Age structured populations
Alfred James
Lotka (18801949)
Vito Volterra
(1860-1940)
First steps in life tables
Fecundity
Age
0
10
20
30
40
50
60
70
80
90
100
N
1000
600
700
595
446
290
180
200
90
50
3
f
0
0
0.1
1.2
0.6
0.3
0.05
0
0
0
0
Mortality Survival
rate
rate
d
0.15
0.09
0.15
0.25
0.35
0.38
0.42
0.55
0.75
0.95
1
l
0.85
0.91
0.85
0.75
0.65
0.62
0.58
0.45
0.25
0.05
0
• N0 is the number of newborns.
• N is the number of females per age
cohort.
• Fecundity f is the average number of
offspring per female.
• d is the mortality rate per cohort.
• l is the fraction of survivors per
cohort.
Pivotal
age
Age
t
0
0
10
5
20 15
30 25
40 35
50 45
60 55
70 65
80 75
90 85
100 95
110 105
120 115
Number of
deaths at
each age
classage
N
1000
850
835
812
777
737
661
551
270
167
7
1
0
The pivotal age is the averge age
per age cohort class
D
d
l
150
15
23
35
40
76
110
281
103
160
6
1
0.150
0.018
0.028
0.043
0.051
0.103
0.166
0.510
0.381
0.958
0.857
1.000
0.850
0.982
0.972
0.957
0.949
0.897
0.834
0.490
0.619
0.042
0.143
0.000
𝐷𝑥
𝑑𝑥 =
𝑁𝑥
Mortality rate
The basic information needed is the
total number of deaths per age cohort.
survival
mortality
𝑁𝑥
𝑙𝑥 = 1 − 𝑑𝑥 =
𝑁𝑥−1
survival rate
First steps in life tables
Fecundity
Age
0
10
20
30
40
50
60
70
80
90
N0
1000
600
700
595
446
290
180
200
90
50
f
0
0
0.1
1.2
0.6
0.3
0.05
0
0
0
100
3
0
Initial age
distribution
𝑁1 10 = 𝑁0 (0) × 𝑙(0)
Death Survival
rate
rate
d
l
N1
0.15 0.85 1148
0.09 0.91
850
0.15 0.85
546
0.25 0.75
595
0.35 0.65
446
0.38 0.62
290
0.42 0.58
180
0.55 0.45
104
0.75 0.25
90
0.95 0.05
23
=E27*
1
0
H27
Age
distribution of
the next
generation
N1(0)
0
0
70
714
268
87
9
0
0
0
=+E28*F28
Population size of each
cohort after
reproduction
𝑁1 0 =
𝑁𝑖 × 𝑓𝑖
If the population is age structured and contains k age classes we get
Fecundities
Survival
rates
Age 0 10 20 30 40 50 60 70 80 90 100
0 0 0 0.1 1.2 0.6 0.3 0.05 0 0 0 0
100.85 0 0 0 0 0 0 0 0 0 0
20 0 0.91 0 0 0 0 0 0 0 0 0
30 0 0 0.85 0 0 0 0 0 0 0 0
40 0 0 0 0.75 0 0 0 0 0 0 0
50 0 0 0 0 0.65 0 0 0 0 0 0
60 0 0 0 0 0 0.62 0 0 0 0 0
70 0 0 0 0 0 0 0.58 0 0 0 0
80 0 0 0 0 0 0 0 0.45 0 0 0
90 0 0 0 0 0 0 0 0 0.25 0 0
100 0 0 0 0 0 0 0 0 0 0.05 0
N0
1000
600
700
595
446
290
180
200
90
50
3
N1
1148
850
546
595
446
290
180
104
90
23
3
Leslie matrix
k
N 0 (1)  b1 N1 (0)  b2 N 2 (0)  ...  bk N k (0)   bk N k (0)
i 1
N0(0) = 1000
N0(1) = 1148
595*0.75=446
The mutiplication of the abundance vector with each row of the Leslie matrix gives the
abundance of the next generation.
Leslie matrix
We have w-1 age classes, w is the maximum age of an individual.
L is a square matrix.
 n0 


 n1 
N t   n2 


 ... 


n
 w 1 
Nt 1  LN t
 f0

 s0
0
L
0
 ...

