Usage of the apparatus of ordinary differential equations in modelling of economic and environmental systems
The ordinary differential equations techniques applying to investigate the economical and ecological systems has been considered in presented article. The interconnected economical complexes development for the countries with the different economical potential has been simulated. The population econ...
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Інститут телекомунікацій і глобального інформаційного простору НАН України
2019
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irk-123456789-1684932020-05-04T01:26:18Z Usage of the apparatus of ordinary differential equations in modelling of economic and environmental systems Oliynyk, A.P. Grygorchuk, G.V. Nezamay, B.S. Feshanych, L.I. Математичні та інформаційні моделі в економіці The ordinary differential equations techniques applying to investigate the economical and ecological systems has been considered in presented article. The interconnected economical complexes development for the countries with the different economical potential has been simulated. The population economical activity influence on the environment pollution and the state of region’s flora has been investigated. The economical efficiency of the new technical diagnostics implementation has been studied. The methods of presented models realization has been presented and investigated, the results of tested calculations have been presented and one’s analysis has been given. The directions of future investigations have been determined. У статті представлені звичайні методи диференціальних рівнянь, що застосовуються для дослідження економічної та екологічної систем. Проведено моделювання взаємозв'язку розвитку економічних комплексів для країн з різним економічним потенціалом. Досліджено вплив економічної активності населення на забруднення навколишнього середовища та стан флори регіону. Вивчено економічну ефективність впровадження нової технічної діагностики. Представлені та досліджені методи реалізації представлених моделей, представлені результати перевірених розрахунків та дано аналіз. Визначено напрямки майбутніх досліджень. 2019 Article Usage of the apparatus of ordinary differential equations in modelling of economic and environmental systems / A.P. Oliynyk, G.V. Grygorchuk , B.S. Nezamay, L.I. Feshanych // Математичне моделювання в економіці. — 2019. — № 3(16). — С. 57-66. — Бібліогр.: 13 назв. — англ. 2409-8876 DOI: 10.35350/2409-8876-2019-16-3-57-66 http://dspace.nbuv.gov.ua/handle/123456789/168493 519.866 en Математичне моделювання в економіці Інститут телекомунікацій і глобального інформаційного простору НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Математичні та інформаційні моделі в економіці Математичні та інформаційні моделі в економіці |
| spellingShingle |
Математичні та інформаційні моделі в економіці Математичні та інформаційні моделі в економіці Oliynyk, A.P. Grygorchuk, G.V. Nezamay, B.S. Feshanych, L.I. Usage of the apparatus of ordinary differential equations in modelling of economic and environmental systems Математичне моделювання в економіці |
| description |
The ordinary differential equations techniques applying to investigate the economical and ecological systems has been considered in presented article. The interconnected economical complexes development for the countries with the different economical potential has been simulated. The population economical activity influence on the environment pollution and the state of region’s flora has been investigated. The economical efficiency of the new technical diagnostics implementation has been studied. The methods of presented models realization has been presented and investigated, the results of tested calculations have been presented and one’s analysis has been given. The directions of future investigations have been determined. |
| format |
Article |
| author |
Oliynyk, A.P. Grygorchuk, G.V. Nezamay, B.S. Feshanych, L.I. |
| author_facet |
Oliynyk, A.P. Grygorchuk, G.V. Nezamay, B.S. Feshanych, L.I. |
| author_sort |
Oliynyk, A.P. |
| title |
Usage of the apparatus of ordinary differential equations in modelling of economic and environmental systems |
| title_short |
Usage of the apparatus of ordinary differential equations in modelling of economic and environmental systems |
| title_full |
Usage of the apparatus of ordinary differential equations in modelling of economic and environmental systems |
| title_fullStr |
