=Paper=
{{Paper
|id=Vol-2516/paper15
|storemode=property
|title=Mathematical Model of Dynamics of Homomorphic Objects
|pdfUrl=https://ceur-ws.org/Vol-2516/paper15.pdf
|volume=Vol-2516
|authors=Olexandr Kuzenkov,Tetiana Serdiuk,Alisa Kuznetsova,Mykola Tryputen,Vitaliy Kuznetsov,Yevheniia Kuznetsova,Maksym Tryputen
|dblpUrl=https://dblp.org/rec/conf/ictes/KuzenkovSKTKKT19
}}
==Mathematical Model of Dynamics of Homomorphic Objects==
https://ceur-ws.org/Vol-2516/paper15.pdf
Mathematical Model of Dynamics
of Homomorphic Objects
Olexandr Kuzenkov1[0000-0002-6378-7993], Tetiana Serdiuk2[0000-0002- 2609-4071],
Alisa Kuznetsova3[0000-0003-4772-683X], Mykola Tryputen4[0000-0003-4523-927X],
Vitaliy Kuznetsov5[0000-0002-8169-4598], Yevheniia Kuznetsova6[0000-0003-2224-8747],
Maksym Tryputen7[0000-0001- 6915-8162]
1Oles Honchar Dnipro National University, 35, D.Yavornitsky Avenue, 4 building of
DNU Dnipro, Ukraine
kuzenkov1986@gmail.com
2Dnipro National University of Railway Transport named after Academician V. Lazaryan
Dnipro, Ukraine
serducheck-t@rambler.ru
3Oles Honchar Dnipro National University, 35, D.Yavornitsky Avenue, 4 building of
DNU Dnipro, Ukraine
karamel75@i.ua
4Dnipro University of Technology, av. Dmytra Yavornytskoho, 19, Dnipro, Ukraine
nikolay.triputen@gmail.com
5National metallurgical academy of Ukraine, Gagarina avenue, 4, Dnipro, Ukraine
wit1975@i.ua
6National metallurgical academy of Ukraine, Gagarina avenue, 4, Dnipro, Ukraine
wit_jane2000@i.ua
7Oles Honchar Dnipro National University, 35, D.Yavornitsky Avenue, 4 building of
DNU Dnipro, Ukraine
triputen2014@i.ua
Abstract. The paper concerns topical problem of mathematical modeling of
dynamics of heterogeneous groups with a logistic function as a basic one. Joint
use of mathematical models of biological systems and computer-based simula-
tion makes it possible to minimize time and save material resources while de-
termining general tendencies of subpopulation progress; and to forecast state of
the system as well as possible consequences of artificial intervention in the en-
vironment. Among other things, it concerns forecasting of genetic abnormali-
ties. The paper proposes a model of dynamics of progress of a population con-
sisting of n subpopulations. The model is represented in the form of differential
equations with transition coefficients within their right sides. The transition co-
efficients mirror the share of species getting from ith subpopulation to jth one.
The proposed system is not Voltairian one since its phase trajectories may cross
coordinate axes. It has been proved that the system of differential equations is
degenerated in the neighbourhood of equilibrium points. Analysis of the system
of differential equations for n=2 has demonstrated a potential for three bifurca-
tions. It has been proved that nine bifurcation types are possible for n=3. Nu-
merical computer-based experiments have shown that the proposed model is
Copyright © 2019 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
stable as for the disturbance of its coefficients, and the obtained characteristics
of the degenerated system are close to real ones.
Keywords: mathematical model, computer-based simulation, differential mod-
el, logistic function, bifurcation characteristics
1 Introduction
Environment is one of the most information-intensive objects. Essentially, it is multi-
component, and experiences constant pressure of human business activities.
Outbreak of deep global ecological changes caused the necessity to analyze their
dynamics, evaluate it, and forecast to make possible decisions aimed at strategy gen-
eration for future community advance. Thus, the development of complex models of
dynamics of natural-industrial processes, and the development of systems for manage-
rial decision support on their basis is the upcoming tendency to develop information
technology, and to implement it.