0

f1
0
s1
0
...
0
f2
0
0
s2
...
0
f3
0
0
0
...
0
...
...
...
...
...
sw  2
fw 1 

0 
0 

0 
0 
0 
N t 1  Lt N 0
Numbers per age class at time t+1 are the dot product of the Leslie matrix with the
abundance vector N at time t
The Leslie model is a linear approach.
It assumes stable fecundity and mortality rates
Going Excel
Age 0 10 20 30 40 50 60 70 80 90 100
0 0 0 0.1 1.2 0.6 0.3 0.05 0 0 0 0
100.85 0 0 0 0 0 0 0 0 0 0
20 0 0.91 0 0 0 0 0 0 0 0 0
30 0 0 0.85 0 0 0 0 0 0 0 0
40 0 0 0 0.75 0 0 0 0 0 0 0
50 0 0 0 0 0.65 0 0 0 0 0 0
60 0 0 0 0 0 0.62 0 0 0 0 0
70 0 0 0 0 0 0 0.58 0 0 0 0
80 0 0 0 0 0 0 0 0.45 0 0 0
90 0 0 0 0 0 0 0 0 0.25 0 0
100 0 0 0 0 0 0 0 0 0 0.05 0
Demographic
low
N0
1000
600
700
595
446
290
180
200
90
50
3
N1
1148
850
546
595
446
290
180
104
90
23
3
N2
1132
976
774
464
446
290
180
104
47
23
1
N3
N4
N5 N6
998 1183 1366 1413
963 848 1005 1161
888 876 772 915
657 755 744 656
348 493 566 558
290 226 321 368
180 180 140 199
104 104 104 81
47
47
47
47
12
12
12
12
1
1
1
1
• At the long run population size increases.
• Diagonal waves in abundances occur.
• The first age cohort increases fastest.
The effect of age in reproduction
Age 0 10 20 30 40 50 60 70 80 90
0 0 0 0.1 1.2 0.6 0.3 0.05 0 0 0
10 0.85 0 0 0 0 0 0 0 0 0
20 0 0.91 0 0 0 0 0 0 0 0
30 0 0 0.85 0 0 0 0 0 0 0
40 0 0 0 0.75 0 0 0 0 0 0
50 0 0 0 0 0.65 0 0 0 0 0
60 0 0 0 0 0 0.62 0 0 0 0
70 0 0 0 0 0 0 0.58 0 0 0
80 0 0 0 0 0 0 0 0.45 0 0
90 0 0 0 0 0 0 0 0 0.25 0
100 0 0 0 0 0 0 0 0 0 0.05
100
0
0
0
0
0
0
0
0
0
0
0
Age 0 10 20 30 40 50 60 70 80 90
0 0
0 0.05 0.3 0.6 1.2 0.1 0
0
0
10 0.85 0
0
0
0
0
0
0
0
0
20 0 0.91 0
0
0
0
0
0
0
0
30 0
0 0.85 0
0
0
0
0
0
0
40 0
0
0 0.75 0
0
0
0
0
0
50 0
0
0
0 0.65 0
0
0
0
0
60 0
0
0
0
0 0.62 0
0
0
0
70 0
0
0
0
0
0 0.58 0
0
0
80 0
0
0
0
0
0
0 0.45 0
0
90 0
0
0
0
0
0
0
0 0.25 0
100 0
0
0
0
0
0
0
0
0 0.05
Reproduction in early age contributes more to population size than later reproduction.
This is caused by the higher number of females in earlier cohorts.
100
0
0
0
0
0
0
0
0
0
0
0
• The effect of the initial age composition
disappears over time
• Age composition approaches an
equilibrium although the whole
population might go extinct.
• Population growth or decline is often
exponential
Age 0 10 20 30 40 50 60 70 80 90
0 0 0 0.1 1.2 0.6 0.3 0.05 0 0 0
10 0.25 0 0 0 0 0 0 0 0 0
20 0 0.91 0 0 0 0 0 0 0 0
30 0 0 0.85 0 0 0 0 0 0 0
40 0 0 0 0.75 0 0 0 0 0 0
50 0 0 0 0 0.65 0 0 0 0 0
60 0 0 0 0 0 0.62 0 0 0 0
70 0 0 0 0 0 0 0.58 0 0 0
80 0 0 0 0 0 0 0 0.45 0 0
90 0 0 0 0 0 0 0 0 0.25 0
100 0 0 0 0 0 0 0 0 0 0.05
100
0
0
0
0
0
0
0
0
0
0
0
High early death rates cause
fast population extinction
and would need high
fecundities for population
survival
Does the Leslie approach predict a stationary point where population abundances
doesn’t change any more?
𝑵𝑡+1 = 𝑅𝑵𝑡
𝑵𝑡+1 = 𝑅𝑡 𝑵0
dN
0
dt
𝑵𝑡+1 = 𝑳𝑵𝑡 = 𝑵𝑡
𝑵𝑡+1 = 𝑳𝑡 𝑵0
We’re looking for the stable state vector that doesn’t change when multiplied with the
Leslie matrix.
This vector is the eigenvector U of the matrix.
Eigenvectors are only defined for square matrices.
𝑵𝑡+1 = 𝑳𝑵𝑡 = 𝜆𝑵𝑡 = 𝑅𝑵𝑡
The largest eigenvalue l of a Leslie
matrix denotes the long-term