Usage of the apparatus of ordinary differential equations in modelling of economic and environmental systems |
| title_full_unstemmed |
Usage of the apparatus of ordinary differential equations in modelling of economic and environmental systems |
| title_sort |
usage of the apparatus of ordinary differential equations in modelling of economic and environmental systems |
| publisher |
Інститут телекомунікацій і глобального інформаційного простору НАН України |
| publishDate |
2019 |
| topic_facet |
Математичні та інформаційні моделі в економіці |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/168493 |
| citation_txt |
Usage of the apparatus of ordinary differential equations in modelling of economic and environmental systems / A.P. Oliynyk, G.V. Grygorchuk , B.S. Nezamay, L.I. Feshanych // Математичне моделювання в економіці. — 2019. — № 3(16). — С. 57-66. — Бібліогр.: 13 назв. — англ. |
| series |
Математичне моделювання в економіці |
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2025-07-15T03:16:00Z |
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2025-07-15T03:16:00Z |
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| fulltext |
~ 57 ~
Математичне моделювання в економіці, №3, 2019. ISSN 2409-8876
UDC 519.866 https://orcid.org/0000–0003–1031–7207
https://orcid.org/0000–0003–1674–9828
https://orcid.org/0000–0003–0402–6040
https://orcid.org/0000–0002–5156–2199
A. OLIYNYK, G. GRYGORCHUK, B. NEZAMAY, L. FESHANYCH
USAGE OF THE APPARATUS OF ORDINARY DIFFERENTIAL
EQUATIONS IN MODELLING OF ECONOMIC AND ENVIRONMENTAL
SYSTEMS
Abstract. The ordinary differential equations techniques applying to
investigate the economical and ecological systems has been considered in
presented article. The interconnected economical complexes development
for the countries with the different economical potential has been
simulated. The population economical activity influence on the environment
pollution and the state of region’s flora has been investigated. The
economical efficiency of the new technical diagnostics implementation has
been studied. The methods of presented models realization has been
presented and investigated, the results of tested calculations have been
presented and one’s analysis has been given. The directions of future
investigations have been determined.
Keywords: differential equation, mathematical modeling method, modeling,
diagnostics.
DOI: 10.35350/2409-8876-2019-16-3-57-66
Introduction
The modeling of economic and environmental systems is performed with the use of
linear and nonlinear systems of ordinary differential equations apparatus. As a
basic, the development of which can be interpreted below, would be a model
"predator – victim", created in 1925 by Alfred Lotka and Vito Volterra [2]. The
basic equation system of this model is written as:
.
)()()(
)()()(
22
11
+−=
−=
tytxgtyk
dt
dy
tytxgtxk
dt
dx (1)
There )(tx – the count of victim’s population, )(ty – the count of predator’s
population, 2,1,, =igk ii – the model’s coefficients, detailed content of which is
given in [2]. Models of economic systems of different nature are considered in
[1, 4–9, 11–13]. The offered research touches a construction and research of three
models of economic and economically – ecological systems, and also them
practical realization and researching.
A.P. Oliynyk, G.V. Grygorchuk, B.S. Nezamay, L.I. Feshanych, 2019
https://teacode.com/online/udc/51/519.866.html
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Математичне моделювання в економіці, №3, 2019. ISSN 2409-8876
1. Model problems formulation
In predicting the development of interconnected economies, the question arises
whether economies with relatively low levels of development may not suffer
significant economic losses at a time when the world's leading economies are
suffering losses as a result of the economic crisis. The methods of mathematical
design are used for research of the indicated question with the use of the systems as
a "predator – victim", that allows to build mathematical models and define their
descriptions that would allow to answer theset questions. The task is reduced to
solution ' Liabilities system of differential equations of the form:
( )
( )
( )
++−=
+−−=
+−−=
214311131039
3
3182172625
2
3142131211
1
xxAxxAxAxA
dt
dx
xxAxxAxAxA
dt
dx
xxAxxAxAxA
dt
dx
, (2)
there 1x and 2x – economically strong countries, 3x a country with a low level of
economy with appropriate initial conditions ( ) 101 0 xx = ; ( ) 202 0 xx = ; ( ) 303 0 xx = .
The coefficients iA could be present as a time function ( )tAA ii = .
