THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 303
ROPELING THE COMMON CRANE (Grus grus) POPULATION WINTERING
IN CBERIA
Juan C. Alonso
Nuseo Nacional de Ciencias Naturales, CSIC, Madrid, Spain
Javier A. Alonso
Facultad de Biologia, Universidad Complutense, Madrid, Spain
Manuel Quintanilla
Instituto de Economia Agraria, CSIC, Madrid, Spain
Ab A field study carried out during the last seven
a wintering population of Common Cranes (Grus grus)
in Spain allowed us to gather accurate field data on certain
relevant demographic parameters: (a) the size of the Western
Palearctic pupulation of this species is estimated to be
around 40000 birds, and (») the age composition in autumn
is estimated for 1979-85 to be around 13.5% young. These
data enabled us to develop and test a population dynamics
model which combines the density- and age-dependent effects
on productivity and survival rates. The parameters of the
productivity and survival functions were varied within
biologically reasonable limits. From the series of possible
combinations we selected those that fitted oue field data
on population size and age structure best. fach variable
was then varied to study their influence on the model.
INTRODUCTION
One of the fields in which mathematical models save’ been
inc:easingly applied during the last years is wildlife ecology
and management (e... Miller and Botkin 1974, Miller 1978,
Johnson 1982). The complexity of animal population dynanics
and the difficulty in measuring the many parameters involved,
such as productivity and mortality rates, longevity, age
of reproductive maturity and the effects of population density
and age on them, make it very diffisult to obtain a series
of data long enough to detect and analyse the effects of
natural or man-induced events that affect these animal popule—
tions. Mathematical models help understand the structures
and function of the latter. This is particularly interesting
in the case of vertebrate species that are K-straterists,
i.e. have long maximum longevities, delayed reuroductive
maturities, low rates of increase and long generation tines
and, therefore, low replacement rates. This pape presents
a first attempt to develop a population ¢ynamic: nodel for
one of these species, the Common Crane (Grus.grus), similar
to those produced for the Sandhill Crane ( 5)
(Willer et al. 1972, Johnson 1979) and the
(Gr a) (Hiller e. 1974).
DENOGRAPHIC PARATIETERS OF THE CRANE POPULATION
While some of the demographic parameters of the Common Grane
eur THE 1900 INTERNATIONAL CONFERENCE Ur THE STOTEM DINAWILS SULIENY. SEVILLA, UU TUPEN, 1900
have -been recorded quite accurately during the last years,
others remain unknown, mainly due to the lack of individually
marked birds.
Field data
Two of the parameters used were obtained by two of us (J.C.A.,
J.A.A.) during 1979-85 at Gallocanta, one of the most important
wintering areas of the Western Common Crane population.
Population size. During the last decades some authors have
tried to census the Western European Crane population: ‘The
results are diverse and confuse: some.are only partial censuses
in staging areas, reaching maximal figures between 25000
and 30000 birds (e.g. Keil 1970, Alerstam and Bauer 1973);
others are’ estimations in more or less wide areas of the
species'breeding range. Only recently has it been possible
to count the migrants while leaving Iberia in spring, with
a total. of 31945 birds (Alonso et al. 1986). Assuming a
certain error in minus, we estimate in around 40000 cranes
the size of this population, which coincides with estimations
recently made in Central Europe (Prange 1984, and in litt.).
Winter age composition. Only two agé classes can be distin-
guished in the field by their plumage, here called adults
and juveniles. In the "adult" category we included all cranes
in nonjuvenile plumage, i.e. more than one year old. The
birds less than one year old were included in the "juvenile"
category. The adult:juvenile ratio was recorded at random,
including unselectively all flocks as they were found in
the field. This age-ratio is only valid and representative
of the whole population if a large number of birds can be
aged each season in the same area and time and under approxima-
tely equal conditions. The study area was surveyed regularly
throughout the whole season. In total we aged over 125000
cranes, of which only the 105973 aged during autumn migration
and early spring migration are used in this study. For a
more detailed sample selection procedure see Alonso &
Alonso (in press). The unweighted averege for the seven
years is 13.54% (Table 1).
Table 1. Age composition of the Western Common Crane population
Year 1979 1980 1981 1982 1983 1984 1985
% juveniles 12.48 12.95, 14.17 12.12 11.67 12.34 19.02,
No. cranes aged 5890 6508 20301 17991 20917 12898 21468
Data from other sources
Some of the data necessary to develop a population dynamics
model are very poorly known or still unknown, like age of
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 305
first ‘breeding, longevity and mortality. It would wu very
time-consuming to significantly increase the accuracy of
these data. Therefore, as some acceptable data for these
parameters are available-of other crane species of the °
yenus, we have used them in the model,
Ape of first breeding. Common Cranes have been reported
to lay eggs at an age of three years, but on the basis of
recent literature revisions (Walkinshaw 1973, Johnsgard
1983) and captive breeding experiences (Archibald pers.
comm.) we assumed that they reach sexual maturity at an
age of four years. .
