INTRODUCTION

Water quality deterioration can be seen now in many countries. It has been reliably identified in some water bodies in Russia; and in the majority of rivers and lakes in economically developed regions in Russia, its regime over the recent two or three decades can be classified as quasi-stationary: it improves or deteriorates in some periods still remaining at about the same level, regretfully, low [2]. A key problem of hydroecology is studying the response of aquatic biota to anthropogenic impacts, which implies quantitative description of the toxic effect produced by a pollutant on aquatic organisms at an increase in the duration of the impact period. Such description, a typical problem of aquatic toxicology, is of particular importance when water quality regime is quasi-stationary. Reliable estimates of the toxic effect of pollutants based on mathematical models of ecotoxicological processes are indispensable in the substantiation of the choice of measures aimed to reduce the input of anthropogenic pollution into water bodies.

The simulation of aquatic toxicology processes is complicated by the fact that the intoxication mechanisms are still poorly known and the experimental data that are accumulated in experiments in vitro (microcosms, laboratories) disagree with the results in vivo (in natural water bodies) [11, 12, 14, 16, 17, 21–23]. Therefore, the quantitative results obtained in laboratories [1, 13, 15, 18, 20, 24] cannot be directly expanded to natural processes. However, it can be argued that the general regularities of the type concentration–effect, reflecting the toxicodynamics regularities, are the same in all cases mentioned above [1, 11–18, 20–24] in terms of the character of the function (but not the values of its parameters) and can bee described by a standard concave/convex logistic function.

As a typical example, Fig. 1 gives the dependence of the death rate of stickleback on the time of the process of its dying at a fixed concentration C of nitric salts in water [13, 18, 24].

Fig. 1.
figure 1

Mortality (the proportion of killed stickleback) as a function of exposure time (day) in solution of (a) Mg (bottom line, C = 300 mg/dm3), Ba (top line, C = 400 mg/dm3); (b) Cu (bottom line, C = 0.2 mg/dm3), Ag (top line, C = 0.04 mg/dm3).

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Note that, although the curves in Fig. 1b have no points of inflexion, the equation of logistic function with appropriate coefficients adequately describes the respective relationships; this allows us to assume that the inflexion points fell beyond the boundaries of the drawing, because the values of the argument for them lie on the abscissa to the left of 0. Hereafter, any segment of the logistic function to the right of its inflection point (as in Fig. 1b) will be referred to as a segment of its convex upward branch.

Clearly, in the case of the curves describing different observed relationships, the graphical similarity alone can serve as a sound argument in favor of the adequacy of the logistic representation of such relationships (if not for all, then at least for a considerable number of real cases) only when the number of observations is large enough. Their number for each pair aquatic organism–toxicant, required to achieve the necessary accuracy, can be evaluated by the methods of mathematical statistics, and the number of such pairs is immeasurable (though it can be reduced by some methods). Another argument can be a logical conclusion derived from a set of substantial assumptions regarding the character of the process under consideration; essentially, this is an axiomatic approach used to construct a phenomenological model of the process. If it yields the same relationship as the empirical model, this is a weighty argument in favor of the adequacy of both the assumptions (axioms) and the obtained mathematical description of the process. This construction with the subsequent comparison of its results with empirical data is the objective of this article.

PHENOMENOLOGICAL TOXICODYNAMIC MODEL

The process of intoxication of organisms by a toxic substance involves a number of poorly known counteracting reactions of protection and damage [1]. However, the similarity of the known typical relationships mentioned above (Fig. 1) allows us to hope that a phenomenological description of toxicodynamic processes can be obtained by proposing some noncontradictory hypotheses regarding the processes taking place in a toxic environment, constructing an appropriate analytical model, and assessing the plausibility of the theoretical results derived from it.

The model of the process can be conveniently represented as an attack of a hazardous substance on the population with a distributed resistant level of the organisms. In the process of the primary intense penetration of the toxic substance into the organisms, the weakest of them die first. The death rate decreases as the most disease-resistant organisms survive in the population. We will introduce some hypotheses to formalize this general representation and to construct a phenomenological model of toxicodynamics.

