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3SIMULATION OF ZOOPLANKTON COMMUNITY'S DYNAMICS IN RELATION TO ISSUES OF BIOSAFETY AND HUMAN ECOLOGY
Abstract
The topicality of a problem under study arises from both its theoretical and applied ecology aspects. Theoretical aspects refer to relation between the ecosystem's stability and Shannon indicators of diversity and evenness, applied ones refer to loss of stability of aquatic ecosystems followed by mass development of toxic cyanobacteria, which is a serious threat for biosafety of drinking and other types of water consumption. The paper aims to highlight the results of mathematical modeling of influence of anthropogenic eutrophication on aspects of zooplankton's structure and dynamics related with possible loss of the lake ecosystem's stability, accompanied by mass development of toxic cyanobacteria. Leading approach of the study is a simulation of dynamics of zooplankton groups' number with the help of original class of discrete dynamical models for analyzing aspects of the structure and dynamics of zooplankton community of lake ecosystem associated with loss of system's stability. As a result of performed studies the idealized systems' trajectories for two periods of anthropogenic eutrophication of Lake Sevan (Armenia), reflecting the cycle of changes in the structure of zooplankton, were analyzed. Material of the paper may be used in development of information systems for decisionmaking in area of biosafety threats' prevention arising from violation of stability in aquatic ecosystems.
Keywords
discrete models of dynamic systems, zooplankton community, homeostasis of ecosystems, eutrophication, biosafety, Shannon diversity index
Elena Vysotskaya
PhD, Professor of Department of Biomedical Engineering Kharkov National University of Radio Electronics Kharkov, Ukraine evisotska@mail.ru
AUTHORS
Grigoriy Zholtkevch
DSc, dean of School of Mathematics and Mechanical Engineering, Head of Department of Theoretical and Applied Informatics V.N. Karazin Kharkiv National University Kharkov, Ukraine g.zholtkevych@gmail.com
Yurii Bespalov
MS, Senior research fellow,
School of Mathematics and Mechanical Engineering V.N. Karazin Kharkiv National University Kharkov, Ukraine bezpalof@bk.ru
Konstantin Nosov
PhD, Research fellow, School of Mathematics and Mechanical Engineering V.N. Karazin Kharkiv National University Kharkov, Ukraine k-n@nm.ru
Sergey Timofeyev
MS, Junior research fellow, School of Mathematics and Mechanical Engineering V.N. Karazin Kharkiv National University Kharkov, Ukraine sergtim2009@yandex.ru
Introduction. One of the most important aspects of human ecology has always been biosafety of drinking and other type of water consumption. In particular, currently, in connection to extreme manifestations of global climate change, this aspect is closely related to the problem of general ecological stability of biological communities of eutrophicated reservoirs. The issues of stability of biological communities in eutrophicated reservoirs are of considerable practical interest in biosecurity. A resonant example of such kind is eutrophication of Lake Kinneret in the Middle East, which is the main source of drinking water for Israel [,,,]. In this case, as in many others, the disturbance of aquatic ecosystem creates a threat to biosecurity in connection with mass reproduction of cyanobacteria — toxic phytoplankton organisms closely related to zooplankton, which saturate the water with products of their life activity. These products are in turn the nutrients for phytoplankton organisms. In addition, in the course of their life activity many representatives of zooplankton (filtrators) contribute to ousting cyanobacteria's competitors by cyanobacteria in phytoplankton. In this regard, stability of zooplankton community of eutrophicated reservoirs is of interest from the point of view of biosafety of drinking and other types of water consumption. In the view of theoretical ecology we can talk about a convenient and practically important object for studying the patterns of relations between indicators of diversity and evenness of the structure of biological objects and their stability.
In particular, we are dealing with indicators of diversity and evenness based on Shannon index [,] which had not lost its popularity among ecologists for more than half a century. It should be said that the problem of relation between diversity of biological objects and their stability has not been solved yet enough satisfactorily. Opinion of R. Margalef expressed in [], that "the ecologist sees in any measure of diversity an expression of the possibilities of constructing feedback systems, or any sort of links, in a given assemblage of species", can be implemented, in particular, with the help of new class of mathematical models developed in V.N. Karazin Kharkiv National University [] — Discrete Models of Dynamic Systems (DMDS). These models allow on the basis of correlations between the system's components to build the structure of relations between the component based on pairwise relationships from the following list: (+ +), (+, -), (- -), (+, 0), (-, 0), (0,0). Besides, each component can interact with itself and allowed self-relationships are symmetrical, i. e. (+,+), (-,-), (0,0).