Longevity and mortality. Although there are some literature
records of several species of cranes living more than 40
years in captivity, authors generally estimate maximum longevi-
ties of 20-25 years for natural populations (Walkinshaw
1973, Johnsgar:| 1983), and Binkley & Miller (1980) recently
utilized 24 age-classes. We also assumed the existence of
24 age-classes.
Most classic bird population studies assumed constant adult
mortality rates (Deevey 1947, Lack 1966). However, these
are not in accordance with longevity data for certain species
for which nowadays enough recoveries of banded birds are
available. It seems more reasonuzle to assume an age-dépendent
mortality effect (Miller et al. 1972, Botkin & Miller
1974). A density-dependent ef. on mortality is also general-
ly accepted, which affects birds of every age and reproductive
condition. Common Cranes have not yet been banded, for which
there are no reliable data on the magnitude and causes of
mortality. Some data are available for the vandhill Cranes
(Jonnsgard 1983) although such values are surely overestimated
due to the effect of hunting. Therefore we have used the
age-specific survivorship values estimated for the non-hunted
population of Whooping Cranes by Binkley 4 ‘tiller (1980).
These are based on annual censuses and age compositions
of that population conducted since 1938.
MODEL DESCRIPTION
The model we have developed is avery simple one that describes
the population dynamics of the species. The program was
written in Basic Apple Soft and has been operated successfully
by one of us (.Q.) on an Apple II computer. Figure 1 presents
a simplified causal diagram which accounts for the density-
dependent effects of the population size on annual . recruitment
and survival rates. Figure 2 shows the complete flowchart
of the model, including (a) the assumed 24 age-classes (see
above, "Demographic parameters"), (b) the distinction between
the first three nonreproductive age-classes and the rest
of sexually mature cranes, and (c) the type of curves (=reverse
logistic functions) that govern the density-dependent effects
on recruitment and survival rates.
The crane population of level 1 (G00, see Appendix) is deter-
mined by the.following two equations:
NAC = GTER * PRD
where NAC = number of births; GTER = number of sexually
306 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
CRANES 0-1 YEARS OLD ne |
+
+O i)
SURVIVAL RATE TOTAL CRANES -———=™ RECRUITMENT RATE
Lee MORTALITY
Figure 1, Simplified causal population model for the Common
Crane.
DENSI TY-DEPENDENT
RECRUITMENT FUNCTION
SURVIVAL No. CRANES
—
CRANES 0-1 YEARS O-1 YEARS
Y
SURVIVAL ea No. CRANES
CRANES 1-2 YEARS L-2 YEARS
’
SURVIVAL flo. CRANES
—
CRANES. 2-3 YEARS 2-3 YEARS
TOTAL CRANES
SURVIVAL Ho. CRANES
CRANES 3-4 YEARS 3-4 YEARS
DENSITY~
SURVIVAL No. CRANES
INDEPENDENT = i ee, OE
AGE-SPECIFIC CRANES 4-5 YEARS 4-5 CEOS \
SURVIVE 9 —nkorunmanunargeanvonicia sere
~ be
Neen eens vote eee ond L CRANES aT __J
y REPRODUCTIVE AGE
SURVIVAL wp Wo. CRANES #
CRANES 22-23 YEARS 22-23 YEARS
!
SURVIVAL No. CRANES
—
CRANES 23-24 YEARS 23-24 YEARS
DENSITY-DEPENDENT
SURVIVAL FUNCTION
|
Figure 2. Flowchart of the Common Crane population model.
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 307
nature cranes; and PRD = productivity, or annual recruitment
rate.
GOO = NAC
where GOO = number of cranes between 0 and 1 year old.
0, reproduction is simplified in the present model to a
single process that depends only on the breeding population
and their productivity expressed as recruitment of young
birds to the winter population. The model ignores hatching
and fledging success, for which no data are available.
The subsequent age-classes or levels are calculated as follows:
For I = 2 to 24, G(1) = G(I-1) * S(I-1)
where I = age-class, or level; G(I) = number of cranes of
age I; and 3(1) = survival rate of cranes at age I.
These density-independent survival rates are assumed to
be different for each age-class considered (see above, “Demo-
graphic parameters") ard each age-specific value is also
affected by 4 reverse logistic function that accounts for
the effect of population density on it: the survival rate
declines at high population levels:
(1 - TM(T)) * (12 - PTMM)
ak © (eToraL ~ GPrx)/1000)
S(t) = PTMM * (1-TH(T)) + =
Le
where S(I) = survival rate of cranes at age 1; PTMM = propor-
tion between minimal and maximal assumed survival rates,
i.e. lower and upper asymptotes of the logistic function;
va (1) density-independent mortality at age I, or natural
rate of deaths duc to accidents, predation, etc.; A = parameter
regulates the rate of decline of the logistic curve
AL = total number of cranes in the population; and GPIX
number of cranes at the inflection point of the curve.