It is assumed that the process of intoxication proceeds at constant concentration of pollutant in water and the conditions of its penetration into organisms (the first hypothesis). This hypothesis simplifies real situations (toxicant concentration can appreciably increase or decrease even over a relatively short time period, and the conditions, i.e., the temperature, the concentration of dissolved oxygen, etc. can change); however, whatever the increase in the complexity of the model with the growing understanding of any process, the basic model practically always is based on an assumption similar to the one formulated above. This hypothesis can be extended to incorporate the assumption that the duration of the intoxication process is not too long; therefore, the increase in the population, if it has taken place over the period under consideration, can be neglected (the more so that the young organisms are more sensitive to the pollution of the medium than the adult ones, and the death rate is much higher among the younger organisms).

From the first hypothesis, it follows that the total amount of the toxicant that enters all (living) organisms in the population per unit time, decreases as some of them die. However, the amounts of the toxicant that enters a single organism per unit time at the beginning and the end of the process are the same; this value is assumed constant (the second hypothesis), while the resistance of even the most tolerant individuals decreases. This biological regularity, i.e., the well-known death of organisms which have lost their congeners (the more so if they live in a toxic medium) has been convincingly described, for example, in [3].

We will refer to the proportion of the dead organisms in the total initial size of the population as the loss of population quality and denote this loss in the time of exposure t by \(u\left( t \right)\), \(\frac{{du\left( t \right)}}{{dt}}\) is the death rate of the organisms. This value is proportional to the amount of the toxic substance that enters the organisms and their resistance, i.e., the values which (in accordance with the general concept of intoxication process) are proportional to population quality (i.e., in fact, its size) and the loss of this quality.

If now we consider u(t) as a relative value, which increases from its initial value, in the best case equal to 0, to the limiting value (all organisms have died), equal to 1, we can use a probabilistic interpretation of the rate to be determined. Now we assume that \(\frac{{du\left( t \right)}}{{dt}}\) is proportional to (a) the probability of the preservation of the quality at the level \(1 - u\left( t \right)\) (the third hypothesis); (b) the probability of the death of individual organisms, which increases with increasing suppression, and hence, u(t) (the fourth hypothesis). In other words, the rate of the toxicodynamics process is \(\frac{{du\left( t \right)}}{{dt}}\sim \left[ {(1 - u\left( t \right)} \right]u\left( t \right)\), or, finally:

$$~\frac{{du\left( t \right)}}{{dt}} = \frac{{u\left( t \right)\left[ {(1 - u\left( t \right)} \right]}}{{{{t}_{0}}}},$$
(1)

here the constant factor \({{t}_{0}}\) characterizes the mean length of the organism survival period in the medium polluted by the toxicant.

The proposed phenomenological approach was used to reduce the multidimensional problem of toxicodynamics to obtain a nonlinear differential equation describing the intoxication process.

Such equations are commonly derived phenomenologically in basic studies on regular dynamics; in this case, the result naturally follows from the accepted hypotheses regarding the gradual poisoning and death of organisms in a toxic medium. It is worth mentioning that the assumptions regarding a linear relationship between toxicant accumulation and resistance, on the one hand, and population quality, on the other hand, have led to the nonlinear toxicodynamics equation (1). It is worth mentioning that the assumptions regarding linear relationships between toxicant accumulation and population quality have led to the nonlinear toxicodynamics equation (1).

The system of accepted hypotheses, which can be used to construct model (1), does not reflect in detail the effect of many physicochemical–biological processes, which cause intoxication and determine its course and effects. However, if the proposed model gives an acceptable description of the general observed toxicodynamic relationships, this will mean that it adequately reflects the joint effect of such processes and can be of use in the substantiation of hygienic standards of chemical concentrations in water and in forecast estimates of the biodiversity, depending on anthropogenic pollution of water objects, and in the scientific–theoretical respect, for assessing the adequacy of the accepted linear hypotheses.

The description of experiments with the obtained model requires an explicit representation of the dependence of the quality deterioration (loss) in the population on the exposure time.

The solutions of nonlinear differential equations similar to that described above are often complex and difficult to describe simple formulas. However, in this case, such solution was found with the use of representation of each part of equation (1) as a product of functions depending on different variables and subsequent integration of both parts of the obtained equality. As the result, we have:

$${\text{ln}}\frac{u}{{1 - u}} = \frac{t}{{{{t}_{0}}}} + {\text{ln}}\left( {{\text{const}}} \right),$$
(2)

where const is integration constant.