Here we describe in short the essence of the DMDS model. Details can be found in the sources [].
Let a natural system (biological, ecological and so on) has N components ...
, An and we assume that components take discrete values 1,2, ..., K, i. e. K values. The value 1 means a minimum amount of a component, the value K means its maximum amount, i. e. a component varies from 1 to K. The time of the system is also discrete, i,
e, t = 0,1,.... Thus, the values of the i -th component Ai at the instants of time t = o,l,...
are numbers A^O), A^X), .... So at the instants of time t the state of the system is an
integer-valued vector (4(0, A2(t), ..., AN(t))T, and the trajectory of the system is an infinitive-right matrix
'4(0) 4(1) 4(2)
4(0) 4(1) 4(2) ...
V4v (0) An(2)
As the system takes a finite number of states, any trajectory like (1) can be presented
by a finite matrix
(A (s)
A2(S)
yAN S
where s > s0 for some integer s0 >0, integer T> 0.
We introduce a notion of relationship between components Ai and Aj as a pair
(®l,ffl2), where gQ = {-,0, +}, that gives us above mentioned relationships (+ +), (+, -), (- -), (+, 0), (-, 0), (0,0). Omitting exact definitions that can be found in cited references, we shall clarify here a sense of relationships on same examples. Say, if a pair
A and Aj is in relationship (+, -), it means that high values of Ai lead to decreasing value of Aj (negative relationship). Reciprocally, high values of Aj lead to increasing value of A (positive relationship). If the pair Ai and Aj is in relationship (-, 0), it means that Ai doesn't influence Aj (neutral relationship), but high values of Aj lead to decreasing value of Ai (negative relationship) and so on.
Revealing of the structure of relationships between all components AUA2) ... f AN is based on the observation table, which includes the values of the components measured at random moments. This procedure can be implemented in different way. We distinguish two approaches: first is based on weighting of all influences on a given component (so called weighted function approach), other is involved a famous Liebig's law of the minimum.
In result we obtain the system of relationships between all components and the trajectory, similar to (2).
With the use of DMDS the structure of relationships and dynamics of systems of different nature were analyzed []. In particular, the structure of relationships between four most important species of zooplankton in lake Sevan (Armenia) has been revealed. This lake was exposed to anthropogenic eutrophication in a way similar to lake Kinneret.
With the help of DMDS it is also possible to investigate the relationships between diversity and evenness of zooplankton community and its stability — by building the corresponding trajectories of systems and analyzing the stability of values of system's component reflecting the number of specified groups of zooplankton and dynamics of Shannon diversity index (calculated on the basis of components' values reflecting the number of specified zooplankton groups). As we obtain idealized trajectories, all values are measured in conventional units.
Corresponding idealized systems' trajectories, that reflect a cycle of zooplankton community's changes at different periods of anthropogenic eutrophication of the lake ecosystem, were built. We are dealing with the periods, which are different according to near future environmental sustainability risk of mass development of cyanobacteria and zooplankton's sensitivity to a weak (such as magnetic storms) external influences. The analysis of differences between these periods by the character of dynamics of Shannon diversity index (calculated on the basis of components' values and reflected the number of specified zooplankton groups) was also carried out as well as the differences between the periods by the character of dynamics calculated in a similar manner regarding total number of zooplankton and its individual members.
Al(s +1) ... A2(s + 1) ...
^0 + 1) ...
+ T -1) ^ A2(S + T -1)
An(s + T -1),
This is the aim of the study.
Materials and Methods. The total period of observation accounts 33 years from 1937 to 1969. This entire observation period (1937 - 1969) of different stages of eutrophication is divided into three sub-periods according to the strength of eutrophication:
• From 1937 to 1957, when Sevan was an oligotrophic reservoir;
• Transitional period between 1958 and 1961;
• From 1962 to 1969, when eutrophication or mass development of cyanobacteria appeared (since 1964).
In the following we consider only first and second sub-periods, which are the periods distinguished by:
1. The fact that during the first period a relatively stable state of the aquatic ecosystem remains, but the second one immediately precedes the violation of stability accompanied with the mass development of cyanobacteria;
2. The sensitivity of zooplankton to such a weak external influence as magnetic storms.
Hereinafter these periods are referred to as "first "and "second", and we use the data on zooplankton number.
Zooplankton community in our study is presented by 9 species. A set of 9 species were divided on the basis of nutrition character. These groups are following:
• Daphniidae. Presented by the only species Daphnia longispina sevanica.