Recruitment rates are also assumed to be density-dependent,
varying from high values at low population densities to
low values as the population grows:
TMP
PRD - -- -
1 ge (8B * (GTOTAL - GPIX)/1000)
where PRD > recruitment rate of the population; TMP = maximal
theoretical recruitment rate (see above, “Demographic parame-
ters"); B = parameter that regulates the rate of declin
and GTOTAL and GPL[X as in the preceeding equation.
This recruitment rate was applied only to sexually mature
birds, i. e. 4 years old.
Each simulation was started with an initial population of
20000 cranes. This initial population was assumed to have
an age-distribution identical to that calculated for the
Whooping Crane by Binkley & Miller (1983), The model was
then run for 50 years.
The initial values of the parameters A and B were estimated
from the figures and equations given by Johnson (1979).
Later, combinations of both parameters between 0.01 and
0.5 were tested. As these parameters govern the form of
the logistic curve, it is virtually impossible to determine
exactly their values with the data available at present.
PTMM regulates the effect of population density on surviva
high values of this parameter indicate low density-dependence
308 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY, SEVILLA, OCTOBER, 1986
of survivorship. We tested values of 0.5 to 0.9.
Yor GPIX, various values around the real size of the erin
population have been considered (35000-60000), mas ine
that the latter is stable,
the values for TMP were estimated by four different ways
(a) If we assume that each mature bird pairs
and each pair produces 2 chicks that would survive until
their first winter, then from 100 cranes, 63.7 birds would
be QB 4 years old (following Binkley & Miller 1943) and
could produce a maximum of 63.7 young, thet would represent
39.1% of the next winter population.
{b) If instead of 2 young per reproductive pair we consider
our field average of 1.35 young per pair, the maximum possible
percentage of juveniles would be 30.i %, also assuming that
all mature birds breed.
{c) One may also assume that the annual recruitment observed
for the Whooping Crane represents a maximal value, provided
that the small population of this endangered species (18
birds in 1938, increasing up to 78 in 1980) should stay
at the left extreme of the density-dependent inverse logistic
function of recruitment. The average for these 43 years
(1938-80) and this species is 14.5 % juveniles in the winter
population.
{d) We could also consider the maximal percentage of juveniles
observed for the Whooping Crane, which was 31.8 % in 1939.
We think, however, that possibility (d) may be influenced
by the stochastic nature of the breeding process and should
be therefore considered as an exceptionally high value,
not representative for TMP in our deterministic ‘model, Possibi-
lity (a) is also hardly representative, as it does not account
for natural, density-independent losses of eggs and young
up to the first winter, moment of the simulation start.
Therefore, we have used as the most realistic values for
TMP those comprised between 0.15 and 0.30.
in spring
RESULTS
We-have tried a total of 221 combinations of the parameters
of the survival and recruitment functions, and have selected
those that yield (a) stabilized populations between 35000
and 45000 cranes, and (b) with percentages of juveniles
of 13.54 + 2.27 ( = mean of the seven annual values measured
in the field + 95% confidence interval ) (Table 2).
DISCUSSION
The model developed may seem mathematically too simple.
This simplicity is mainly a consequence of the scarce data
available on the demography of crane populations, particularly
of the Common Cranc. We are thus unable to narrow the variation
margins of the multitude of possible combinations of these
parameters sufficiently. Therefore, we must consider a relati-
vely large series of plausible combinations of survival
and recruitment curves that are consistent with the initially
accepted conditions of population size and percentage of
juveniles ( (a) and (b), see Results).
A further limitation of the model is its deterministic nature,
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 309
Table 2. Values of the model variables that yield stabilized
population sizes and age ratios similar to those observed
in the field.
= GPIX
Trial No. PIMM = K tigo0 TPB GTOTAL —% JUV
4. 0.5 0.2 50 0.2 0.2 37200 13.20
2 0.5 0.2 50 0.2 0.15 36400 12.75
3 0.7 0.15 50 0.2 0.15 38875 12.86
4 0.7 0.15 50 0.2 0.2 36850 13.27
5 0.7 0.15 50 0.2 0.3 37780 13.72
6 0.7 0.2 50 0.2 0.15 38440 12.39
7 0.7 0.2 50 0.2 0.2 39400 12.87
8 0.7 0.2 500.2 0.3 40380 13.43
9 0.8 0.15 50 0.2) 0.15 38000 12.50
10 0.8 0.15 50 0.2 0.2 39100 12.91
a 0.8 0.15 50 0.2 0.3 40380 13.43
12 0.8 0.2 50 0.2 0.15 39900 12.06
13 0.9 0.15 50 0.2 0.15 40900 11.79
49.5 Fi 3 E 1
ig. 3. Examples
3 aa.s FAA IAL or” camatattone
8 YANN “> Cerset number et
37.5. ye (ASS ARARASRAR RAR AE 63 right side of the
x diagrams) showing
331.5 the effect of chan
Py | ging parameter PTMM
2 25.5 from 0.7 (trial no.
2 i 7, accepted as one
19.5 of the best simula
10 20 40 50 tions tested, see
YEARS Table 2) to 0.5
: (trial no. 63) or
16.0 (trial no. 51) in
I the size of the
crane population
4 (left diagram),
63 and the percentage
ae Cott 7 oof juveniles (right
diagram).
