Calculation of function for population dynamics model with continuous-time age
In this article, we present an approach to identify functions in population process model with continuous-time age. The main consideration was given to functions identification of birth, mortality, migration and specific function “becoming mature”. Such functions could be defined analytically or by...
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Інститут кібернетики ім. В.М. Глушкова НАН України
2008
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| Назва видання: | Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
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| Цитувати: | Calculation of function for population dynamics model with continuous-time age / Yu. Zagorodni, V. Voytenko // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2008. — Вип. 1. — С. 61-67. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-186762011-04-08T12:03:51Z Calculation of function for population dynamics model with continuous-time age Zagorodni, Yu. Voytenko, V. In this article, we present an approach to identify functions in population process model with continuous-time age. The main consideration was given to functions identification of birth, mortality, migration and specific function “becoming mature”. Such functions could be defined analytically or by using specific numeric procedure. A numerical example is given to demonstrate the method, along with computer application to population analysis in Canada and Ukraine. В статті запропонований підхід до визначення функцій – параметрів моделі популяційної динаміки з неперервним віком. Головна увага приділяється ідентифікації функцій народжуваності, смертності та спеціальної функції швидкості дорослішання. Ці функції можуть бути визначенні аналітично, або за допомогою спеціальної чисельної процедури. Для демонстрації моделі розглянутий приклад порівняльного аналізу моделей динаміки населення України та Канади. 2008 Article Calculation of function for population dynamics model with continuous-time age / Yu. Zagorodni, V. Voytenko // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2008. — Вип. 1. — С. 61-67. — Бібліогр.: 9 назв. — англ. XXXX-0060 http://dspace.nbuv.gov.ua/handle/123456789/18676 en Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки Інститут кібернетики ім. В.М. Глушкова НАН України |
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In this article, we present an approach to identify functions in population process model with continuous-time age. The main consideration was given to functions identification of birth, mortality, migration and specific function “becoming mature”. Such functions could be defined analytically or by using specific numeric procedure. A numerical example is given to demonstrate the method, along with computer application to population analysis in Canada and Ukraine. |
| format |
Article |
| author |
Zagorodni, Yu. Voytenko, V. |
| spellingShingle |
Zagorodni, Yu. Voytenko, V. Calculation of function for population dynamics model with continuous-time age Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
| author_facet |
Zagorodni, Yu. Voytenko, V. |
| author_sort |
Zagorodni, Yu. |
| title |
Calculation of function for population dynamics model with continuous-time age |
| title_short |
Calculation of function for population dynamics model with continuous-time age |
| title_full |
Calculation of function for population dynamics model with continuous-time age |
| title_fullStr |
Calculation of function for population dynamics model with continuous-time age |
| title_full_unstemmed |
Calculation of function for population dynamics model with continuous-time age |
| title_sort |
calculation of function for population dynamics model with continuous-time age |
| publisher |
Інститут кібернетики ім. В.М. Глушкова НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/18676 |
| citation_txt |
Calculation of function for population dynamics model with continuous-time age / Yu. Zagorodni, V. Voytenko // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2008. — Вип. 1. — С. 61-67. — Бібліогр.: 9 назв. — англ. |
| series |
Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
| work_keys_str_mv |
AT zagorodniyu calculationoffunctionforpopulationdynamicsmodelwithcontinuoustimeage AT voytenkov calculationoffunctionforpopulationdynamicsmodelwithcontinuoustimeage |
| first_indexed |
2025-07-02T19:37:18Z |
| last_indexed |
2025-07-02T19:37:18Z |
| _version_ |
1836565176089837568 |
| fulltext |
Серія: Технічні науки. Випуск 1
61
4. Дутка В. А. Комп’ютерне формування температурного поля твердосплав-
ного різця для його індукційного паяння та гартування // Сверхтвердые
материалы. – 2008. – № 2. – C.72-78.
5. Попова Л. Е., Попов А. А. Диаграммы превращения аустенита в сталях и
бета-раствора в сплавах титана: Справочник термиста. – М.: Металлур-
гия, 1991. – 503 с.
6. Лахтин Ю. М. Металловедение и термическая обработка металлов. – М.:
Металлургия, 1984. – 360 с.
