 2 = 0.270, 3 = 2.780, 4 = 0.058. 3.3. Testing of mathematical models and results of modelling the interaction of microalgae with phosphates Consider the example when 0 = 120, 0 = 14. The results of calculation are shown in Table 1 and Figure 2.
The proposed system of diferential equations (8) with initial conditions (1) in general describes the dynamics of changes in the concentrations of euglena and phosphorus impurities during their interaction quite qualitatively. In particular, the graphs of the functions = () and = () have a clearly expressed S-shape. The curve = () adequately reflects the dynamics of changes in the concentration of phosphorus impurities during all days of interaction, except, perhaps, the second day. Thus, during the second day, the calculated concentration of phosphorus impurities in Examples 1-3 decreased by 50.6-53.2 %, while according to experimental data, it decreased by 62.8-92.9 %. During the third day, according to the calculations, a faster decrease in the concentration of phosphorus impurities was observed (more than 4 times compared to the previous day).
According to the calculations on the first two days of interaction, the obtained concentration of () of microalgae practically coincided with the experimental concentrations. On the third day, the calculated results slightly exceeded the experimental results - by 15.9%, 9.9%, 12.3%, respectively. On the fourth day, the diference between the calculated and experimental concentrations of microalgae was almost levelled.
The authors have not employed any Generative AI tools.