Sl
POPE eer rer
10.0
10 20.~—«-30 40 50
310 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
70
Fig. 4. As in Fig. 3
° ! examples showing the
oe 86 effect of changing
a \ | Parameter TMP from
wg G2 7 7 0.2 (trial no. 7) to
a } i | | | | 38 0.1 (trial no. 41)
a 28 { [ ] Wi / i or 0.3 (trial no. 38.
é
ei
2
aR UNPAUAT .
10 20 40 50
YEARS
50
wn 40
3 t
wm 30
2 38
5 20
2 iY ty fot,
we (10 oa aL
°
10 20 30 40 50
YEARS
44
Fig. 5. As in Fig. 3
S 39 7 examples showing the
8 effect of changing
can parameter GPIX from
x 50000 (trial no. 7)
4 29 17 to 40000 (trial no.
Pt 17)
a "
& 24
oS
19
10 20 30 40 50
YEARS
16.0
wm 14.8
ra
a
13.6 +
a on % 47
212.4
5
5
ye 11.2
10.0
10 20 30 40 50
YEARS
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 311
As claimed by Miller et_al. (1972) and Johnson (1979), stochas-
tic variability of biological processes related with crane
demography cannot be modeled due to the lack of information
on causal intercorrelations between environmental and popula-
tional variables.
Nevertheless, the model remains interesting and valid as
a description of the population dynamics of this potentially
threatened species, provided the difficulty of significantly
improving the accuracy of most demographic parameters in
anear future. It helps, as well, discover the kind of data
that need further study most urgently, in order to try to
manage the species properly. Although a sensitivity analysis
remains to be done, the study of varying the model parameters
already suggests some interesting preliminary conclusions.
The value given to parameter PTMM, that governs the density-
dependence effect on survival, depends entirely on the author
criterion. So, the models of Miller et al. (1972) and Johnson
(1979) differ in the relative importance of this parameter.
In our model, as the value of PTMM increases, the population
(GTOTAL) and the % juveniles present higher oscillations,
the first around higher than normal values and the second
around lower values (Fig. 3). Thus, a decrease in the relative
importance of the density-dependence (= increase of .PTMM)
reduces, as expected, the self-regulating ability of the
population. The best value of PTMM is 0.7 (Table 2 and Fig. 3),
and 20% changes around this value determine changes of only
approximately 5% in the size of the population and 3% in
the percentage of juveniles.
Values of TMP lower than 0.10 determine the extintion of
the population in less than 20 years, for every numerical
combination of the other four parameters (Fig. 4). On the con-
trary, values higher than 0.20 determine high oscillations of
the population size and percentage of juveniles, It is interes-
ting to observe tha all values of annual recruitment measured
in the field in the crane population and the most realistic
values inferred from our species as well as from other crane
species fall within these limits. This suggests similar
demographic structures in the various crane species for which
populational data are available. Also interestingly, the
annual recruitment values measured are closer to the lower
limit than to the higher of those inferred, perhaps suggesting
a conservative behaviour of the crane population in the
selection of the optimal productivity rate.
The parameter GPIX directly affects the stabilization level of
the population size. Changes in the parameter between 40000 and
50000 change GTOTAL in the same sense and by identical magnitu-
des, but do not alter the % juveniles (Fig. 5). Higher values
of GPIX determine very high oscillations of GTOTAL and % juve-
niles. Although we have used the same GPIX for both survival
and recruitment functions, if different points are used
in the same simulation, the results are different,
Changes in parameters A and B change GTOTAL and % juveniles
in the same sense, although by lower magnitude.
CONCLUSIONS
We have developed a simple model that serves as an appropriate
instrument for the description of the Common Crane population
312 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
dynamics. The series of curves obtained do not exclude the
possibility that other combinations exist. This model may
be considered as a first attempt, with only preliminary
results. The main limitations derive from the difficulty
of obtainingcertain demographic parameters. The main advantage
is the possibility of generating hypothesis and undertaking
“natural experiments" on this basic model (determination
of optimal population size depending on carrying capacity
of breeding and wintering areas, predicting future trends
of the population, etc.) and readjusting it to mew data.
A sensitivity analysis is necessary to determine the relative
importance of the parameters used. The possibility remains
of future incorporation of submodels that account for the
stochastic variability due to natural events or human activi-
ties. The model presented could provide a basis for future
research as well as guidelines for the management and conserva—
tion of the: Common Crane in W-Europe.
ACKNOWLEDGEMENTS
We sincerely thank J.S. Martinez-Vicente for writing the
program and discussing the results, significantly contributing
at different stages of the work, Financial support while
in the field was obtained from ICBPS the MOPU, and CAYCIT
Project No. 22107-01.
REFERENCES
Alerstam, T. & Bauer, C.A. (1973).A radar study on Crane
(Grus grus) spring migration over the Southern Baltic
area. Vogelwarte 27:1-16.
