=Paper=
{{Paper
|id=Vol-1851/paper-7
|storemode=property
|title=Descriptive Modelling of System Dynamics at Different Stages
|pdfUrl=https://ceur-ws.org/Vol-1851/paper-7.pdf
|volume=Vol-1851
|authors=Illia Otlev
}}
==Descriptive Modelling of System Dynamics at Different Stages==
https://ceur-ws.org/Vol-1851/paper-7.pdf
Descriptive Modelling of System Dynamics at
Different Stages
Illia Otlev
School of Mathematics and Computer Science,
V.N. Karazin Kharkiv National University
4, Svobody Sqr, Kharkiv, 61022, Ukraine, tomasgrey@mail.ru
Abstract. The characteristic problem of modern time that appears be-
fore researchers in the various areas of knowledge is the informative su-
persaturating that is largely caused by the systematic use of internet-
technologies. Which mean that modern researcher, regardless of object
of his researches, that studies the difficult system, runs into the enormous
volumes of data, that require rapid treatment, analysis and practical con-
clusions.
Keywords: dynamical systems, mathematical modelling, descriptive dy-
namics
1 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 organ-
isation, proceeding in the skipped information and construction of the
model, is an intricate scientific and technical problem. It is well-known
that the problem to expose terms of stability in at the natural and ar-
tificial 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 observa-
tional data and then to determine terms of system stability.
Fortunately, today there is a lot of datasets for the different areas of
science (biology, medicine, ecology and other) and these datasets are ac-
cessible to the researchers. For example, data from International Council
for the Exploration of the Sea (ICES) [2] that were used in previous re-
searches 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.
42
The problem of homoeostasis (aspiring of the system is to keeping
balance) and stability is closely constrained with the problem of dynamic
firmness 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 pa-
rameters.
Despite of the different ways to find out the connections between pa-
rameters of the system in biology and ecology a general method offers
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 be-
tween them. This approach gives to us an opportunity to show the struc-
ture 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 find
the marked dependences by means of statistical methods. For example,
a cross-correlation analysis is used for the estimation of connections be-
tween 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 mod-
elling of system dynamics. It is a convenient method for the description
of the difficult systems, that consists from solving two important tasks.
The first 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 first 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.
43
2 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 eco-
nomic system can be described through N components like A1 , A2 , . . . , AN .
These components can be of different nature. For example, they can be
number of animals or amount of biomass of different 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 (1), . . ..
The trajectory of the system is described by an infinite-right matrix
as
A1 (0) A1 (1) A1 (2) . . .
A2 (0) A2 (1) A2 (2) . . .
(1)
.. .. .. ..
. . . .
AN (0) AN (1) AN (2) . . .
This trajectory includes all states of the system at the moments t =
0, 1, . . .. Hence, the state of the system at the instant of time t is repre-
sented by the vector (A1 (t), A2 (t), . . . , AN (t))T where the sign T is the
sign of the matrix transposition. It is supposed that the system is strictly
determined, and it‘s state at the moment of time t is completely deter-
mined by the state at the moment t − 1.
The system has only finite number of states, namely, K N . In this case,
there exists τ > 0 such that Ai (t + τ ) = Ai (t) for all t ≥ t0 where t0 is
some positive integer. This τ is called a period of the trajectory.
We can extract a minor from the matrix (1) formed by the columns
t (t ≥ t0 ), (t + 1)th , and up to (t + τ − 1)th . The obtained minor
th
A1 (t) A1 (t + 1) A1 (t + 2) . . . A1 (t + τ − 1)
A2 (t) A2 (t + 1) A2 (t + 2) . . . A2 (t + τ − 1)
(2)
.. .. .. .. ..
. . . . .
AN (t) AN (t + 1) AN (t + 2) . . . AN (t + τ − 1)
presents full description of the behaviour of the system.
In this context, we say that dynamics is the complex of {1, 2, . . . , K}-
44
valued mappings fi (s, a1 , . . . , aN ) where i = 1, 2, . . . , N , a1 , . . . , aN belong
to {1, 2, . . . , K}, and s = t, t + 1, . . . , t + τ − 1 such that
Ai (s + 1) = fi (s, A1 (s), . . . , AN (s)) for t ≤ s < t + τ − 1. (3)
We study only the case when all fi in (3) do not depend on s and call this
case the case of stationary dynamics. In this case, the equations Eq (3)
take the form
Ai (s + 1) = fi (A1 (s), . . . , AN (s)) for t ≤ s < t + τ − 1. (4)
2.2 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 Ω =
{=, 0, +} then the relationship between components Ai and Aj is deter-
mined as a member from the set Ω × Ω and denoted as Λ(Ai ; Aj ) =
(ω1 , ω2 ) ∈ Ω × Ω.