Rewriting the right part of (2) in the form ln\(\left( {K{\text{exp}}\frac{t}{{{{t}_{0}}}}} \right)\), and taking into account the initial condition \(K = \frac{1}{{u\left( 0 \right)}}\), where \(~u\left( 0 \right) > 0\), it can be readily shown that the toxicodynamics process can be described by the following logistic equation:

$$u\left( t \right) = \frac{{u\left( 0 \right)}}{{u\left( 0 \right) + {\text{exp}}\left( { - \frac{t}{{{{t}_{0}}}}} \right)}}.$$
(3)

The obtained expression (3) shows that \(u\left( t \right)\) is an S-shaped function, the governing parameters of which are the mean survival time of population organisms in the polluted environment \({{t}_{0}}\) and the initial quality loss of the population \(~u\left( 0 \right)\), which, under natural conditions, is related with the involvement of organisms in food chains and under laboratory conditions, with their response to placing in the microcosm.

MODEL TESTING

The obtained expression for \(~u\left( t \right)\) gives a satisfactory description of the relationships given in Fig. 1, at least, within the limits of the experimental error, as it is shown in Fig. 2 at t0 = 1.7 day, u(0) = 0.19 (the top line in Fig. 2); t0 = 1.5, u(0) = 0.02 (the bottom line in Fig. 2a); t0 = 1.2, u(0) = 0.4 (the top line in Fig. 2b); u(0) = 0.03, t0 = 1.4 (the bottom line in Fig. 2b).

Fig. 2.
figure 2

Simulation of toxicodynamic relationships given in Fig. 1 with added experimental measurement error: (a) Mg (bottom line), Ba (top line); (b) Cu (bottom line), Ag (top line). Denotations of the axes are the same as in Fig. 1.

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The proposed model can be used to obtain a quantitative estimate of toxicodynamic processes. For example, in the case of relatively weak toxicants (salts of Mg and Ba), we have t0 >1.5 day, and in the case of strong toxicants (Cu and Ag salts), it is slightly greater than 1 day. In addition, the organisms taken for the experiments were, most likely, in different initial states: they were nearly safe for Mg and Cu salts, because the value of \(~u\left( 0 \right)\) is close to zero, while in the case of Ba and Ag salts, they were weakened as \(u\left( 0 \right)\) reaches the values of 0.2 and 0.4, respectively.

The obtained toxicodynamic function (3) gave a satisfactory description for other experimental data.

One more illustration was based on experimental results of bioassay tests of the toxicity of aqueous ethanol solution and the antioxidant SkQ1 derived from it [11]. The tests were made with the use of the widespread crustacean organisms―ceriodaphnia, which are often used to assess freshwater quality.

The results of the experiment are given in Fig. 3. It can be seen that they can be adequately described by the obtained equation of toxicodynamics with the following parameters: in Fig. 3a: u(0) \(~ = ~\) 0.05, \({{t}_{0}}~\) = 1.6 for the bottom line and u(0) = 0.10, \({{t}_{0}}~\) = 2.1 for the top line, in Fig. 3b: u(0) \(~ = ~\) 0.15, \({{t}_{0}}~\) = 1.1 for the bottom line and \(~u\left( 0 \right) = 0.25,~{{t}_{0}}\) = 0.7 for the top line. In both cases, full S-shaped relationships were obtained for spirit solution, while in the case of more toxic SkQ1 solution, these were segments of convex upwards branch of logistic curve.

Fig. 3.
figure 3

The decrease in the proportion of the survived daphnia (taking into account experimental error) in spirit solutions, C = 0.001 mg/dm3 (bottom lines in both panels); SkQ1 (top lines): C = 55 nM (5а) and C = 280 nM (5b). Denotations of axes are the same as in Fig. 1.

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Calculations showed that the survival time of test-organisms in antioxidant solution is shorter than that in spirit solution by a factors of 1.3 at C = 55 nM and 1.6 at C = 280 nM. In this case, almost five-fold increase in toxicant concentration was accompanied by a decrease of the estimated time \({{t}_{0}}\) by a factor of one and a half in spirit solution and more than by half in antioxidant solutions, whatever the value of \(u\left( 0 \right)\).

Overall, the model calculation shows that the increase in pollutant toxicity level is accompanied by a change from the complete S-shaped toxicodynamics relationship to a segment of convex upwards branch of logistic curve. This result corresponds to experimental conclusions for various groups of test-organisms belonging to practically opposite links of the trophic chain, as can be seen from the comparison of Figs. 1 and 4.

Fig. 4.
figure 4

An example of the dependence of population quality loss on pollutant toxicity level and exposure time.

