The characteristic problem of modern time that appears before researchers in the various areas of knowledge is the informative supersaturating that is largely caused by the systematic use of internettechnologies. Which mean that modern researcher, regardless of object of his researches, that studies the di cult system, runs into the enormous volumes of data, that require rapid treatment, analysis and practical conclusions.
Introduction The vast majority of data that are accessible via the Internet do not have the formalised structure, contain errors or are the results of incomplete observations. This means that analysis of large data arrays, their organisation, proceeding in the skipped information and construction of the model, is an intricate scienti c and technical problem. It is well-known that the problem to expose terms of stability in at the natural and arti cial systems, which are needed to provide existence and development of these systems is a very important problem. Unfortunately, we often do not have any formal theory when we study complex systems, and we are forced to build qualitative dynamic models based on available observational data and then to determine terms of system stability.
Fortunately, today there is a lot of datasets for the di erent areas of science (biology, medicine, ecology and other) and these datasets are accessible to the researchers. For example, data from International Council for the Exploration of the Sea (ICES) [2] that were used in previous researches conducted by Department of Theoretical and Applied Computer Science at V.N. Karazin Kharkiv National University were sent to the study of the indicated task.
The problem of homoeostasis (aspiring of the system is to keeping balance) and stability is closely constrained with the problem of dynamic rmness of this system. Study of this problem is closely constrained with the research of relations that determine the dynamics of system states that is described by the complexes of system parameters and their changes (transitions) in dependence on interrelations between these system parameters.
Despite of the di erent ways to nd out the connections between parameters of the system in biology and ecology a general method o ers based on the binary relations of next types : (+; +), ( ; ), ( ; +), ( ; 0), (+; 0), (0; 0). Such description suits to many systems. For the systems with many parameters this set describes all pair wise relationships between them. This approach gives to us an opportunity to show the structure of relations in the obvious form of intercommunications between the parameters of the natural system.
It is important to understand that not always is possibility to nd the marked dependences by means of statistical methods. For example, a cross-correlation analysis is used for the estimation of connections between two or more variables, but covers only statistical dependences and does not give to description of case- consequence relations.
A perspective method for description of the system is descriptive modelling of system dynamics. It is a convenient method for the description of the di cult systems, that consists from solving two important tasks.
The rst task is the construction of descriptive model of dynamics of the system. A construction gives an empiric idea about the state of the system transitions. It`s only a description that based on the discrete eventual scale of the states of the system, but not exact presentation of the system. For example: the degenerate state, low-spirited state, normal state, state higher norms, state of complete satiation. A scale gives us an opportunity to describe the value of parameters of the system and consider time-histories of vector of the states of the system. A solution of this problem is useful and practical, because a descriptive model does not need high-cube of data and time for the construction and gives an opportunity to lay down an idea about tendencies state of the system transition, to check and cast aside inadequate hypotheses.
The second task is a problem of authentication, or, in other words, task of synthesis of the system. Based on the conclusions that we got at the rst step we can assume there is a mechanism of state of the system transition to that, to try to build him self-reactants description, to pick up to basis parameters, to expect the error of our suppositions.
Modelling Framework The study is based on the modelling framework proposed in [4]. 2.1
States, System Trajectories, and Dynamics
Practically it looks like the following. Consider that the biological or
economic system can be described through N components like A1; A2; : : : ; AN .
These components can be of di erent nature. For example, they can be
number of animals or amount of biomass of di erent species, etc. Discrete
values are assumed for components, such as 1; 2; : : : ; K. Here 1 is the least
meaning, and K is the greatest. Thus, the value 1 means the minimum
amount of a component and the value K means the maximum amount.
The value of each component is observed and measured at discrete instant
of time t = 0; 1; : : :. Thus, the value of the component Ai at the instants
of time t = 0; 1; : : : is numbered as Ai(0); Ai(
The trajectory of the system is described by an in nite-right matrix
as
0 A1(0) A1(
AN (0) AN (
The system has only nite number of states, namely, KN . In this case, there exists > 0 such that Ai(t + ) = Ai(t) for all t t0 where t0 is some positive integer. This is called a period of the trajectory.
We can extract a minor from the matrix (
AN (t) AN (t + 1) AN (t + 2) : : : AN (t + 1) 1 1) C
C C
A 1) presents full description of the behaviour of the system.