Another model is related to a system described by three differential equations
for functions: x(t) – population in the region; y(t) – the level of pollution and other
non–harmful effects on the environment caused by the economic activity of the
population; z(t) – the level of flora of the region (trees, agricultural products,
forests, gardens, etc.), however, the equation system looks like:
.
+−=
−=
+−=
FzGyHx
dt
dz
EzDx
dt
dy
CzByAx
dt
dx
(3)
Initial conditions must be specified for the correct formulation of the modeling
task:
=
=
=
0
0
0
)0(
)0(
)0(
ZZ
YY
XX
. (4)
The third model describes a situation for which the functions x(t), y(t); z(t) are
introduced with the following meaning: x(t) – costs for implementation of new
technical diagnostics and control standards; y(t) – costs for elimination of
emergencies consequences; z(t) is the efficiency of the studied industrial system’s
element. When a mathematical model is constructed, a differential equation system
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that describes how to modify the corresponding variables per unit of time in
assuming the nature of the relationship between the quantities is recorded. As a
result, the following system of ordinary differential equations that binds the
variables х(t); y(t); z(t) is obtained:
( )
( ) ( )
−=
+−+−=
+−−=
yKxK
dt
dz
zKyByKxAxK
dt
dy
zKyKxAxK
dt
dx
87
654
321 , (5)
with appropriate initial conditions.
Systems of type (2) – (5) are a certain extension of the known predator–prey
model. The algorithms for finding the coefficients of systems (2) and (3) by the
method of expert estimation are proposed, and in the modeling of system (2)
additional conditions for its coefficients in terms of obtaining asymptotically stable
solutions are established.. Introducing nonlinear components into systems (2), (3),
and (5) allows us to obtain solutions that more accurately reflect the essence of the
phenomena and processes being modeled. In the implementation of the mentioned
models, the following approach was used: in the first step, all the mentioned
models were selected as linear, suitable calculations and analysis of the obtained
results were carried out.. If there were doubts about the correspondence of these
results to the characteristics of real systems, then new nonlinear terms describing
the level of interaction of the relevant factors were introduced into the respective
systems. If the qualitative behavior of the solutions satisfied the researcher, then
methods of practical evaluation of the coefficients of the systems based on the
results of their statistical studies, real data on the characteristics of their functioning
were created. When model solutions, that meet certain economic requirements and
environmental standards, are obtained, recommendations to optimize the systems
under consideration by the necessary criteria, which are responsible for the stable
operation of the respective systems with the fulfillment of their functions. All
models are brought to numerical realization in the form of software complexes by
Runge-Kutta methods [3], it allows to carry out a wide class of calculations in
order to estimate the dynamics of the process development depending on the
suitable coefficients of the model.
Special kind of models is the simulation of an advertising campaign for goods
and services is an important element in ensuring that they hit the market. Often,
advertising is carried out haphazardly or using certain empirical methods, which
can have the opposite effect when product advertising begins to act as a counter–
advertisement. At the same time, mathematical methods, in particular, the
apparatus of ordinary differential equations, to build a model of an advertising
campaign are promising for studying the features of the advertising process.
The advertising model is based on the following assumptions. It is believed
that ( )
dt
tdN – the rate of change in the number of consumers who know about the
product and are ready to buy it ( ( )tN – the number of informed customers),
function ( ) ( )( )tNNt −⋅ 01α – characterizes the intensity of the advertising
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Математичне моделювання в економіці, №3, 2019. ISSN 2409-8876
company, ( ) 01 >tα characterizes the cost of advertising, 0N – the total number of
potential buyers. It is also believed that people who know about the product, in one
way or another, disseminate information about the product to customers who do
not know about it (potential customers). This contribution is characterized by an
addition ( ) ( ) ( )( )tNNtNt −⋅⋅ 02α . The value ( )t2α characterizes the degree of
communication of clients with potential clients.
Based on the made assumptions made, the equation [1] is obtained:
( ) ( ) ( ) ( )[ ] ( )( )tNNtNtt
dt
tdN
−+= 021 αα (6)
with initial condition ( ) 10 NN = , 01 NN < . If ( ) ( )tt 21 αα >> , the next equation can
be received (matches classic advertising campaigns):
( ) ( ) ( )tNt
dt
tdN α= . (7)
Otherwise, it is possible to receive the equation that describes the so-called
"network marketing":
( ) ( ) ( )( )
( ) .