Anthropogenic pressure results in the worsening of the global environmental im-
pact as well as in the increased number of pathologies of biological objects (i.e. im-
mune retrogression, decreased reproductive function etc.). In the context of unstable
environmental factors and mosaicism of areas of species, genetic inhomogeneity of
species and certain populations increase significantly. Features of the situation should
be involved while planning environmental protection measures, performing ecological
monitoring, and solving problems concerning forecast of future of the populations. It
is extremely important to study a level of genetic heterogeneity of people. Accumula-
tion of pathologic recessive genes may be latent for a long time, and starting from a
certain moment it may be manifested in the form of rapid growth of the number of
definite hereditary diseases.
Use of mathematical models of dynamics of heterogenic populations as well as
their application for computer-based simulation makes it possible to identify efficient-
ly and with minimum time consumption the common tendencies for subpopulation
progress, to forecast the system state as well as possible consequences of artificial
intervention in the process [1-4]. Among other things, mathematical modeling helps
forecast formation of genetic abnormalities.
In its classic view, the system of dynamics of arbitrarily isolated populations is rep-
___
resented as xi f i x , i 1, n where xi is a size of i th population, and f i x deter-
mines its reproductive potential. The paper proposes to consider a model of subpopu-
n ___ ____
lation dynamics in the form of xi Aij f j x , i 1, n , i.e. when f j x , j 1, n
j 1
function is a part of a right-hand member of an equation with some coefficient
Aij 0;1 which will be defined below as a transition coefficient. Similar problems
were considered in [5], and in [6] they were further developed.
2 Mathematical model of dynamics of homomorphic objects
with logistic function as a basic one
1. The research is based upon the concept of a population as a set of species which
can be divided conventionally into n subpopulations; from the genetic viewpoint,
they are more or less homogenous while differing from each other significantly.
They are not isolated reproductively; thus, there is certain degree of probability
that inheritors of species from ith subpopulation will get to jth population. In gen-
eral, differential model of the system may be expressed as follows:
dx j n
Aji f i ( x ), j 1, n (1)
dt i 1
where xj is a size of jth subpopulation, fi(x) is a function, describing general repro-
ductive potential of ith subpopulation, and Aji is a share of inheritors of ith population
getting to jth one. Assume that Aij 1. in terms of any i. fi(x) function mirrors the
j
commonly known logistic law
1 n
f i ( x ) ai 1 xl xi (2)
K l 1
where a i mirrors reproductive potential of subpopulation with i index, and K is a
capacity of the population survival areal. According to (1), (2), population growth
nears zero when its size nears zero or when total size of all subpopulations nears max-
imum possible ecological capacity of K environment.
(1), (2) system is not Voltairian one to the extent that its trajectories may cross co-
ordinate axes and, for instance, local behaviour of the system in the neighbourhood of
the reference point depends upon its characteristics not only in the first quarter. Such
results have found their application in other industries [7], as well as in such works as
[8-11].
3 Equilibrium points of the system
To study equilibrium points of the system (1), (2) apply standard analysis on Lya-
punov [9]. It is seen easily that zero point (i.e. reference point) is one of the equilibri-
um points. Moreover, there is also endless number of equilibrium points lying in the
plane
x K .
i
i (3)
Character of arrangement of equilibrium points is rather natural from ecological
viewpoint. It is known that if there are no representatives of the certain species, they
cannot originate from nothing. If subpopulations inhabit one and the same ecological
niche consuming the same resources, their random distribution within the niche in
terms of their sizes is equivalent.
Theorem 1: (1), (2) system is degenerated system in the neighbourhood of specific
points of stationary hyperplane (3).
Proving: general view of ijth component of Jacobian matrix J of (1), (2) system is
n n
X k Aip a p x p
J ij Aij a j 1 k 1
p 1 . According to (3), general view of ijth component
K K
of Jacobian matrix J of (1), (2) system within planes of stationary hyperplane will be
1 n ____
J ij Aik ak X k . Since J ij does not depend on j directly, J pi J hi , p, h 1, n
K k 1
is true; it follows that vectorial columns of Jacobian matrix are linearly dependent,
Det J 0 ; hence, the system is degenerated within specific points of stationary
n
hyperplane x K .
i 1
i
The theorem has been proved.