Alonso, J.A, & Alonso, J.C. (in press), Demographic parameters
of the Common Crane Grus g. grus population wintering
in Iberia. Aquila.
Alonso, J.C., Alonso, J.A. & Cantos, F.J. (1986). On the
size of the Common Crane Grus grus population migrating
through Western Europe. Ornis Fennica 63.
Binkley, C.S. & Miller, R.S. (1980). Survivorship of the
Whooping Crane, Grus americana. Ecology 61: 434-437.
Binkley, C.S. & Miller, R.S. (1983). Population charachteris-
tics of the Whooping Crane, Grus americana. Can. J.
Zool, 61: 2768-2776.
Botkin, D.B. & Miller, R.S. (1974). Mortality rates and
survival of birds. Amer. Nat. 108: 181-192.
Deevey, E.S. Jr. (1947). Life tables for natural populations
of animals. Quart. Rev. Biol. 22: 238-314.
Johnsgard, P.A. (1983). Cranes of the World. Croom Helm.
London.
Johnson, D.H. (1979). Modeling Sandhill Crane population
dynamics. U.S. Fish Widl. Serv., Spec. Sci. Rep. Wildl.
222 (10 pp.).
Johnson, D.H. (1982). Population Modeling for Furbearer
Management. In G.C. Sanderson (Ed.), Midwest Furbearer
Management. North Central Section, Central Mountains
and Plains Section, and Kansas Chapter of the Wildlife
Society, pp 25-37.
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Keil, W. (1970). Untersuchungen fiber den Zug des Kranichs
von Herbst 1966 bis Frithjahr 1970. Emberiza 2: 49-60.
Lack, D. (1966). Population Studies of Birds. Clarendon.
Oxford. (341 pp.).
Miller, R.S. (1978). Population modeling as an aid to designing
management programs. In S.A. Temple (Ed.), Endangered
Birds: Management Techniques for Preserving Threatened
Species. The Univ. of Wisconsin Press, Madison, pp.413-
417.
Miller, R.S. & Botkin, D.B. (1974), Endangered species:
models and predictions. Amer. Sci. 62: 172-181. .
Miller, R.S., Botkin, D.B. & Mendelsshon, R. (1974). The
Whooping Crane (Grus ina) population of America.
Biol. Conserv. 6: 106-111.
Miller, R.S. Hochbaum, G.S. & Botkin, D.B. (1972). A simulation
model for the management of Sandhill Cranes. Yale Univ.
Sch. For. Environ, Bull. 80: 49 pp.
Prange, H. (1984). Der Kranichzug in Thiiringen und seine
Einordnung in die Mitteleuropdische Flugroute. Thi.
Orn, Mitt. 32: 1-16.
Walkinshaw, L.H. (1973). Cranes of the World. Winchester
Press, New York.
314 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
APPENDIX I, PROGRAM
##*MODELO GRULLAS-S##*
*#4CODIGO DE VARIABLES#**
GOO=NUMERO DE GRULLAS ENTRE
GO1=NUMERO DE GRULLAS ENTRE
GO2=NUMERO DE GRULLAS ENTRE
GO3=NUMERO DE GRULLAS ENTRE
GO4=NUMERO DE GRULLAS ENTRE
GOS=NUMERO DE GRULLAS ENTRE
GOS=NUMERO DE GRULLAS ENTRE
GO7=NUMERO DE GRULLAS ENTRE
GOB=NUMERG DE GRULLAS ENTRE ANOSCINDIVIDUOS> ,N, HOF
GO9=NUMERO DE GRULLAS ENTRE 9 @ ANOSCINDIVIDUDS) ,N #10