,
2
0
dttd
tNNtN
d
tdN
ατ
τ
=
−= (8)
Obviously, equations (7) and (8) are squared:
( ) tN
tN
e
eNtN
0
0
1
0
+
= , (9)
In this case, the number of informed consumers if ∞→t remains constant:
0
0
0
0
1
limlim N
e
eNN tN
tN
tt
=
+
=
∞→∞→
(10)
and is equal to the number of potential customers.
When the value ( ) ( ) ( )[ ]tNtt 21 αα + becomes negative ( ( ) 02 <tα – negative
evaluation of consumers of the quality of the goods) – manufacturers should further
analyze and evaluate the possibilities of direct advertising in the promotion of
products in the market.
Depending on the values ( )t1α , ( )t2α and ( )tN , product promotion activities
can be aimed at improving the results of both direct advertising ( ( )t1α ) and
promoting indirect advertising ( ( )t2α ).
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Математичне моделювання в економіці, №3, 2019. ISSN 2409-8876
If 0NN << (a little-known product), and as a result, ( ) ( )tNt 12 αα <<⋅ , then
the equation (6) comes to view
( ) ( ) 01 Nt
dt
tdN α= and has the solution
( )∫=
t
dttNN
0
10 α . (11)
As for (8), the solution (6) with conditions ( ) 11 αα =t і ( ) 22 αα =t
( )
( )
( )120
120
1
2
1
00
αα
αα
α
α
+
+
+
⋅−
= N
N
e
NeN
tN (12)
And, based on (10):
0lim NN
t
=
∞→
. (13)
Thus, the solution (1) is stable, which substantiates the correctness of the
proposed models.
In the general case, integration (6) is performed using numerical methods [3] –
for example, fourth-order Runge-Kutta methods. The order of precision of a
method is made on the basis of solving model equations.
Based on the proposed model, the following tasks are solved:
– the task of estimating the profit of an advertising company,
– the dependence of the number of potential clients on the methods and
intensity of the advertising campaign is investigated.
Methods for determining or selecting ( )t1α , ( )t2α based on data on a planned
(or ongoing) advertising campaign, have been developed to evaluate its
effectiveness. The areas of further research may be related to the processing of
statistics on various advertising campaigns, their effectiveness, duration over time
in order to restore the analytical structure of the features introduced, and the
possible correction of the model (1).
2. Analysis of the obtained results
In the implementation of model (2) the values of coefficients at which periodic
crisis manifestations of countries with higher levels of economic development
(series 1, 2 in Figure 1) do not affect the level of economy of a country with
relatively weaker economic indicators (row 3) were established.
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Математичне моделювання в економіці, №3, 2019. ISSN 2409-8876
Figure 1 – Countries economics indicators dynamics
Implementation of the model (3), (4) allows you to set these coefficients, in
which the stability of solutions with the desired asymptotic values (Figure 2) is
provided.
Figure 2 – The change of economic and ecological system indicators dynamics
The implementation of model (5) allows us to set the values of coefficients in
which the stability of the solutions with the desired asymptotic values of the
indicators and the dynamics of their change over time is ensured (figure 3):
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Математичне моделювання в економіці, №3, 2019. ISSN 2409-8876
Figure 3 – The change of cost indicators for standards of technical diagnostics
implementation, for elimination of emergencies consequences and efficiency of gas
transmission system’s work
The results of advertising campaign intensity model calculations are presented
in Figure 4. The results of the calculations can determine the time at which the
advertising campaign can be rolled – further investment has no proper effect.