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Haber’s formula [13, 18] characterizes the effect of intoxication as equally depending on pollutant concentration and exposure time \(u\left( {C,t} \right)\sim Ct\). In this context, it appears reasonable to mention also the similarity between the dependences of quality loss on exposure time and the level of pollutant toxicity l. Anyway, the hypotheses used in the derivation of the time dependence of quality loss \(u\left( t \right)\), are quite acceptable in the derivation of function \(u = u\left( l \right)\). Therefore, it is convenient to transform Haber’s relationship into \(u\left( {l,t} \right) = l~t\), after which, the toxicodynamics model of the population will take the form

$$\begin{gathered} u\left( {l,t} \right) = u\left( l \right)u\left( t \right) \\ = \frac{{u\left( 0 \right)}}{{u\left( 0 \right) + {\text{exp}}\left( { - \frac{t}{{{{t}_{0}}}}} \right)}}\frac{{u\left( 0 \right)}}{{u\left( 0 \right) + {\text{exp}}\left( { - \frac{{l~}}{{l{{~}_{0}}}}} \right)}}. \\ \end{gathered} $$
(4)

The appropriate three-dimensional S-shaped plot of dependence (4), given in Fig. 4, demonstrates the behavior of a resistant population under toxic load. One can see that at the initial stage, intoxication does not cause a considerable response effect. However, biological stress appears after that (the effect becomes much stronger at smoothly increasing duration of the impact or pollutant toxicity) followed by complete degeneration of the controlled vital functions of the population.

To make the use of the obtained toxicodynamics model more convenient, Figs. 5, 6 give dependences of the intoxication process at parameters typical of the water-ecological systems described in the literary sources mentioned above.

Fig. 5.
figure 5

Quality loss determined by pollutant toxicity level in the case of u(0) = 0.05. Left to right: t0 = 0.8, 1.05, 1.33, 1.67, 2.0. Denotation of axes is the same as in Fig. 1.

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Fig. 6.
figure 6

The process of intoxication depending on the initial loss of population quality at t0 = 1.7. Left to right: u(0) = 0.50, 0.25, 0.10, 0.02, 0.005. Denotations of axes are the same as in Fig. 1.

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POSSIBLE PRACTICAL USE OF THE MODEL

Such organization of water use and exploitation of aquatic organisms, which does not disturb natural ecosystems is a key objective of the state and corporate management of the resources of water objects. To achieve it requires the implementation of a system of various measures, including investment, organizational, regulatory-economic, and juridical, as well as those indirectly aimed in the remote future at the progress to this objective, i.e., scientific, educational, and cultural measures.

If an action is of ecotechnological character and there are at least assumptions regarding the way in which its implementation can affect the rate of population quality losses (clearly, toward its decrease, otherwise there is no reason to consider it), the toxicodynamic model proposed in this article can be of use for assessing its expedience.

The calculation method for such assessment is convenient to consider with the exposure time \({{t}_{{\text{0}}}} = 1\), which causes no loss of generality, because it is equivalent to a change of time unit. This allows us to rewrite (1) in the form: \(\frac{{du\left( t \right)}}{{dt}} = u\left( t \right) - {{u}^{2}}\left( t \right)\), where the first term in the right-hand part describes the suppression of the population, and the second term accounts for its resistance.

Let an ecotechnological operation causes a population quality loss by \(p\), such that \(\frac{{du\left( t \right)}}{{dt}} = \left[ {u\left( t \right) - p} \right] - {{u}^{2}}\left( t \right)\). Particular cases of the graphic form of this equation are given in Fig. 7. As can be seen, at \(~p~\) = 0 (no ecotechnological measures), we have \(\frac{{du\left( t \right)}}{{dt}} > 0\) all over the range of the values of \(u\left( t \right),\) except for two singular points, of which the left one is unstable (in the sense that whatever small deviation of \(u\left( t \right)\) from zero will cause deterioration of the population quality). Similarly, at \(p = 0.12\), the population quality will deteriorate in the majority of the values of \(~u\left( t \right)\), thus demonstrating the insufficient intensity of ecotechnological measures.

Fig. 7.
figure 7

The rate of loss of population quality as a function of the actual loss level under nonregulated regime at \(p = 0\) (full line), 0.12 (dashed line), \(0.25\) (dash-and-dot line), \(0.4\) (dashed line).

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The same conclusion will follow for the entire interval \(0 < p < 0.25\).

The population quality will improve only at \(p > 0.25,{\text{ when }}\frac{{du\left( t \right)}}{{dt}} < 0\). However, in such case, the cost of the ecotechnological measure can be found to be unacceptably high, and the resources required for its implementation to be spent inefficiently. Moreover, the excessive measures, as well as insufficient ones, may be not useful in terms of the required result.

