Page 437
MACROSYSTEM APPROACH FOR MODELLING OF REGIONAL DYNAMICS
Yu.S.POPKOV
The Institute for System Studies
of the USSR Academy of Sciences
9,Prospect 60 let Oktyabria
Moscow, 117312, USSR.
telex 411237 POISK SU
tel. 135-42-22
fax. 007/095/938- 2209
Abstract
Formulation of the general scheme for developing dynamic
models of macrosystems with selfreproduction and resources ex-
change,using the local equilibria principle,is proposed. Regional
system, where reproduction and exchange processes have signifi-
cantly different relaxation times,is considered.The modelof
regional system,where dynamic exchange processes are described
as the evolution of local stationary states,is presented. Calcu-
lation,classification and bifurcation of equilibrium points of
regional dynamics are studied.Results of computational experi-
ments are given.
1. Introduction
During last decades we have observed growth of interest in
re-gional dynamics problems,stimulated by the accelerated rates
temporal -spatial regional development.The regional system is a
system of human settlements (centers),linked by network, through
which exchanges of goods,manpower,information, intellectual re-
sources are realised intensively.The specific properties of re-
gional system are the existence of close links among settlements
and their resource interdependence.
Interraction between regional settlements (centers) is rea-
lized in different forms.We consider one of them,when reproduc-
tion of some resources at centers is accompanied by their redis-
tribution among centers.
The problem of regional dynamic modelling is very popular.
It may be seen from large number of papers dealing with various
aspects of this problem (see,for example ,Prigogine, Herman, Allen
1977; Allen, Sanglier 1979;Wilson 1981;Roy,Brotchie 1984; Weidlich,
Page 438 System Dynamics '91
Haag 1988;Popkov 1989).
The approach to regional systems modelling pursued below
is based on the natural phenomenology of interaction between
selfreproduction at centers and resources exchanges among them.
We use the general procedure for describing dynamics of macro-
systems with selfreproduction and local-stationary
distribution, proposed by Popkov(1988, 1989).
2. General scheme
Consider the macrosystem consisting of n interlinked
blocks, in which system elements are located. The macrosystem
elements are of two types: specific for each block and nonspe-
cific, i.e. common for all blocks. The structure of links bet-
ween blocks may be different for both types of elements. Exam-
ples of these structures are shown at fig.1. Two interdepen-
dent processes are realised in the macrosystem: selfreproduc-
tion at blocks and distribution among blocks. The specific ele-
ments of each block and of blocks linked to it, as well as the
nonspecific elements participate in the selfreproduction pro-
cess. A state of this process is characterized by some parame-
ters, considered as the state variables. These variables may be
the numbers of specific and nonspecific elements at block,
their concentration or distribution over some scale. Below the
deterministic selfreproduction processes are considered.
The distribution process is related to the exchange bet-
ween blocks of universal (the same for all blocks) product, re-
produced in these blocks. Such a product may include nonspeci-
fic elements, set of goods, some universal good, which is used
for pricing the other goods.. Below stochastic distribution
processes are considered.
The state of each macrosystem block at time t is characte-
rized by generalized state variables for specific elements
zit) = {Z,,(1),.4.52, (b)}, for nonspecific elements xi(t) =
{x5 CE), 06 x CED and the average flows of universal good
y(t) = {y,,(t),...5y, Cb).
Assumption 1. The relaxation time of distribution process
is much less then the relaxation time of the selfreproduction
process.
This assumption makes possible to apply the local equilib-
ria principle (De Groot, Mazur, 1964) and to characterize the
System Dynamics '91 Page 439
distribution process by its local stationary state Y(t) =
{y, Do 4, JL}, where yj; s(t) is a local-stationary flows of
the universal good.
The selfreproduction process is the result of interaction
among specific and nonspecific elements (including consumption
of the universal good). The flow G(z,x,Y") and the flow
F(z,x,Y") constitute the aggregate characteristic of the
selfreproduction process.
Then the dynamics of the selfreproduction process may be
described with the following system of differential equations:
Zz = G(z,x,Y"); x = F(z,x,Y") (1)
In applied studies the system
z = G(z,x) + AY"; x = F(z,x) + BY’, (2)
is widely used, where A,B - constant matrixes.The matrix of
flows y*, characterizing the local-stationary state of stochas-
tical distribution of the universal good among blocks, is incl-
uded into right sides of the equations (1,2).
Consider the phenomenology of stochastic process of the
local-stationary state formation. Let wb) = At yD) be the
volume of the universal good, transfered from block i to block
j during time interval At, u,(t) - the summary volume of the
universal good, transfered from i during At. Since the univer-
sal good is produced at macrosystem blocks, it appears natural
to assume that u,(t) depends on the states of block i and other
blocks, which exchange the specific elements with block i, i.e.
u(t) = uj(2'¢t), 2° ¢t),x4(t)). Distribution of the univer-
sal good among macrosystem blocks during time interval At, cha-
racterized by the matrix W(t) = {w, (03, is accompanied by co-
nsumption of r resources. Let their stores be Q(t) = At q(t),
k=T,r, where q(t) is the. maximal volume of resource k, which
may be consumed in unit of time (maximal flow of resource k).