If n=2, then the system is represented as:
dx1 x1 x2
dt 1a1 x1 1 2 a2 x2 1 K
. (4)
dx2 a x 1 a x 1 x1 x2
1 1
dt 2 2 2 1
K
Following symbols were introduced for (4) system: parameter i Aii mirrors a
part of subpopulation i growth belonging to parental one according to its phenotypic
n
characteristics. In terms of circulation system, parameters are Aij 1 i ,
i 1,i j
___
j 1, n .
Subpopulation may vary in reproductive coefficients аi, initial size, and in transi-
tion coefficient i (i.e. certain share of growth of a subpopulation belonging to “paren-
tal” one according to its phenotypic characteristics).
Figures 1 and 2 demonstrate samples of phase patterns in the context of two-
dimensional case for two sets of parameters. From ecological point of view, we are
interested mainly in the first quarter where subpopulation sizes are positive.
Fig. 1. Phase pattern of system (4) with following parameters: 1 0.95; 2 0.95;
a2 0.3; a1 0.1.
Fig. 2. Phase pattern of system (4) with following parameters:
1 0.95; 2 0.95; a1 5; a2 2.
Taking into consideration the fact that we are interested in the first quarter, i.e.
x1 , x1 [0, K ] , the system behaviuor near (3.3) line depends completely upon ai indi-
ces: if a1 0 , a2 0 then all points of the section are attracting; if a1 0 , a2 0
they are repelling; if a i have unlike signs, the section will consist of two parts – with
attracting points, and with repelling ones.
Theorem 2: degenerated stationary line x1 x2 K of (4) system consists of
points of attracting ray (i.e. attractor) with x1 a2 a1 K a2 0 general view, and
points of repelling ray (i.e. repeller) with x1 a2 a1 K a2 0 general view. Coor-
a2 a1
dinates of a point, connecting rays of attractor and repeller are K; K;
2 1
a a a1 a 2
within the point, characteristic equation of Jacobian matrix has zero root of 2 nd order.
Proving: according to theorem 1, system (4) is degenerated in the neighbourhood
of points of stationary line x1 x2 K . Elements of Jacobian matrix system are:
x x a x 1 2 a2 x2
J 11 1a1 1 1 2 1 1 1 ;
K K
x x a x 1 2 a2 x2
J 12 1 2 a2 1 1 2 1 1 1 ;
K K
x x 1 1 a1 x1 2 a2 x2
J 21 1 1 a1 1 1 2 ;
K K
x x 1 1 a1 x1 2 a2 x2
J 22 2 a2 1 1 2 .
K K
Within points of stationary line x1 x2 K , Jacobian matrix is
1a1 x1 1 2 a2 x2 K 1a1 x1 1 2 a2 x2 K
J , and its characteristic
1 1 a1 x1 2 a2 x2 K 1 1 a1 x1 2 a2 x2 K
equation is t t a1 x1 a2 x2 K 0 where due to degeneration of Jacobian matrix,
2
t1 0 , and t2 a1 x1 a2 x2 K .
Within points of stationary line x1 x2 K , nonzero root of the characteristic
equation can be represented as t2 a2 a1 x1 K a2 ; then, if a2 a1 x1 K a2 0
it is obvious that points of the stationary line are attracting, and
if a2 a1 x1 K a2 0 , they are repelling. It is understood that the two rays will be
a a x K a2 0
combined within the point meeting 2 1 1 condition also having
x1 x2 K
a2 a1
K; K coordinates. Within the point where attractor and repeller are
a2 a1 a1 a2
combined, according to x1 a2 a1 K a2 0 condition, root of the characteristic
equation is t2 0 .
The theorem has been proved.
The system trajectory may be directed either to endlessness or to equilibrium
points lying on a plane (3). In the latter case, finite point of the phase trajectory de-
pends upon initial conditions. The dependence is not linear one; and while approach-
ing attractor, trajectory density is not constant. In some specified sense, one can state
that probability of getting to different points of the attractor is not similar.