7. Жук Я. О., Червінко О. П., Васильєва Л. Я. Уточнена модель структурних
перетворень в тонкому сталевому циліндрі при тепловому опроміненні
торця. – Доп. НАН України. – 2007. – № 4. – С.53-58.
The numerical technique for estimation of quench hardness and thick-
ness of hardened case of tool holder after induction brazing and case-
hardening in solution of salt and alkali is proposed. The results for cases of
case-hardening in quencher at temperature 300, 200 and 100 ºC are pre-
sented. It is shown that by variation of the cooling properties of quencher
the high hardness of hardened surface can to obtain.
Key words: computer simulation, case-hardening in solution of salt
and alkali, quench hardness, hardened case.
Отримано: 05.06.2008
Yu. Zagorodni1, V. Voytenko2
1Kyiv National University of Taras Shevchenko
2King’s University College, Canada
CALCULATION OF FUNCTION FOR POPULATION
DYNAMICS MODEL WITH CONTINUOUS-TIME AGE
In this article, we present an approach to identify functions in po-
pulation process model with continuous-time age. The main conside-
ration was given to functions identification of birth, mortality, migra-
tion and specific function “becoming mature”. Such functions could
be defined analytically or by using specific numeric procedure. A
numerical example is given to demonstrate the method, along with
computer application to population analysis in Canada and Ukraine.
Key words: population dynamics, the birth rate, functions
identification, continuous-time age.
1. Model description of population dynamics with continuous-time age
Population models are used in biology and ecology to model the dy-
namics of wildlife or human populations. In common, population is as an
aggregate of elements of one biological type subject to particular changes
[6]. For social systems, element of population is a person or group of per-
© Yu. Zagorodni, V. Voytenko, 2008
Математичне та комп’ютерне моделювання
62
sons (for example, a family, region, county, state, etc). Demographic event
means any changes in person status as an element of social system. These
changes can cause transfer a particular person from one to another social
group. Group usually means aggregate of population elements with similar
features (for example, age, income, nationality, etc). Then element transfer
from one group to another group defines a process of migration. We can
name the group open, if a number of elements could be changed dramati-
cally [7]. In common, we can consider any cells aggregate as population.
Some other samples from biological population are cultural plants, wild
animals, bacterial cells, virus particles, etc.
Most of publications provide useful theoretical basis for analyzing
the sensitivity of stable population distribution and rate of growth to
changes in the fundamental parameters [2, 3, 4]. In most cases, features
which formed population group are continuous by their nature, for exam-
ple age, weight and other values (measured by real numbers) [6, 7]. They
are interesting from practical and theoretical points of view. In contrast of
features described by finite functions, we consider continuous functions
for description density of distribution for particular feature.
From a mathematical point of view the model is a linear system of
partial differential equations, where the state variables are the population
densities in each spatial patch, together with a boundary condition of inte-
gral type, the birth equation [1, 5]. A population is divided into Np ∈ N
groups which correspond to the special characteristics of population (for
example, features of welfare, professional features, etc). The main func-
tion (state variable) of this model is continuous piecewise differentiate
function np (t, τ), which characterises a density of population in time t by
age τ on the set of time parameters [ ][ ] pNpTVt ,1,,0,0),( max ==∈ ττ .
Then a common number of elements for all groups by age features for
particular group p could be defined by the next equation:
∫=
max
0
),()(
τ
ττ dtntN pp .
Then a model could be described using the next system equations for
each group pNp ,1= :
=
=
−
∂
∂
−=
∂
∂
∫
).(),0(
,),(),()0,(
),,(),(
),(
)(
),(
max
0
τϕτ
ττ
τ
ττ
τ
p
pp
pp
p
p
p
n
dssthstQtn
thtd
tn
tl
t
tn
(1)
Серія: Технічні науки. Випуск 1
63
Also consider the main model parameters as functions which charac-
terise population dynamics:
– function of mortality in time t of people in age τ – ),( τtd p , as an
addition of 2 processes mortality and migration: ( ) ( ) ( )τττ ,,, tmtstd ppp += ;
– function of new beings birth rate for parents in age τ – ),( τtQ ;
– auxiliary function of “becoming mature” speed (motion down the
axis of age [ ]max,0 ττ ∈ ) )(tl p
– function of migration speed ),( τtmp by condition of group open-
ness pNp ,1= .