average net reproduction rate.
The right (dominant) eigenvector
contains the stable state age
distribution.
𝑳𝑼 = 𝜆𝑼
Important properties:
1. Eventually all age classes grow or shrink
at the same rate
2. Initial growth depends on the age
structure
3. Early reproduction contributes more to
population growth than late
reproduction
Leslie matrices in insect populatons
Age Eggs Larva 1 Larva 2 Larva 3 Imago
Eggs 0
0
0
0
2000
Larva 1 0.25
0
0
0
0
Larva 2 0
0.15
0
0
0
Larva 3 0
0
0.15
0
0
Imago 0
0
0
0.1
0
Age Eggs Larva 1 Larva 2 Larva 3 Imago
Eggs 0
0
0
0
200
Larva 1 0.25
0
0
0
0
Larva 2 0
0.15
0
0
0
Larva 3 0
0
0.15
0
0
Imago 0
0
0
0.1
0
Largest eigenvalue
r = l = 1.02
2000 female eggs per individual are cause
a steady population increase. This relates
to 4000 eggs when including males.
Leslie matrices deal with effective
populations sizes.
Largest eigenvalue
r = l = 0.65
The population steadily
declines.
𝑵1 = 𝑳𝑵0
N0
100000
25000
3750
563
56
Eggs
11250
25000
3750
562.5
56.25
Larva 1
11250
2813
3750
562.5
56.25
Larva 2
11250
2813
421.9
562.5
56.25
Larva 3 Imago
11250 11250
2813 2812.5
421.9 421.875
63.28 63.2813
56.25 6.32813
Eggs
1265.6
2812.5
421.88
63.281
6.3281
The diagonal matrix
elements show how
many individuals
survive.
Stable age distribution
Age Eggs Larva 1 Larva 2 Larva 3 Imago
Eggs 0
0
0
0
2000
Larva 1 0.25
0
0
0
0
Larva 2 0
0.15
0
0
0
Larva 3 0
0
0.15
0
0
Imago 0
0
0
0.1
0
𝑵𝑡+1 = 𝑳𝑵𝑡 = 𝜆𝑵𝑡
The largest eigenvalue l of a Leslie matrix denotes the
long-term net population growth rate R.
The right (dominant) eigenvector contains the stable state age distribution.
U
0.970859
0.237064
U=
0.034732
0.005088
0.000497
l = 1.02
Sum
Age
Nstable
U
Eggs
0.970859 0.777782
Larva 1
0.237064 0.189919
Larva 2
0.034732 0.027825
Larva 3
0.005088 0.004077
Imago
0.000497 0.000398
1.248241
1
Stable age class distribution
For the population to
survive the number of first
instars has to be
0.189919/0.000398 = 477
time larger than the
number of imagines.
Remaining in the same age class
Age Eggs Larva 1 Larva 2 Larva 3 Imago
Eggs 0.10
0
0
0
2000
Larva 1 0.25 0.15
0
0
0
Larva 2 0
0.15 0.05
0
0
Larva 3 0
0
0.15 0.05
0
Imago 0
0
0
0.1
0.5
𝑵𝑡+1 = 𝑳𝑵𝑡 = 𝜆𝑵𝑡
The probability that an egg survives and
remaines in the egg state is 0.10
The probability that an imago survives and
reproduces in the next generation is 0.5.
This is the case in biannual insects
(for instance some Carabus)
Largest eigenvalue
R = l = 1.21
U
0.972869
0.229415
U=
0.029662
0.003835
0.00054
Sum
l = 1.21
l = 1.02
Age
Nstable
U
Eggs
0.972869 0.786907
Larva 1
0.229415 0.185563
Larva 2
0.029662 0.023992
Larva 3
0.003835 0.003102
Imago
0.00054 0.000437
1.236321
1
Stable age class distribution
Nstable
0.777782
0.189919
0.027825
0.004077
0.000398
Without
staying
the same
Sensitivity analysis
Age Eggs Larva 1 Larva 2 Larva 3 Imago
Eggs 0
0
0
0
2000
Larva 1 0.25
0
0
0
0
Larva 2 0
0.15
0
0
0
Larva 3 0
0
0.15
0
0
Imago 0
0
0
0.1
0
Age Eggs Larva 1 Larva 2 Larva 3 Imago
Eggs 0
0
0
0
2.5
Larva 1 0.95
0
0
0
0
Larva 2 0
0.91
0
0
0
Larva 3 0
0
0.93
0
0
Imago 0
0
0
0.95
0
l = 1.14
l = 1.02
High mortality, high fecundity
r strategist species