4 Analysis of bifurcation characteristics of the system
Analysis of system (4) demonstrates possibility of three bifurcations. Suppose that
equality of zero of a real part of no less than one of proper meaning of Jacobian ma-
trix is the required bifurcation condition.
Theorem 3: system (4) is regenerated in the neighbourhood of singular point (i.e.
a a 1 0
reference point), if a1 0 a2 0 1 2 1 1 2 1 2 .
1a1 2 a2 0
Proving: Jacobian matrix of system (4) within singular point (i.e. reference point)
1a1 1 2 a2
is J ; its characteristic equation is
1 1 a1 2 a2
t 2 Tr J t Det J 0 where Tr J 1a1 2 a2 , and Det J a1a2 1 2 1 .
a1 0
It is obvious that while fulfilling any of a2 0 conditions, Jacobian is
1 2 1
Det J 0 ; thus, the coordinate system is degenerated in the neighbourhood of the
Det J a1a2 1 2 1 0
reference point. If , then reference point is elliptic
Tr J 1a1 2 a2 0
point, and real part of the pair of complex roots of characteristic equation
Tr J 1a1 2 a2 is equal to zero. Hence, the system is degenerated in the neigh-
Det J a1a2 1 2 1 0
bourhood of reference point, if .
Tr J 1a1 2 a2 0
The theorem has been proved.
Theorem 4: in the neighbourhood of a reference point, system (4) is of a saddle
1a1 2a2
2
type, if a1a2 1 2 1 0 . If 0 a1a2 1 2 1 , then reference
4
1a1 2 a2
2
point is a node; if a1a2 1 2 1 , then it is a focus. In the context
4
of the two latter cases, point is stable if 1a1 2 a2 0 ; it is unstable,
if 1a1 2 a2 0 .
1a1 1 2 a2
Proving: Jacobian matrix of system (4) is J at the
1 1 a1 2 a2
origin, and its characteristic equations are t 2 1a1 2 a2 t a1a2 1 2 1 0 . It
is obvious, that if Det J a1a2 1 2 1 0 , then discriminant of characteristic
equation being D J Tr J 4 Det J is more than zero, and the product of roots
2
of the characteristic equation (according to Vieta theorem) is less than zero. Hence,
the origin is less than zero and the origin is of a saddle type by definition. Therefore,
consider that Det J 0 . Discriminant D J of the characteristic equation is
D J 1a1 2 a2 4a1a2 1 2 1 .
2
It is understood that if
1a1 2 a2
2
a1a2 1 2 1 , then singular point (i.e. reference point) is elliptic
4
one and real part of a pair of complex roots has stable focus; if 1a1 2 a2 0 , it is
1a1 2 a2
2
unstable. If a1a2 1 2 1 , then the singular point is hyperbolic
4
one. According to the assumption, Det J 0 . Hence, roots of the characteristic
equation (as a result of Vieta theorem) have similar sign; consequently, if
1a1 2 a2 0 , then the origin has a type of a stable node; if 1a1 2 a2 0 it is of
unstable node type. The theorem has been proved.
Below, the results, represented in theorems 3 and 4, are illustrated. Fig. 4 demon-
strates phase patterns in the process of reproductive coefficient a1 transition through
zero value. Other parameters have been selected as follows:
1 0,8; 2 0,8; a2 5; K 100 (5)
a b c (5)
Fig. 3. Phase pattern of system (4) with parameters and a) a1 1 , b) a1 0 , and c) a1 1 .
In Fig. 3, the origin is unstable nodes and all points of x1 x2 K line are attractive.
Such behavior is typical for the system under standard conditions when two progress-
ing subpopulations complement each other by a certain share of their inheritors while
increasing system-wide population biomass. When overall size of the series achieves
maximum acceptable edge K, the subpopulation growth comes to an end in the sense
that the number of newborn species is equal to the number of died ones.
In the context of the degenerated case (Fig. 3, b) all the phase trajectories are
straight ones, zero equilibrium point almost decays and line (3) stays to be attractor.