• Diaptomidae. Includes species Acanthodiaptomus denticornis, Arctodiaptomus bacillifer, Arctodiaptomus spinosus fadeevi.
• Copepoda. Presented by the only species Cyclops strennus sevani.
• Rotatoria Includes species Keratella quadrata, Filinia longiseta, Pedalia mira, Synchaeta pentinata.
For groups consisting of more that one species their number were calculated as average (mean) value of all included species.
For using the DMDS model and analyzing the results of simulation we introduced the concept of "efficiency" for the components (i.e. the groups of zooplankton) in the system. It is assumed that efficiency is measured in conventional units taking the values 1, 2, 3. According to the model described above the number of levels, which components' values can take is to be K = 3. These values can be interpreted as low, median and high levels of efficiency correspondingly.
In trajectories built for the system at each moment of time the overall efficiency as a sum of individual efficiencies for all groups and Shannon diversity index were calculated.
Shannon diversity index was calculated by formula
k
H = -£ Pi log2 Pi,
i=1
where p, — the ratio of i-th component efficiency to overall efficiency, k — numbers of groups, i. e. 4 for our case.
Results. On the base of above mentioned literature data [] on the dynamics of zooplankton's number the idealized trajectories for two sub-periods representing behavior of identified dynamical system were built. Numerical values of the trajectories, i. e. a series of system's states at sequential moments, are shown in Tables 1, 2. As mentioned, we additionally calculated for each moment the overall efficiency as a sum of individual efficiencies and Shannon diversity index. Comparison of Tables 1 and 2 shows that there is a systemic effect for idealized trajectories built for both the first and second periods, which consists in observed combinations of maximum values for overall efficiency and Shannon diversity index (steps 4-5 for the first period and 6-7 for the second). The
differences between the periods regarding the manifestation of this systemic effect are insignificant and, to our opinion, this fact restricts the possibilities for use of this system effect for determination of different stages of eutrophication. As usual, we mean the stages that create or don't create threats for biosafety related to mass development of toxic cyanobacteria.
In reaching this conclusion the first stage of the analysis of the idealized trajectories of the system, we have not specified the role of the aforementioned groups of zooplankton in the formation of high or low values of Shannon diversity index and evenness - index which, according to the paradigm of using his concepts, in many cases determines the stability of ecological systems.
Meanwhile, obtained using DMDS results provide such specificity, which was done at later stages of the analysis of the form of the trajectories of the system. In the first period, the combination of the maximum values of total efficiency and Shannon diversity index and evenness (the fourth and fifth steps) does not coincide with the maximum values Daphniidae.
In the second period, such a coincidence is observed on the sixth and seventh steps. This suggests the presence of the second period of stable and highly productive state of aquatic ecosystem, characterized by two important opportunities for mass development of cyanobacteria aspects.
On the one hand, this saturation of water waste products of zooplankton, which are used as nutrients by all groups of phytoplankton, which creates conditions for its outbreak biomass. On the other hand - grazing Daphniidae, which got great development of phytoplankton organisms that are competitors of cyanobacteria. Pose a threat to drinking water consumption Biosafety species of cyanobacteria do not eat away the live form.
Daphniidae and other planktonic organisms, filter feeders. In the simulated within the present work the zooplankton community of Lake Sevan only Daphniidae are filter feeders power in nature, which in the process of supply free water from seston -suspension of organic matter. As part of this suspension are present: dead organic matter (detritus), processing this dead matter decomposers bacteria and microalgae, constituting as photosynthetic organisms, cyanobacteria competition.
Cyclopidae occupy a niche in the zooplankton community of predators, limiting biomass and abundance of other zooplankton groups (including not considered in the literature data we use Inhuzoria). When comparing the data in Tables 1 and 2 are not found significant in terms of the differences between the two study periods in the dynamics Cyclopidae and the nature of its connection with the dynamics of other groups, as well as the dynamics of the total efficiency of all groups and Shannon diversity index and evenness.
Diaptomidae serve as food for larger, compared with seston particles of dead organic matter. During power Diaptomidae saturate water with nutrients, but not free from micro-algae, phytoplankton - competitors cyanobacteria. When compared in Tables 1 and 2 the trajectories of the system can conclude Diaptomidae greater efficiency in the second period. This should cause a corresponding hydrochemical aspect of eutrophication greater saturation of water with nutrients photosynthetic organisms. Which will result in increased development of all groups of phytoplankton - and cyanobacteria and their competitors.