So, function ),( τtnp is differentiate on the set of Zd VVV /= , where
VVZ ∈ – terminal set of transition points, where the system can be
changed (functions )(tl p and ),( τtd p could have breaks). Let’s assume
{ }zN
zzzZ TTTV ,...,, 10= , where 00 =zT , and 0≥zN – number of breaks on
a time set [ ]Tt ,0∈ . In the break points, next conditions should apply:
( ) ( )
( ) ( )
.,10,
,,0,0
,00
zii
i
zpi
i
zp
i
zpi
i
zp
Nidc
TdzTd
TlcTl
=≥
−=+
−=+
ττ (2)
Let’s assume )(),( tdtd pp ≡τ . Then analytical solution of system (1)
could be presented like that:
( )
.)()(
,),(),(
,)(),(
0
0
0)(0
∫
∫
=
=
−=
−
t
pp
t
pp
pp
tD
pp
dssltL
dssdstD
tLFeNtn p
τ
ττ
(3)
Let’s consider the next function as a solution (3):
( ) ( ) ( )( )
( ) ( )( ).)()(exp),(
,)()(exp)(
2),(0
20
τττ
τττ
τ
−+−−=
−+−−=−
− tLbtLaeNtn
tLbtLatLF
pppp
tD
pp
pppppp
p
(4)
Then edge condition will be described as:
( )∫ −−=
max
0
2)(2exp),(1
τ
dssbsastLastQ pppp . (5)
Математичне та комп’ютерне моделювання
64
It is possible with the next function of new beings birth rate:
( ) ( )
≥−
<
=
−− .,
;,0
),(
000
0
0 ssessQ
ss
stQ ssq (6)
Assume 160 =s – the average age of reproductive period start for
human population. If we make substitution (6) to the edge condition (5),
we will get the next ratio:
( ) ( )
( )( ) ( )( ).expexp
exp2
,)(2
2
max
2
0
0
0 τrasra
qsra
Q
btLaq
pp
p
ppp
+−−+−
−+
=
−=
(7)
where rba pp ,, – constant positive values.
We can study population dynamics trajectories using equations (5-7).
In this case, very important role plays numerical and analytical calcula-
tions of integral functions )(tLp and )(tDp .
2. Target setting of model functions identification
and its solution algorithm
As we can see from described model above (1-7), functions ),( τtd p ,
),( τtQ , )(tl p and their parameters characterise population dynamics. We
will try to calculate these functions in order to identify particular popula-
tion dynamics. First of all, we should apply the next restrictions:
maxmaxmax ),(0,)(0,),(0 QtQltldtd ppp ≤≤≤≤≤≤ ττ .
Functions could be define explicitly (for example, as functions (8)),
or in a numerical form using defined computational procedure. After such
functions definitions, population dynamics, as functions np (t, τ) could be
defined on the set of [ ][ ] pNpTVt ,1,,0,0),( max ==∈ ττ using functions
(3,5,7).
But it isn’t true for all possible types of model (1). In this case we
have to build a difference scheme for definition of mesh functions xp (ti, τj)
– discrete analog of np (t, τ)functions which define on the grid
( ){ }τττ jhihtMjNitW jtiji ===== ;;,1,,1:, ,
where
M
h
N
Tht
max, τ
τ == . Difference scheme will become as:
( ) ( ) ( )11 ,,1, −+ +
−−= jipi
t
jipiti
t
jip txl
h
htxdhl
h
htx τττ
ττ
, (8)
where ( ) ( ) piiii NpNiMjtddtll ,1,1,0,,1,, =−==== .
Серія: Технічні науки. Випуск 1
65
Edge and initial condition will be:
( ) ( ) ( )
( ) ( ) .,
,,,,
0
0
0
jjp
M
j
jipjipip
tx
txtQhtx
τϕτ
τττ τ
=
= ∑
=
A scheme (8) will work correctly if:
01 ≥−− iti
t dhl
h
h
τ
or maxmax dhl
h
ht
τ
τ
+
≤ .