The case is realistic from the practical point of view since when subpopulation one
has zero coefficient of reproductive coefficient, its species are available owing to the
growth of subpopulation two. In this context, certain share of species of subpopula-
tion x 2 belonging to parental one according to their phenotypical characteristics, is 2
and share of inheritors will belong to x1 ; thus, 1 2 . It is obvious that in such a
case, ratio of size of two subpopulations will be stable, i.e. phase trajectories will be
straight lines.
In the case represented in Fig. 3, c, saddle is a reference point. Certain share of
points within a line (3) are repelling; certain share of trajectories tend to infinity de-
spite the fact that in the neighborhood of the reference point, where realistic values of
initial size of the subpopulations, the system remains unstable as before. Such a “sce-
nario” for the system development is widely used under the real conditions since in
the context of one system degradation it can support sufficient level of its size owing
to the subpopulation being developed; when total biomass achieves its critical value,
the former can preserve the ratio.
Fig. 4 represents another bifurcation case, when 1 parameter goes through critical
value 0.8. Other system parameters were selected as follows:
2 0.2; a1 1; a2 1.5; K 100 (6)
a b c
Fig. 4. Phase pattern of system (4) with parameters (6) and a) 1 0.7 ; b) 1 0.8 ; c)
1 0.9 .
The case is not realistic from biological point of view but it is interesting mathemati-
cally. In terms of the selected coefficients, subpopulation one is at a disadvantage: its
reproductive coefficient is negative and it exists owing to subpopulation two produc-
ing mainly species of subpopulation one. In Fig. 4, reference point is a stable focus; if
the initial number of species is insufficient, both subpopulations die out. In the con-
text of other initial conditions, the trajectories tend to the upper part of a line (3). Fig.
4, b represents transition degenerated case when two lines (i.e. attractor and repeller)
are available. Boundary trajectory points are focused on two half-lines and certain
trajectories tend to infinity. After subsequent magnification of 1 in Fig. 4, it becomes
obvious that reference point becomes a saddle. The majority of trajectories tend to
the upper part of a line (3); moreover, their density is maximal within lower part of
the half-line. As before, in the context of sufficiently large size of population two,
trajectories may tend to infinity.
In addition to the considered cases, another bifurcation when (1 2 1)a 1 a2 0
may take place. In this context, critical values of parameters are determined using the
equation
1a1 2 a2 0 (7)
It is understood that in the neighborhood of point (7), discriminant of characteristic
equation is negative, i.e. singular point is a focus. In this context, bifurcation is a
change of a stable focus for unstable one or vice versa. Fig. 5 shows such a bifurca-
tion when condition (7) is fulfilled and discriminant of characteristic equation is nega-
tive:
1 0.8; a1 0.25; 2 0.1; K 100 (8)
a b c
Fig. 5. Phase pattern of system (4) with parameters (8) and a) a2 1 ; b) a2 2 ; c) a2 3 .
When reference point is a stable focus, in terms of small x2 the system cannot achieve
its equilibrium within a line (3), sizes of the both subpopulations tend to zero. Taking
into consideration the fact that in the context of ecological system, xi coordinates can-
not have negative values, the case of unstable focus cannot be considered as essential-
ly different though area of initial values x2 , in terms of which the system is in equilib-
rium, is quite broader.
Fig. 6. Bifurcation diagram of system (4) in terms of fixed 2 =0.1 and a 2 =2
Bifurcation diagram represents general information concerning potential bifurcations
of system (4) (Fig. 6). As it has been demonstrated (Fig. 3-5), bifurcations of node-
saddle, focus-saddle, and stable focus-unstable focus types are observed within the
system. Fig. 7 demonstrates a segment of bifurcation diagram (Fig. 6); namely, it is
the segment where density of bifurcational curves is maximal.
Fig. 7. Segment of bifurcation diagram demonstrated in Fig. 6.
As it is seen, degenerated point M, within which bifurcation of type three originates,
is available within parametric space of system (4). Bifurcation diagram (Fig. 6)
proves graphically the results, represented in theorems 5 and 6.
Features of such a point are as follows: ordinary system bifurcations are possible
within the point as well as such more complicates transitions as “stable focus-unstable
node”, “stable focus-unstable focus”,“focus-saddle”, “unstable focus-stable node”,
“node-centre”, and “centre-saddle”.