If Λ(Ai ; Aj ) = (ω1 , ω2 ) then the mean of the relation is the following
1. if ω1 = − then the larger value of the component Aj is, the lower
value of the component Ai would be;
2. if ω1 = 0 then the value of the component Aj would not influence on
value of the component Ai ;
3. if ω1 = + 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 Λ, i.e. we claim the
satisfaction of the condition
Λ(Ai , Aj ) = (ω1 , ω2 ) implies Λ(Aj , Ai ) = (ω2 , ω1 ).
All the combinations (ω1 , ω2 ) correspond to relationships (interspecific
interactions) of neutralism, amensalism, predation, commensalism, and
mutualism widely used in ecology and biology [3].
We can associate each matrix Λ with the N × N -matrix S consisting
of elements equals ether −1, or 0, or +1 in the following manner
−1 if Λ(Ai , Aj ) = (ω1 , ω2 ) and ω1 = −
Sij = 0 if Λ(Ai , Aj ) = (ω1 , ω2 ) and ω1 = 0
+1 if Λ(Ai , Aj ) = (ω1 , ω2 ) and ω1 = +
45
Below we denote by SN the class of such N × N -matrices.
Now we describe some general approach to determine system dynam-
ics based on a matrix from the class SN .
To do this let us consider some positive real-valued function ψij of
two arguments from {1, 2, . . . , K} that specifies the influence degree of
component Aj onto component Ai in the corresponding states. We require
only that each function ψij does not decrease in the second argument.
Also, let us consider some folding function π : RN → R and determine
fi (a1 , . . . , aN ) =
inc ai if π(Si1 · ψi1 (ai , a1 ), . . . , SiN · ψiN (ai , aN )) > δ
ai if |π(Si1 · ψi1 (ai , a1 ), . . . , SiN · ψiN (ai , aN ))| ≤ δ (5)
dec ai if π(Si1 · ψi1 (ai , a1 ), . . . , SiN · ψiN (ai , aN )) < −δ
where inc x = min(x + 1, K) and dec x = max(x − 1, 0) for 1 ≤ x ≤ K.
Let us set the following problem.
Problem 1. Let ψij , δ > 0, and π be given then for each matrix S of class
SN and a dataset to find 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 ψij and π. Namely, let
N
(k) P
ψij (ai , aj ) = ψij where k = |ai − aj | and π(x1 , . . . , xN ) = xs .
s=1
In this case, Eq (5) is rewritten in the following form
N
(|a −a |)
Sis · ψis i s > δ
P
inc ai if
s=1
N
(|a −a |)
Sis · ψis i s < δ
P
fi (a1 , . . . , aN ) = ai if (6)
s=1
N
(|a −a |)
Sis · ψis i s < −δ
P
dec ai if
s=1
It is easily seen that Eq (6) determine uniquely the transition operator
T = (f1 , . . . , fN )T by the tuple of parameters
(k)
hδ, ψij | 1 ≤ i, j ≤ N, 1 ≤ k ≤ Ki.
In this context, it is evident that different parameter tuples can determine
the same transition operator. Therefore, we have the following problem.
46
(k)
Problem 2. For two tuples hδ 0 , ψij | 1 ≤ i, j ≤ N, 1 ≤ k ≤ Ki and
(k)
hδ 00 , φij | 1 ≤ i, j ≤ N, 1 ≤ k ≤ Ki determine are the corresponding
transition operators equal or no.
Decidability of this problem is ensured by the theory of linear inequalities
developed by S. Chernikov [1] But the principal task is to develop effective
computational method to solve the problem.
3 Conclusion
In the paper, the modelling framework has been established. This frame-
work is appropriate for mathematical modelling of descriptive dynamics
of complex natural and artificial systems. We have demonstrated that
all stages of building mathematical models are too complicated, but the
most difficult 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 mod-
els, which will be useful in solving different 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.
References
1. Chernikov, S.: Linear Inequalities (in Russian). M.: Nauka (1968).
2. ICES Datasets. Dataset Collections. ICES. http://www.ices.dk/marine-data/
dataset-collections/Pages/default.aspx
3. Terentyev, P.: Mathematization of biology (in Russian). Trudy Leningradskogo
obschestva estesvoispytateley. Uspehy Biomentrii. 75(5), 5–8 (1975).
4. Zholtkevych, G., Bespalov, Yu., Nosov, K., Abhishek M.: Discrete Modeling of Dy-
namics of Zooplankton Community at the Different Stages of an Antropogeneous
Eutrophication. Acta Biotheoretica. 61(4), 449–465 (2013).
47