Rotatoria are by the nature of his power and sedimenter like Daphniidae free water from the seston. But the most important difference in their power play bacterial decomposers, so Rotatoria role in the release of water from the competition cyanobacteria may not be significant. When compared in Tables 1 and 2 the trajectories of the system can conclude Rotatoria greater efficiency in the first period. This may cause a decrease in the concentration of bacteria in the water-reducers, which, as zooplankton organisms, nutrients water saturated photosynthetic organisms. Which will result in a reduction of all groups of phytoplankton - and cyanobacteria and their competitors. Less
the same development of all groups of phytoplankton in the first period, compared to the second corresponds to a lesser extent in the first period of development processes anthropogenic eutrophication of Lake Sevan.
It follows that to understand the implications for ecosystem stability differences between the first and second periods should be considered role Daphniidae. We draw attention to the trajectories of the system built for both periods, a previously upomyatye areas characterized by maximum values of total coincidence promises more effective all groups of zooplankton and Shannon index of diversity and evenness. Index, high values which in many cases demonstrate the stability of the system. On these sites you can expect consistently high emissions into water waste products of zooplankton, which are nutrients for all groups of phytoplankton. As already mentioned above in the second period in this area there is also a pronounced maximum Daphniidae. Recall that this group is capable of zooplankton in the course of its power to remove from the water microalgae, cyanobacteria are competitors.
Therefore, a high number of Daphniidae is an additional factor in phytoplankton dominance of cyanobacteria.
In the first period, a state in which both said present aspect offline So we can see that using DMDS approach with a concrete character of the structure of zooplankton reflected Shannon index of diversity and evenness, reveals aspects of the system, creating the possibility of mass development of cyanobacteria.
TABLE 1. IDEALIZED TRAJECTORY OF THE SYSTEM, BUILT FOR THE ZOOPLANKTON
COMMUNITY OF THE FIRST PERIOD OF EUTROPHICATION OF LAKE SEVAN; ROWS - THE COMPONENT VALUES IN POINTS, COLUMNS - CONDITIONAL TIME STEPS
Daphnidae, 1 1 1 2 2 2 3 3 3 2
Oyclopidae 1 1 1 2 3 3 3 3 3 2
Rotatoria 1 2 3 3 3 3 3 2 1 1
Diaptomidae 1 1 1 2 2 1 1 1 1 1
Oumulative effect 4 5 6 9 10 9 10 9 8 6
Shennovsky index of diversity and 2,0 1,9 1,7 1,9 1,9 1,8 1,8 1,8 1,8 1,9
evenness 2 9 7 7 9 9 9 1
Non-conventional time steps 1 2 3 4 5 6 7 8 9 10
TABLE 2. IDEALIZED TRAJECTORY OF THE SYSTEM, BUILT FOR THE ZOOPLANKTON
COMMUNITY OF THE SECOND PERIOD OF EUTROPHICATION OF LAKE SEVAN; ROWS - THE COMPONENT VALUES IN POINTS, COLUMNS - CONDITIONAL TIME STEPS
Daphniidae, 1 1 1 1 1 2 2 1 1 1
Cyclopidae 1 2 3 3 3 3 3 2 1 1
Rotatoria 1 1 1 1 1 2 3 3 3 2
Diaptomidae 1 1 1 2 3 3 3 3 3 2
Oumulative effect 4 5 6 7 8 10 11 9 8 6
Shennovsky index of diversity and 2,0 1,92 1,79 1,84 1,81 1,97 1,98 1,89 1,81 1,91
evenness
Non-conventional time steps 1 2 3 4 5 6 7 8 9 10
Discussion.
As can be seen from these results, the use of DMDS allows you to make a formalized description of ecological situations using Shannon index of diversity and evenness additional information, in part similar to that obtained by using indicator organisms, but at the same time giving more opportunities for meaningful interpretation by analyzing the impact on the stability of the system structure feedbacks in it. In the case studies we are talking about a very important from the point of view of the violation of human ecology of the aquatic ecosystem stability with a mass of toxic cyanobacteria, which poses a threat to biosecurity and other types of drinking water consumption.
Both in terms of theoretical ecology, and based on the practical problems of human ecology, are of some interest shown in this paper, new approaches to the diagnosis and prediction of creating the above threats Biosafety unstable states of aquatic ecosystems. We are talking about an approach using a new class of mathematical models - DmDs.
In conclusion, we can conclude about the prospects of this application DMDS both in terms of theoretical ecology and solutions for safety-related water consumption applied problems of human ecology.
Can recommend the use of presented results for use in information systems to support decision-making in the field of prevention of threats biosafety arising from violations of the stability of aquatic ecosystems
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