Specify full-circle trajectory Г within certain neighbourhood of point M in bifurca-
tion diagram represented by Fig. 7. The trajectory, just as a part of the bifurcation
diagram in the neighbourhood of point M, is shown schematically. Actually, radius of
trajectory Г is 0.01. Trajectory Г passes through all segments of the bifurcation dia-
gram; thus, the system behaviour in the neighbourhood of point M is quite sufficient
to describe in full every possible state graph of variations of proper values as well as a
discriminant of the characteristic equation of system (4) while anticlockwise move is
taking place along the trajectory Г. For definiteness, locate K , N , P , T , and Z
points within the diagram.
Fig. 8. Dynamics of characteristic numbers and Jacobian matrix discriminant of the system (4)
in the context of movement along trajectory (Fig. 7).
For the diagram in Fig. 8 following symbols were used: D - discriminant of the char-
acteristic equation; t1 , and t2 - proper numbers of Jacobian matrix of the system (4);
b - real part of a pair of complex proper values of Jacobian matrix of system (4).
Discretization interval applied for the diagram is 0.01.
Every of K , N , P, T , Z points of trajectory of the bifurcation diagram shown in
Fig. 7 is separate bifurcation. KN arch of trajectory is within the part of paramet-
ric field, where topology of phase space on the neighbourhood of singular point is a
saddle, which corresponds to a positive value of a discriminant of characteristic value
as well as to negative production of proper values of the characteristic matrix.
NP arch corresponds to a type of a singular point (i.e. reference point) unstable node
depending upon the variation from negative proper parameter to positive one. Bifur-
cation transition through point P is stipulated by the transition of proper values of
Jacobian matrix to imaginary plane. Analytical condition for such a bifurcation origin
is Tr 2 J 4 Det J 0 . Discriminant of characteristic equation takes negative val-
ues when proper values are found out within imaginary axis. In this context, the graph
mirrors only the real part of a pair of the complex characteristic numbers of Jacobian
matrix being b Tr J . Tr J 0 condition is fulfilled within point T of the sys-
tem (4) stipulating loss of stability of elliptic equilibrium point as well as variation of
the phase space topology in the neighbourhood of the singular point (i.e. reference
point) from stable focus to unstable one.
System (4) is degenerated system in the sense that it involves continuous set of
equilibrium points of measurability 1. Generally, such systems are unstable structural-
ly. To analyze characteristics of the system from the point of view, it is possible to
introduce to it additional parameter K which reflects certain difference in ecological
capacity for the two subpopulations. Express system (4) for two-dimensional case in
such a way:
dx1 x1 x2 x1 x2
dt 1 a1 1 K X 1 1 2 a2 1 K K x2
(9)
dx2 a 1 x1 x2 X 1 a 1 x1 x2 x
2 2 1 1
dt
2
K K
1
K
Fig. 9 shows phase patterns of system (9) in terms of different values of coefficient
K ; other parameters were selected as follows:
1 0.3; a1 0.5; 2 0.5; a2 0.3; K 100. (10)
a b c (5)
Fig. 9. Phase pattern of the system (4) with parameters (10), and а) K 30 , b) K 0 ,
c) K 30 .
Computer simulation demonstrates significant stability of the system. As it is seen
from the figures, even in the context of substantial disturbance of K nature of the
trajectories does not experience essential changes within the major part of the phase
space. Topology variation, taking place objectively, is seen in the slow motion of the
system along a line (3) after its arriving towards (K,0) point or (0,K) point.
Another disturbance type is connected with discretization interval effect being al-
ways available in the context of computer simulation. Actually, discrete model form
is more adequate to the reality since population size cannot vary continuously. In the
context of the degenerated systems, discretization may result in the breakdown trajec-
tories and the system behaviour may experience qualitative variations. System (4)
demonstrates stability to such disturbances as well. In terms of rather high coefficients
of reproductive functions, cases are possible when phase trajectories pass attractor
(3); generally, the trajectory returns to its stationary line during the next interval. Fig.
10 shows a fragment of a line (3) and trajectories with sufficiently large discretization
interval in its neighbourhood.