For analytical solution (5) functions )(tLp and )(tDp should be de-
fined as:
( ) ( )
( ) ( ),
,
0
0
∑
∑
=
=
=
=
i
i
M
i
iptip
M
i
iptip
tdhtD
tlhtL
where
=
t
i
i h
tM .
3. A numerical example of parameters definition
for Canada-Ukraine demographic model
The data taken from population censuses [8, 9] are among the basic
sources of demographic figures which constitute the fundament of a number
of analyses. Compared with others, they have the advantage of providing
results from the past based on real data. Using this data, we tried to identify
functions in models of demographical processes with continuous-time age.
Taking a model described above into consideration, we suggested the
next set of model functions (1):
( )0110 sin)( wtwlltl p ++= ; ( )τττ twdddtd p 3210 sin),( ++= . (9)
Using computer application of such simulation process, we obtained
the next parameters value, for Ukraine and Canada respectively:
Table 1
Model parameters value for Ukraine and Canada
№ Parameter Ukraine Canada
1 N0 899 373,6
2 ap 0,00023 0,00015
3 bp 0,00027 0,00012
4 l0 0,726 1,3
5 w0 0,436 0,00016
6 w1 0,0506 0,00016
Математичне та комп’ютерне моделювання
66
Continuation of table 1
№ Parameter Ukraine Canada
7 D 0,31952 0,117
8 d2 0,00667 0,00003
9 d1 0,00017 0,00001
10 d2 0,00002 0,00003
11 w3 0,6 0,04899
12 Tz
1 2,547 68,75
13 c1 0,004 0,08
14 d1 1,525 0,43515
Fig. 1. Dynamics of birth-rate level in Canada
(line 1 – statistics, line 2 – experimental)
In conclusion, we have to admit one transfer point for Ukraine. Point
of break of functions is 547,21 =zT , which is approximately a year of
1994. Such point of break in Canada was observed at 18,71 =zT , when
function )(1 td was decreased, probably because of migration wave. Coef-
ficients of population loss functions are considerably small, which ex-
plained by compensation mortality by migrations.
Thus, using model (1) and function (2-7) we can investigate qualita-
tive and quantitative characteristics of population dynamics and make
comparable analysis on this basis.
Literature:
1. Caswell, H. Sensitivity analysis of transient population dynamics // Ecol. Lett.
– 2007. – 10. – P.1-15.
Серія: Технічні науки. Випуск 1
67
Fig. 2. Dynamics of mortality level in Ukraine
(line 1 – statistics, line 2 – experimental)
2. Caswell, H. Matrix Population Models – Construction, Analysis and Interpre-
tation // 2nd ed. Sinauer Associates. – 2001. – Sunderland, MA.
3. Metz J. A. J. & Diekmann,O. The Dynamics of Physiologically Structured
Populations. – Springer-Verlag, Heidelberg, 1986.
4. Michod, R. E. &Anderson, W. W. On calculating demo-graphic parameters
from age frequency data // Ecology. – 1980. – 61. – P.265-269.
5. Ovide Arino, Eva Sanchez, Rafael Bravo de la Parra, Pierre Auger. A Singular
Perturbation in an Age-Structured Population Model // SIAM Journal on Applied
Mathematics. – 2000. – Vol. 60. – No. 2 (Dec., 1999 – Feb., 2000). – P. 408-436.
6. Poluektov O. Dynamic theory of biological populations. – Moscow: Science,
1974. – 456 с.
7. Staroverov O. O. Basics of mathematical demography. – Мoscow: Science,
1997. – 158 p.
8. Statistic Canada, http://www.statcan.ca
9. Statistics Ukraine, http://www.ukrstat.gov.ua
В статті запропонований підхід до визначення функцій – парамет-
рів моделі популяційної динаміки з неперервним віком. Головна увага
приділяється ідентифікації функцій народжуваності, смертності та
спеціальної функції швидкості дорослішання. Ці функції можуть бути
визначенні аналітично, або за допомогою спеціальної чисельної про-
цедури. Для демонстрації моделі розглянутий приклад порівняльного
аналізу моделей динаміки населення України та Канади.
Ключові слова: популяційна динаміка, темп народжуваності,
ідентифікація функцій, неперервний вік.
Отримано: 05.06.2008
http://www.statcan.ca
http://www.ukrstat.gov.ua
|