Advertising campaign intesity
Figure 4 – The results of advertising campaign intensity model calculations
The conducted researches testify high efficiency of economic systems
mathematical modeling method for their study, description and optimization. In
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Математичне моделювання в економіці, №3, 2019. ISSN 2409-8876
fact, the modeling problem is divided into two stages – at the first, the relationships
between its elements are studied in order to obtain qualitative indicators that
adequately reflect its behavior, and the second, in fact, the inverse problem of
selecting of the model coefficients that allow to determine the quantitative indices
of the simulated systems is solved. The areas of further research are related to the
adaptation of the developed models to real economic systems.
In particular, for the model of interconnected economies it is necessary to
establish coefficients that quantitatively characterize the level of interconnection
between each pair of economic complexes, and in the first stage of modeling these
coefficients are normalized, their numerical values are chosen at the interval (0; 1)
and are determined by the method of expert estimates, however in the analysis of
real economic systems, it is necessary to switch to the dimensional values of these
coefficients, which requires the interconnected efforts of specialists in the fields of
economics and applied mathematics.
For the second model, which describes the functions x(t) – population in the
region; y(t) – the level of pollution and other non–harmful effects on the
environment caused by the economic activity of the population; z(t) – the level of
flora of the region (trees, agricultural products, forests, gardens, etc.) to determine
the model coefficients it is necessary to take into account the peculiarities of each
of the studied regions – it should be noted that the values of the coefficients will
depend on the characteristics of the studied region – the coefficients for the
industrial regions will differ from the coefficients determined for the Carpathian or
Bukovina region. It is also advisable to use the method of expert estimation for
model calculations, having in mind again the scaling of coefficients by the interval
(0; 1).
To practical realization the third model with investigated functions x(t) – costs
for implementation of new technical diagnostics and control standards; y(t) – costs
for elimination of emergencies consequences; z(t) is the efficiency of the studied
industrial system’s element in determining the model values of the system
coefficients by the method of expert assessments, an important point is the
selection of experts who should be specialists in the operation and evaluation of the
technical condition of the systems under study and on economic issues to find the
best ways to distribute investments. At the same time, it is important to ensure the
objectivity of expert assessments – the opinion of experts on only one issue should
not prevail.
For the practical implementation of the proposed model of the advertising
company it is necessary to take into account the fact that only in this model its
coefficients are functions of time. This necessitates the need for a wide-ranging
survey of advertising campaigns for different types of advertising items – although
widely used consumer electronics campaigns, the number of such campaigns
provides a great deal of material for determining the appropriate features – model
equation coefficients, based on analysis of relevant statistics.
The peculiarity of each of the proposed models is the fact that they are most
effective in predicting the behavior of the simulated systems, since the construction
of sufficiently accurate forecasts allows you to solve the following problems:
– forecasting the development of interdependent economies based on the study
of economic trends characteristic of previous periods;
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Математичне моделювання в економіці, №3, 2019. ISSN 2409-8876
– forecasting of economic and ecological characteristics of regions taking into
account their peculiarities:
– predicting the technical and economic efficiency of implementing new
diagnostic systems and the effectiveness of advertising campaigns.
Conclusions
1. The technique of ordinary differential equations can be successfully applied to
the simulation of the interconnected economies development, to estimation the
change of economic and ecological system indicators dynamics and the change of
cost indicators for standards of technical diagnostics implementation, for
elimination of emergencies consequences and efficiency of gas transmission
system’s work and to predictions the advertising campaign efficiency.
2. All this models can be realized using the simple linear and quadratic type
relationships, which, however, allow to receive the numerical results that are
sufficiently accurate in terms of practical needs.
3. Construction and implementation of models 1 – 3 is carried out in two
stages – the first of them is a simulation of the desired behavior of the system
(prediction) b and in the second – the model is corrected by developing a
methodology for determining its coefficients (correction).
4. Model 4 uses only one differential equation, which in most cases can be
solved analytically. The obtained solutions are stable, which is a confirmation of
the correctness of the proposed models.
5. The above four models do not exhaust the entire class of problems of
modeling environmental, economic and other types of systems – for the scope of
this work, the results of the authors' work on modeling systems in medicine, as
well as models of dimensions above three are presented. However, these models
have been successfully used in the approaches presented in the presented work.
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Стаття надійшла до редакції 24.07.2019.
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