Fig. 10. Trajectories of the system (4) with t = 5 interval in the neighbourhood of attractor (3).
5 Bifurcations of comeasurability 1 system
Theorem 5: in the neighbourhood of the singular point (i.e. reference point), sys-
tem (4) may have only three bifurcations of comeasurability 2 arising if
a1 0 a1 0 a2 0
.
a2 0 1 2 1 1 2 1
Proving: according to the results of theorem 3, system (4) is degenerated in the
neighbourhood of a reference point, if
1 2 1 2
a a 1 0
a1 0 a2 0 1 2 1 . It is obvious that in the context
1a1 2 a2 0
general type of bifurcations of comeasurability 2 of (3.4) system should be С42 6 ,
namely
a1 0 a1 0 a2 0
, , (11)
a2 0 1 2 1 1 2 1
a1 0 a1 0 1 2 1
a1a2 1 2 1 0 , a1a2 1 2 1 0 , a1a2 1 2 1 0 . (12)
a a 0 a a 0 a a 0
1 1 2 2 1 1 2 2 1 1 2 2
As it is understood, zero-dimensional set is the solution for systems (12); thus, only
three bifurcations (11) of comeasurement 2 of system (4) within the singular point
(i.e. reference point). The theorem has been proved.
Theorem 6: in the neighbourhood of the singular point (i.e. reference point), sys-
tem (4) experiences the only bifurcation of comeasurement 3 arising, if
a1a2 1 2 1 0
.
1a1 2 a2 0
Proving: topological structure of a phase pattern in the neighbourhood of a refer-
ence point of the system (4) may be in such nonequivalent states as “stable node”,
“unstable node”, “saddle”, “stable focus”, “unstable focus”. It is commonly supposed
that “node” and “focus” are equivalent topologically. We emphasize that in this con-
text, two-dimensional set “saddle” is transitional state between “stable node” and
“unstable node”; one-dimensional set “centre” is transitional state between “stable
focus” and “unstable focus”. Relying upon the abovementioned, differ conditionally
topological structures of a phase pattern of “node” type and “focus” type. Analytical
condition to transfer from “stable node” to “stable focus”, and from “unstable node”
to “unstable focus” zero equality of a discriminant of characteristic equation of the
D J Tr 2 J 4 Det J 1a1 2 a2 4a1a2 1 2 1 .
2
system (4) being
Basing upon general view of determinant of characteristic matrix
Det J a1a2 1 2 1 as well as real part of complex roots of a characteristic
equation Tr J 1a1 2 a2 , we can see that if the conditions of the theorem
Det J 0
are fulfilled, then discriminant of the characteristic equation is equal to
Tr J 0
a a 1 0
zero; hence, if 1 2 1 2 then system (3.4) will experience bifurcation of
1a1 2 a2 0
comeasurement 3 at the reference point. The theorem has been proved.
6 Conclusions
The paper represents results of analysis of mathematical model of dynamics of heter-
ogeneous groups with logistic function as a basic one for n 2 cases. Such a model
is not Voltairian one to compare with classic model; i.e. its trajectories may cross
coordinate axes. Moreover, it depends heavily on transition coefficients proposed by
the paper. Analysis of a model of metapopulation dynamics, including several sub-
groups competiting for common resource, has helped demonstrate rather diverse po-
tential system dynamics. Notwithstanding the degenerated, in some specified sense,
nature of the model, its dynamics is not trivial since in the context of variation of the
system parameters, three bifurcation types are possible if n 2 and nine bifurcation
types are possible if n 3 . As the numerical experiments have shown, the system is
stable sufficiently stable in relation to its coefficient disturbance; and characteristics
of nondegenerated systems, which can be obtained in such a way, are close to the
system under analysis. However, in the case of stability of nontrivial equilibrium,
finite state of the system depends upon initial conditions and it cannot be considered
as absolutely random one. Following of the system trajectories towards equilibrium
has certain regularities which will be described in detail during analysis of bifurcation
characteristics of the system.
Models of metapopulation dynamics and regularities of directivity of the system
trajectories to equilibrium may be used by IT while developing systems to support
managerial decision making.
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