Resources consumption and their stores are linked as
$,CW(E)) = Qt), kel, r; (3)
where CW) is a function of consumption of resource k. Below
we consider the linear consumption functions:
n
XZ Ci Vig sq(t), k é1,r; (4)
j=
i,
Page 440 System Dynamics '91
where CF i is a resource k consumption per unit of time.
We assume that the universal good is produced in portions
and Wy, (t) is the number of these portions, transfered from i
to j during At. Let Py be the probability of producing one por-
tion of the universal good at block i; b., - probability of
transfering one portion of universal good from block i to block
j. From the definition of macrosystem state variables it fol-
lows that the set of states for the distribution process is the
‘set of pairs (i,j) - ’producer - consumer of good’. Then
a, PbS, the probability of a portion getting to the state
(i, j).The states (i,j) characterize by capacities Gy
Assumption 2. The parameters of distribution processes are
a, (2.x); Co Os jue 22 X73 GEG, (Zs X)5 G, FG, (2.x).
Thus, numerous random distributions of the universal pro-
duct portions over states (i,j) determine flows wt) =
(w(t), 4, jeDm » Which characterize the distribution process
macrostate. Due to the Assumption 1 the distribution W(t) is
established during short time interval At while the blocks
state variables z(t),x(t) remain the same.
The local stationary macrostate of the distribution pro-
cess may be obtained as the state with maximal entropy under
resources constrains (see Wilson 1967;Popkov 1980).In this case
the time interval At is assumed sufficiently less than the
selfreproduction process relaxation time, but long enough
for the macrostate to be formed from multiple realizations
of the microstates (in physical sence, At is infinitezimal).
Since the local stationary state is given by the point of
entropy maximum, and not by its value,the entropy function may
be determined by flows Y:
aj
n Vij
H(Y,Z,x) = - LY,; In , (5)
i, j=l ~
a, (2%)
where
a, (2, X04, (Z,X)G, (2, x). (6)
Then the model of the distribution process local stationa-
ry states becomes
HCY,Z xX) > max, (7)
YeD(z,x). (8)
System Dynamics '91 Page 441
The set of feasible states is
n
D(z,x) = OS Be Piel = g,(2,%), k=T,r} = (9)
Thus the dynamic model of the macrosystem with selfrepro-
duction and local-stationary redistribution (DMSR) becomes
z= G(z,x,Y'(z,x)); xX = F(z,x,Y"(z,x)), (10)
Y"(z,x) = argmax (H(Y,z,x) YeD(z,x)). (11)
The structural scheme of DMSR is shown at fig.2.
3.Dynamic model of regional system
Consider the regional system consisting of n interlinked
centres with capacities Eyseees EB. We characterize the system
state- by- population x,t) at--center-i--and- migration flow Vij
from center i to center j. The capacities Eyer EL and popula-
tion volumes satisfy the following unequalities:
0s x,(t) = E,, ieT,n. (12)
Assume that the biological reproduction of centers’ popula-
tion takes place significantly slower, than the mechanical mig-
ration among centers, This is a hypothesis, and for some count-
ries it is confirmed by the empirical studies.
The migration from center i to center j is stimulated by
the attractiveness of center j for the residents of center i.
The attractiveness is determined by many factors and various
methods are proposed to estimate it (for example, Weidlich, Ha-
ag, 1988). We use the stochastic model of migration and estima-
te attractiveness via probability aij of resident choosing the
pair of centres: i - "old" residence, j - "new" residence. The
probability a; depends on population allocation, i.e.
aj Fa j(%).
This dependence may take different forms but certain pro-
perties are typical. So, if the population of "new" residence
equals zero or maximal capacity of this center, then the
attractiveness of center j equals zero:
1X70, (13)
per XP =0s i, jeT,n. (14)
Assume, that only fixed share hy of population in center
iparticipates in migration. This share of population will be
called mobile.
Ay Xp eer X O Xie
ay (Cpr e ry EX
Page 442 System Dynamics '91
Introduce the notation:
n n
PED =P yt» O10 .= F
i % Jt i j=l
Considering the environment as the additional center num-
bered O we obtain
yD. (15)
n n
DP.t) = Lact.
i =0 i=1
The process of changes population centres may be described by
the following system of differential equations:
Loa T Do a * elm
x, = Ti (x) * En yy 46h.05
where T(x) is the reproduction function; T (x, )20; T,(0)=0.