G10=NUMERO DE GRULLAS ENTRE 10 Y 11 ANOSCINDIVIDUOS) ,N, #11
G11=NUMERO DE GRULLAS ENTRE 11 Y 12 ANOS INDIVIDUOS) ,N,#12
GI2=NUMERO DE GRULLAS ENTRE 12 Y 13 ANOS<INDIVIDUOS) ,N,#13
GI3=NUMERO DE GRULLAS ENTRE 13 Y 14 ANOSCINDIVIDUOS) ,N,#14
GI4=NUMERO DE GRULLAS ENTRE 14 Y 15 ANOSCINDIVIDUOS) ,N,H15
GIS=NUMERO DE GRULLAS ENTRE 15 ¥ 16 ANOS¢INDIVIDUOS) ,N, #16
G16=NUMERO DE GRULLAS ENTRE 16 Y 17 ANOSCINDIVIDUOS) ,N,#17
B17=NUMERO DE GRULLAS ENTRE 17 Y 18 ANOS¢INDIVIDUDS) ,N, #18
GIG=NUMERO DE GRULLAS ENTRE 18 ¥ 19 ANOS<INDIVIDUOS) ,N, #19
G19=NUMERO DE GRULLAS ENTRE 19 Y 20 ANOS¢INDIVIDUOS) ,N,#20
G20=NUMERO DE GRULLAS ENTRE 20 Y 21 ANOS INDIVIDUOS) ,N,#21
G21=NUMERO DE GRULLAS ENTRE 21 Y 22 ANOSCINDIVIDUOS) ,N,#22
G22=NUMERO DE GRULLAS ENTRE 22 Y 23 ANOSCINDIVIDUOS) ,N, #23
G23=NUMERO DE GRULLAS ENTRE 23 ¥ 24 ANOS(INDIVIDUOS) ,N,#24
SOO=TASA DE SUPERVIVENCIA DE GOG<TANTG POR 1),F,#25
SO1=TASA DE SUPERVIVENCIA DE GO1<TANTO POR 1) ,F,#26
SG2=TASA DE SUPERVIVENCIA DE G02¢TANTO POR 1),F,#27
SO3=TASA DE SUPERVIVENCIA DE GO3(TANTO POR 1) ,F,#28
SO4=TASA DE SUPERVIVENCIA DE GO4(TANTO POR 1),F, #29
SOS=TASA DE SUPERVIVENCIA DE GOS<(TANTO POR 1),F #30
SO¢=TASA DE SUPERVIVENCIA DE GOS(TANTO POR 1),F, #31
SO7=TASA DE SUPERVIVENCIA DE GO07¢TANTO POR 1) ,F,#32
SOB=TASA DE SUPERVIVENCIA DE GOS¢TANTO POR 1),F, #23
SO9=TASA DE SUPERVIVENCIA DE GO9<TANTO POR 1),F,834
SIG=TASA DE SUPERVIVENCIA DE G10(TANTO POR 1),F,#35
SLI=TASA DE SUPERVIVENCIA DE G11(TANTO POR 1),F,#36
S12=TASA DE SUPERVIVENCIA DE G12(TANTO POR 1),F,#37
SI3=TASA DE SUPERVIVENCIA DE G13¢TANTO POR 1) ,F, #36
S14=TASA DE SUPERVIVENCIA DE Gi4<(TANTO POR 1),F, 439
S15=TASA DE SUPERVIVENCIA DE G1S(TANTO POR 1),F,#40
S16=TASA DE SUPERVIVENCIA DE G1é(TANTO POR 1),F, #41
Si7STASA DE SUPERVIVENCIA DE G17(TANTO POR 1),F,#42
SIS=TASA DE SUPERVIVENCIA DE G18(TANTO POR 1),F,#43
S19=TASA DE SUPERVIVENCIA DE Gi9(TANTO POR 1) ,F, #44
SZ0=TASA DE SUPERVIVENCIA DE G20CTANTO POR 1),F, #45
S21=TASA DE SUPERVIVENCIA DE G21(TANTO POR 1) ,F #46
S225TASA DE SUPERVIVENCIA DE G22¢TANTO POR 1),F #47
Y 1 ANOCINDIVIDUOS) ,N,HO1
YY 2 ANOSCINDIVIDUDS) ,N, #02
Y 3 ANOSCINDIVIDUOS) ,N, #03
Y 4 ANOSCINDIVIDUDS) ,N,#04
Y 5 ANOS< INDIVIDUOS) ,N, #05,
Y 6 ANOS<INDIVIDUOS) ,N, #06
Y 7 ANOS(INDIVIDUOS) ,N, #07
Y 8 ANOSCINDIVIDUOS) ,N, #08
Yo
WL
aVauaenHo
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
5510
5520
5530
5540
5550
5560
5570
5580
5590
5600
5610
5626
5630
5640
5450
5660
5670
5680
5690
5708
5710
5720
5730
5740,
5750
5760
5770
57380
5790
Seog
5810
5820
5830
Seat
5850
5860
5870
5820
5890
5900
5910
5920
5930
5940
5950
5960
5970
3920
5990
4000
6010
4020
6030
04g
6050
REM
REM
8
REM
REM
REM
REM
REM
REM
REM
REM
REM
LA TASA DE SUPERVIVENCIA DE 623 ES 0!