In this context, we say that dynamics is the complex of f1; 2; : : : ;
Kg(
Ai(s + 1) = fi(s; A1(s); : : : ; AN (s)) for t
s < t +
1:
(
Ai(s + 1) = fi(A1(s); : : : ; AN (s)) for t
s < t +
1:
(
Relationships between Components In physics, the dynamics of a system is usually determined by the sum of contributions corresponding to the dynamics of pairwise interactions of system components. Now, we give some generalisation of this idea for our study.
Introduce the concept of relationships between components. Let = f=; 0; +g then the relationship between components Ai and Aj is determined as a member from the set and denoted as (Ai; Aj) = (!1; !2) 2 .
If (Ai; Aj) = (!1; !2) then the mean of the relation is the following 1. if !1 = then the larger value of the component Aj is, the lower value of the component Ai would be; 2. if !1 = 0 then the value of the component Aj would not in uence on value of the component Ai; 3. if !1 = + then the lower value of the component Aj is, the larger value of the component Ai would be.
We require also antisymmetric of the relationship , i.e. we claim the satisfaction of the condition
(Ai; Aj) = (!1; !2) implies (Aj; Ai) = (!2; !1): All the combinations (!1; !2) correspond to relationships (interspeci c interactions) of neutralism, amensalism, predation, commensalism, and mutualism widely used in ecology and biology [3].
We can associate each matrix with the N N -matrix S consisting of elements equals ether 1, or 0, or +1 in the following manner 8 1 if (Ai; Aj) = (!1; !2) and !1 = Sij = < 0 if (Ai; Aj) = (!1; !2) and !1 = 0 : +1 if (Ai; Aj) = (!1; !2) and !1 = + Below we denote by SN the class of such N N -matrices.
Now we describe some general approach to determine system dynamics based on a matrix from the class SN .
To do this let us consider some positive real-valued function ij of two arguments from f1; 2; : : : ; Kg that speci es the in uence degree of component Aj onto component Ai in the corresponding states. We require only that each function ij does not decrease in the second argument. Also, let us consider some folding function : RN ! R and determine fi(a1; : : : ; aN ) = 8 inc ai if (Si1 <
ai if j (Si1 : dec ai if (Si1 i1(ai; a1); : : : ; SiN i1(ai; a1); : : : ; SiN i1(ai; a1); : : : ; SiN iN (ai; aN )) > iN (ai; aN ))j iN (ai; aN )) < where inc x = min(x + 1; K) and dec x = max(x
Let us set the following problem. 1; 0) for 1 x
K.
Problem 1. Let ij , > 0, and be given then for each matrix S of class SN and a dataset to nd an operator T = (f1; : : : ; fN )T such that the trajectory obtained with using the operator T is well coincide with the given dataset. 2.3
Weight Function Approach This approach assumes a special form of functions ij and . Namely, let N ij (ai; aj ) = i(jk) where k = jai aj j and (x1; : : : ; xN ) = sP=1 xs. In this case, Eq (5) is rewritten in the following form fi(a1; : : : ; aN ) = 8 N >> inc ai if P Sis >>>>< s=N1
ai if P Sis >>>> Ns=1 >>: dec ai if P Sis s=1 (jai asj) > is (jai asj) < is (jai asj) < is (5) (6) It is easily seen that Eq (6) determine uniquely the transition operator T = (f1; : : : ; fN )T by the tuple of parameters h ; i(jk) j 1 i; j
N; 1 k
Ki: In this context, it is evident that di erent parameter tuples can determine the same transition operator. Therefore, we have the following problem.
i; j N; 1 k Ki and
Ki determine are the corresponding Decidability of this problem is ensured by the theory of linear inequalities developed by S. Chernikov [1] But the principal task is to develop e ective computational method to solve the problem. 3 In the paper, the modelling framework has been established. This framework is appropriate for mathematical modelling of descriptive dynamics of complex natural and arti cial systems. We have demonstrated that all stages of building mathematical models are too complicated, but the most di cult task among them is the model parameter estimation for identifying the structure of the studied system.
This paper starts the development of the class of mathematical models, which will be useful in solving di erent problems, especially some environmental and healthcare problems.
At this paper, the few problems that need to be solved have been drawn. Firstly, it is a choice of operators that are adequate to our rules of transitions. And secondly, it is a problem of data visualisation.
Both of these problems are closely related to the problem of large data, and we hope that our study will contribute to both modelling of complex systems and processing of big data areas of knowledge.