According to the general scheme of DMSR formulation; the
matrix Y"=Ly;] is determined as
Y"(x) = argmax { H(Y,x) YeD(x) }, (16)
where
H(Y,x) = - r y,, In i } (17)
iz1,jeo 19 4; (07
DOIAY, 5 r yh D 7 C5 1 gM; jie1,n; ke1,ry_
j=0 i=1, j=0
(18)
Considering the structure of the feasible set (18), notice
that the first group of equalities is implied by the assumption
that all mobile population of center i participates in the
migration. The second group of equalities accounts for
resources consumption. For example, these expenditures may be
stipulated by travel costs, new housing construction, creation
of additional working places. Here it is assumed that resources
consumption is proportional to the flows y,; and the coeffici-
ents of proportionality Cs ik do not depend on distribution of
the population over centres. Such a dependence may take place
only for the total stores Cm of resources.
Due to the special structure of the feasible set D(x) (18)
‘ Degg ‘|
x, = F(x.) + Dy, , ieTn;
i is“ j=t Jji
where
T(x) = T,Qd-h,x,.
System Dynamics ‘91 Page 443
We assume that T,(x,) < h,x,.
Consider the problem (15-18). Using Lagrange method we ob- °
tain the following system of differential equations:
. n
x, = Tx) * Bah O00, (19)
r
exp (-Y Avec, .)
wo K idk
$; OOF 20, ieI,n, je0,n, (20)
EPs OEE Moi sk?
r
, hx, exp. i Ay Ci i )
gee - aay (x) Ci ie TH = - g¢ ,,(x)=0,
a = (x,,) eep(-Z A Gi)
s=0 k=1
kel,r (21)
Hence, the equations (19-21) describe the dynamical model
of population reproduction and migration in the regional system.
4.Model of regional system with linear reproduction
Consider the regional system, where reproduction function
T(x, )=0x,, mobile population share h,=h and resources const-
raints in the model of migration are omitted.Then the equations
(19). come to:
aj,(%)
—__— hx, - (h-a)x,, ieT,n. (197)
1 ep tan
Consider the case when emigration exceeds reproduction
(a<h) and assume that the attractiveness of center j depends
only on its own population:.
a, O05 Vix.) = x (Box), jelsn Vo2Ee.
Transform the system (19’), substituting the variables
x,Ey for Xy> t/h for t. We obtain
n
. x,(e-X) OX pew
x(t) = Kel - —— x,, i=T (22)
where e=E,/E,.
The system (22) belongs to the class of differential equa-
tions x, = x,9.(x), for which the nonnegative ortant
RP = {x € R”| x2 0, i=T,m} constitutes the invariant set.
Page 444 System Dynamics '91
We study the singular points of this system and their evo-
lution depending on change of centers capacity.It is shown that
system (22) have 2n, positive solutions
e
ag = e- + i ve ?-te(™ 17, ieT,n;
e™ =1[2(h-a)n, al!/2,
n, is the number of peeneLve coordinateot singular points.
The positive singular points xj sexist if the. following unequa-
lities are valid:
_fa, 1 in, 22 (23)
e,> 3 t75 if ee - (em) ] ,iel,n,
In (Popkov, Shwetsov, 1990) the analysis of situations connected
with existing of positive singular points fordifferent centers
capacity is realized.
The inequalities (23) divide the (Oyri vise, ) area of pa-
rameters values into several subareas with different numbers of
equilibria. The smooth variation of parameters within one of
these areas provides the smooth shift of equilibrium position.
If the (Oys +0 @) parameter point crosses an area boundary,
the equilibrium bifurcations occur, such as birth, disappearance
or merging of equilibrium points. The transformations of traje-
ctories structure, implied by the equilibrium bifurcations,
provide the qualitative changes in system behaviour. Thus the
global properties of system are changing through the equilibri-
um bifurcations.
In (Popkov, Shwetsov, 1990) full description of the bifurca-
tion of equilibrium positions in this system is obtained. On
fig.3 is shown the results of the investigation of equilibrium
positions of the regional system consisting of two centers. The
phase pictures illustrate the evolution of equilibrium points
for various centers capacity. When the capacities are small,
system degrades due to population emigration to environment
(fig.3a,b).For some value of capacity determined by conditions
(23) the nontrivial equilibrium points arise, at the beginning
System Dynamics '91 Page 445
- one point (fig.3c) and then - two points (fig.3d). The point
"A" on this picture is stable, and the point "B" is not stable.
Notethat the phase plane is divided into two parts. One of them
is area of the attraction of the trivial point and the other -
of point "A". When the center capacities increase then size of
"A" attractiveness-area increases too (fig.3e,f).
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Q—O G27") 2.x |H(Verom| y*
d= FeV) | Ye 2(z,2)
oS
Fig 1 Fig 2
System Dynamics '91
Page 446
1.6284 2 = 1.2000
r= 5.0000, > 4.5
ES
-0000
1.5000 e2
7.7500 @2= 1,5000
ey =
@ = 2.5000 & = 1.5000
1.6500
2.1000 “a2
ey=
BL
By
Fig 3.