GTER=TOTAL DE GRULLAS EN EDAD DE REPRODUCIRSE( INDIVIDUOS) ,A, #4
PRD=PRODUCTIVIDAD A, #49
GTOTAL=NUMERO TOTAL DE GRULLAS(INDIVIDUOS) ,A, #50
JZ=PORCENTAJE DE GRULLAS JOVENES(TANTO POR 1),A,#51
##2CODIGO DE PARAMETROS 0 TASAS¥#*
PTMM=PROPORCION DE LA FUNCION DE SUPERVIVENCIA, #01
A=PARAMETRO DE LA FUNCION DE SUPERVIVENCIAC..) ,#02
GPIX=TAMANO DE LA POBLACION EN EL PUNTO DE INFLEXION¢INDIVIDUO
Ss), #03
REM
REM
REM
REM
REM
REM
REM
REM
REM
REM
REM
REM
REM
REM
REM
REM
REM
REM
TMPSTASA MAXIMA DE PRODUCTIVIDAD(TANTO POR 1),#04
B=PARAMETRO DE LA FUNCION DE PRODUCTIVIDAD<..) #05
TMOO=TASA
TMO1=TASA
TMO2=TASA
TMO3=TASA
TMO4=TASA
TMOS=TASA
TMO6=TASA
TMO7=TASA
TMO8=TASA
TMO9=TASA
TM10=TASA
TMi 1=TASA
TMi 2=TASA
TM13=TASA
TMI 4=TASA
TM15=TASA
TMI 6=TASA
TM17=TASA
TMI 8=TASA
TM19=TASA
TM20=TASA
TM21=TASA
TM22=TASA
#e2S1STEMA
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
OE
DE
DE
DE
OE
G02=G01 #801
03=G02*502
GN4=G03*S503
G05=G04*S04
GOS=G054S05
GO7=6046*S06
MORTALIDAD
MORTALIDAD
MGRTALIDAD
MORTALIDAD
MORTAL I DAD
MORTAL1 DAD
MORTALIDAD
MORTALIDAD
MORTALIDAD
MORTALIDAD
MORTALIDAD
MORTAL IDAD
MORTALIDAD
MORTALIDAD
MORTAL IDAD
MORTAL IDAD
MORTALIDAD
MORTALIDAD
MORTALIDAD
MORTALIDAD
MORTALIDAD
MORTALIDAD
MORTALIDAD
ECUACIONES
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
DE
G00(TANTO
G01 (TANTO
GO2(TANTO
GO3<¢ TANTO
G04 (TANTO
GOS¢TANTO
GOS¢TANTO
GO7< TANTO
GOBCTANTO
GOOKTANTO
GIO<TANTO
G11 (TANTO
GI2¢TANTO
G13¢TANTO
G14¢TANTO
GIS(TANTO
G16< TANTO
G1 7(TANTO
GIB(TANTO
G19¢TANTO
620¢TANTO
G21 (TANTO
G22¢TANTO
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
POR
ANALITICASES*
1), #08
1) 807
1) ,808
1) ,#O9
1) #10
1,811
1),#12
1) #13
10,814
1) H15
1) 816
1) #17
1) ,H18
1) HI?
1),820
4) ,#21
4), #22
1) 4823
1) 5824
1) ,825
1) #26
1) 827
1), #28
315
316 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
6060 REM G1é=615*S15
6070 REM G17=G616%516
6080 REM G18=G17*S17
6090 REM G19=G18x518
6100 REM 620=G19*S19
6110 REM G2i=620%520
6120 REM G22=G21%S21
6130 REM G23=G622%522
6140 REM — SOO=PTMM*(1-TMOO)+((1-TMO0) #¢1-PTMM)/¢ 1 +EXP<AR<GTOTAL~GPIX2/1
G00)))
6150 REM — SO1=PTMM*¢1~TMO1)+¢¢1-TMO1)*(1-PTMM)/( 1 +EXP(A®(GBTOTAL-GPIX)/1
000)>)
6160 REM — SO2=PTMM¥C 1-TMO2)+( (1-TMOZ)*(1-PTMM)/(1+EXPCAXCGTOTAL-GPIXI/1
6170 REM | SOS=PTMM¥(1-TMO3)+¢C1-TMO3) #¢1-PTMM)/(1 4EXP(AX(GTOTAL-GPIX)/1
4180 REM — SQ4=PTMMx(1-TM04)+¢¢1-TMO4) #(1-PTMM)/(1+EXP(AX(GTOTAL-GPIX)/1
6190 REM — SQS=PTMMx(1-TMOS)+<(1-TMOS)#¢1-PTMM)/< 1 +EXPCA®(GTOTAL~GPIX)/1
6200 REM — S0=PTMM¥¢1-TMO4)#¢(1-7M0.4)#(1-PTMM)/ (1 +EXPCAX(GTOTAL-GPIX)/1
210 REM — SO7=PTMM*(1-TMO7)+¢(1-TMO7)¥(1-PTMM)/¢ 1 4EXPCAX(GTOTAL-GPIX)/1
6220 REM — SOB=PTMMx< 1 -TMOB) +4 (1-TMOB) #(1-PTMM) /( 1 +EXPCAX(GTOTAL-GPIXO/1
6230 REM = SO9=PTMM#4 1-TMO9) +6 (1 +TMO9) ¥¢ 1-PTMM)/¢ 1 +EXPCAR(GTOTAL-GPIX)/1
6240 REM = S1Q=PTMM¥(1~TM10)40¢1-TM10)*(1-PTMM)/¢ 1+EXP(AX(GTOTAL-GPIX)/1
6250 REM Si 1=PTMM#C1-TM11)+(C1-TM11)8¢1-PTMM)/¢ 1 EXPCAR(GTOTAL-GPIX)/1