An acceptable solution could be the choice of the value of \(p\) at an intermediate level \(p = 0.25\); however, in this case, the condition \(\frac{{du\left( t \right)}}{{dt}} = 0\) is satisfied in a single singular point u(t) = 0.5 (Fig. 7), which is unstable, because even small changes in the value of \(~p\) will cause considerable deterioration of ecosystem quality.

The best variant of solving the problem requires the control of population quality loss with establishing the time dependence of the value \(~p = p\left( t \right)\) at the varying level \(p\left( t \right) = hu\left( t \right)\) \((h\) is the coefficient to be controlled). Now the equation of toxicodynamics becomes \(~\frac{{du\left( t \right)}}{{dt}} = u\left( t \right)\left( {1 - h} \right) - {{u}^{2}}\left( t \right)\), such that the rate of quality loss crosses \(u\left( t \right)\) axis in points 0 and \(u\left( t \right) = 1 - h\). Both these points are stable, and we note that in the domain \(u\left( t \right) > 1 - h\) the rate \(\frac{{du\left( t \right)}}{{dt}}\) becomes negative, i.e., we see an increase in the population quality, rather than its decrease.

This creates the possibility of ecotechnological control by specifying the values of \(h\). If, for example, in some case, it is desirable not to allow the population quality drop to a level of >0.6, 0.5, 0.4, or 0.3, the value of \(~h = 1 - u\left( t \right)\) should not be < 0.4, 0.5, 0.6, 0.7, as it is illustrated by the graphical representation of equation for \(\frac{{du\left( t \right)}}{{dt}}\) (Fig. 8).

Fig. 8.
figure 8

The rate of loss of population quality under regulated regime at \(h = 0.4\) (full line), 0.5 (dashed line), 0.6 (dash-and-dot line), 0.7 (dashed line).

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CONCLUSIONS

The predominant dependences of suppression of aquatic ecosystems on exposure time are as follows:

logistic S-shaped function, the second derivative of which changes its sign from plus to minus in an inflection point;

a segment of a convex (upward) branch of the logistic S-shaped function (the second derivative of which on this branch is positive).

In this study, noncontradictory hypotheses regarding intoxication mechanisms are proposed to describe these dependences:

the conditions of intoxication development (including toxicant concentration in water) do not change during the process;

the amount of the toxicant that enters the body of an aquatic organism per unit time remains constant throughout the intoxication process;

the total volume of the toxicant that enters the organisms of the population is directly proportional to the total number of the organisms that have not died;

the growth rate of the share of the organisms in the population that have died is directly proportional to this share (i.e., the number of the organisms that have died).

These hypotheses are based on the changing proportion of the levels of resistance/susceptibility of the organisms to the toxicant, depending on the level of population quality loss (a decrease in its size) in accordance with the concept of intoxication as an attack of a hazardous substance on the population, where the effectiveness of the attack is determined by the distribution of the resistance power in such a manner that the decrease of the resistance and an increase in susceptibility are described by logistic equations.

The proposed model has two control parameters:

the level of quality loss (suppression) of the population at the moment of the start of intoxication process \(u\left( 0 \right)\);

the mean survival time t0 of population organisms in the polluted medium.

It is shown that if the relative of \(~u\left( 0 \right)\) ≤ 0.1 (which under the accepted model ensures high initial resistance), and t0 ≥ 0.6 (which corresponds to a moderate level of pollutant toxicity), then the toxicodynamic dependence time–effect will have a standard S-shape. On the other hand, if the first value is ≥0.3, and the second is ≤0.7, the S-shape of the dependence will be reduced, and the trajectory will have a high initial rate of population quality decrease and subsequent asymptotic decline of this effect.

The proposed phenomenological toxicodynamics model can contribute to the substantiation of the admissible concentration of chemicals in water objects and to extension of the notions about the adaptation potential of populations, which can be used to forecast the risk of their abrupt suppression [4]. Based on this, an equation of controlled toxicodynamics was derived for the organization of water use, which does not cause disturbance in ecosystems.

Estimating model parameters in each case will make it possible to reasonably regulate water removal and timely organize measures for protection of water and hydroecosystems aimed to support weak links of trophic chains and biodiversity as a whole and to forecast hydroecological catastrophes. A necessary condition for this is to take into account the proposed toxicodynamic relationships when introducing water-ecological regulations given, in particular, in documents [5–10].