6260 REM $1 2=PTMM#¢1~-TM12)+¢(1-TM12) (1 -PTMM)/( 1 +EXP(AX(GTOTAL-GPIX)/1
4270 REM $4 3=PTMM#( 1-TM1304¢.C1-TM13) 81 -PTMMD/(14EXP(A#( BTOTAL-GPDO/1
$280 REM 814=PTMM¥C 1-TM14)4¢C1-TM14) *¢ 1 -PTMM)/(1+EXP(A®(GTOTAL-GPIX)/1
6290 REM S{S=PTMM®C1-TM15) + C1-TM15) #¢1-PTMM)/ (1 +EXP(A<GTOTAL-GPIX)/1
6300 REM S14=PTMM¥<1-TM14)+¢¢1-TM16) (1 -PTMM)/ (1 4EXPCAX(GTOTAL-GPIXI/1
4310 REM — S17=PTMM¥¢1~TM17)+¢(1-TM17) #¢1-PTMM)/(1+4EXPCAX(GTOTAL-GPIX)/1
6320 REM S{8=PTMM#<1-TM18) +¢(1-TM18) *(1-PTMM)/( 1 EXPCAX(GTOTAL~GPIX)/1
6830 REM $19=PTMM¥C1-TM19)4¢C1-TM19) #41 -PTMM)/( 1 4EXP(AX(GTOTAL~GPIX)/1
6340 REM «© $20=PTMM#(1-TM20)4¢¢1-TM20) (1 -PTMM)/(1 4EXPQA#(GTOTAL-GPIX)/1
4350 REM S21=PTMM#(1-TM21)+(C1-TM21)¥(1-PTMM)/( 1 #EXP£AX(GTOTAL-GPIX)/1
6860 REM — S22=PTMM#<1-TM22)+¢(1-TM22) #1 -PTMM)/(1+EXPCAX(GTOTAL-GPIX)/1
000)9)
‘THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
6370 REM — PRD=(TMP/< 1 +EXP(BXCGTOTAL-GPIX)/1000)))
6380 REM GTOTAL=GTER+G00+G01+602
6390 REM
7000 REM #*xSISTEMA DE ECUACIONES MDS*x*
7010 IF T< > TI THEN 7030
7020 GOTO 7070
7030 VN(1) = INT <UN(48) * VN(49) + .5)
7040 FOR I = 1 70 23
7050 VN(I + 1) = INT <UNCID * VNC24 + 1D + 5D
7060 NEXT I
7070 YNC48) = 0
7080 FOR I = 4 TO 24
7090 VN¢48) = UNC48) + YNCT?
7100 NEXT
7105 YN(S0) = VN¢4B) + UNC1) + YNC2) + WNC3)
7110 FOR 1 = 1 TO 23
7120 VN¢I + 24) = TAC1) * (1 - TAC] + 5)) # (1 = TAC] + 59) # (1 - TACEDD
/ 1 + EXP (TAC2) # CCYNC5O2 = TAC3)> / 100009)
7130 NEXT I
7140 YN649) = TAC? 7 C1 + EXP CTACS) * CCVNCSO) ~ TAC3)? / 1000)))
7160 VNCS1) = INT (<UNCL) 7 YN(50) + .00005) * 100009 / 190
8000 RETURN 3
30000 REM
30010 REM ***DATOS ESTRUCTURALES#*
30020 REM
30030 DATA 0001 ,0050,1,1
30040 DATA 24,28,51
30050 REM
30060 REM *#*CONDICIONES INICIALES=x*
30070 REM
30080 DATA 2700,1620,1520,1420
30090 DATA 1340,1240,1140, 1080
30100 DATA 1000,0920,0840,0760
30110 DATA 0480,0620,0540, 0480
30120 DATA 0420 0360 ,0200 0240
30130 DATA 0200,0140,0092,0044
30140 REM
30150 REM **#VALORES DE PARAMETROS G TASAS#=*
30160 REM
30170 DATA .?
20180 DATA .2
30190 DATA 4a000
30200 DATA .20
30210 DATA 42
30220 DATA .3750,.0183,.0210,.0238
30230 DATA .0222, .0347,.0344, .0376
30240 DATA .0416,,0464,.0517,.0576
20250 DATA .0647,.0725, 0826, 0939
30260 DATA .1090,.1270,.1510,,1860
20270 DATA .2370,.3220,.4910
317
318 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
31000 REM
31010 REM *#*SIMBOLOS DE VARTABLES##x
31020 REM
31030 DATA G00,G01,602,603,604
31040 DATA G05,G06,G07,608,G09
B1050 DATA G10,G11,G12,613,614
31060 DATA G15,G15,617,618,G19
B07 DATA G20,G21,622,623
31080 DATA $00,S01,802,803,S04
31090 DATA $05,$06,S07,808,809
31100 DATA $10,911,512,813,814
BI110 DATA $15,816,$17,818,319
31120 DATA $20,S21,S22
BI130 DATA GTER,PRD,STOTAL
31135 DATA JX
31140 DATA PTMM,A,GPIX,1MP,B
31150 DATA TMOG,TMO1,7M02,7M03,7M04
31160 DATA TMOS,TM0é,TMO7,TMO8, TMOF
31170 DATA TMI0,TMi1,TM12,7Mi3,1Mi4
Bl1ga DATA TM1S,TMI6,7M17,TMI8,TMi9
31190 DATA TMZ0,7M21,1M22