Low mortality, low fecundity
K strategist species
l > 1 → effective population size increases
How robust is l with respect to changes
in survival and fecundity rates?
l = 1.01
The lowest possible fecundity is
1.4 female eggs per female.
Age Eggs Larva 1 Larva 2 Larva 3 Imago
Eggs 0
0
0
0
1.4
Larva 1 0.95
0
0
0
0
Larva 2 0
0.91
0
0
0
Larva 3 0
0
0.93
0
0
Imago 0
0
0
0.95
0
Sensitivity analysis
Age Eggs Larva 1 Larva 2 Larva 3 Imago
Eggs 0
0
0
0
2.5
Larva 1 0.95
0
0
0
0
Larva 2 0
0.91
0
0
0
Larva 3 0
0
0.93
0
0
Imago 0
0
0
0.95
0
l = 1.14
Increasing mortality rates until the population stops increasing
Age Eggs Larva 1 Larva 2 Larva 3 Imago
Eggs 0
0
0
0
2.5
Larva 1 0.86
0
0
0
0
Larva 2 0 0.819
0
0
0
Larva 3 0
0
0.837
0
0
Imago 0
0
0
0.855
0
l = 1.05
Mortality rates might be 10% higher to remain
effective population sizes still increasing.
Survivorship tables
Number of
death
Death
rate
Survival
rate
Age
N
0 1000
10 850
20 835
30 812
40 777
50 737
60 661
70 551
80 270
90 167
100
7
110
1
120
0
D
150
15
23
35
40
76
110
281
103
160
6
1
l
d
Average Cumulative
number number
alive in a alive in a Average life
cohort
cohort
expectation
L
925
843
824
795
757
699
606
411
219
87
4
1
SL
6168
5243
4401
3577
2783
2026
1327
721
310
92
5
1
e
61.7
61.7
52.7
44.1
35.8
27.5
20.1
13.1
11.5
5.5
6.4
5.0
0.85 0.15
0.98 0.02
0.97 0.03
0.96 0.04
0.95 0.05
0.90 0.10
0.83 0.17
0.49 0.51
0.62 0.38
0.04 0.96
0.14 0.86
0.00 1.00
+H25/H +J25/H +(H25+H +SUMA(L$2 +M24/H24*
24
24
24)/2
4:L24)
$G$14
𝐿 𝑥 =
𝑁 𝑥 + 𝑁(𝑥 + 1)
2
𝑚𝑎𝑥
Σ𝐿 =
𝐿(𝑥)
𝑥
Σ𝐿𝑥
𝑒𝑥 =
𝑘
𝑁𝑥
k = length
of cohort
(10 years)
The female life table of Polish women 2012 (GUS 2013)
0.00426
0.00021
0.00015
0.00011
0.00010
0.00010
0.00009
0.00009
0.00009
0.00008
0.00009
0.00010
0.00011
0.00013
0.00016
0.00018
0.00021
0.00023
0.00024
0.00024
L
99787
99563
99544.5
99531
99520
99510.5
99501
99491.5
99483
99474.5
99465.5
99456.5
99446
99433.5
99419.5
99402.5
99383
99361.5
99338
99314
99290
SL
8095890
7996103
7896540
7796996
7697465
7597945
7498434
7398933
7299442
7199959
7100484
7001019
6901562
6802116
6702683
6603263
6503861
6404478
6305116
6205778
6106464
e
80.9589
80.30312
79.32076
78.33264
77.34202
76.34974
75.3566
74.36413
73.37081
72.37667
71.38317
70.38958
69.39591
68.40422
67.4131
66.4232
65.4358
64.44887
63.46367
62.47889
61.49387
0.260914
0.281214
0.302288
0.324249
1
3710
2634
1814
731.5
0
8889.5
5179.5
2545.5
731.5
0
2.059185
1.669191
1.175751
0.5
0
Age
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
N
100000
99574
99552
99537
99525
99515
99506
99496
99487
99479
99470
99461
99452
99440
99427
99412
99393
99373
99350
99326
99302
D
l
d
426
21
15
11
10
10
9
9
9
8
9
10
11
13
16
18
21
23
24
24
0.99574
0.99978
0.99985
0.99988
0.99990
0.99991
0.99990
0.99991
0.99992
0.99991
0.99991
0.99991
0.99988
0.99987
0.99985
0.99981
0.99980
0.99977
0.99976
0.99976
98
99
100
>100
>120
4317
3103
2165
1463
0
1524
1214
938
702
1463
0.739086
0.718786
0.697712
0.675751
0
Average life
expectancy at birth
𝐿 𝑥 =
𝑁 𝑥 + 𝑁(𝑥 + 1)
2
𝑚𝑎𝑥
Σ𝐿 =
𝐿(𝑥)
𝑥
Σ𝐿𝑥
𝑒𝑥 =
𝑘
𝑁𝑥
Polish survivorship curve 2012
Type I
Type I, high survivorship of young
individuals: large mammals, birds
Type II, survivorship independent of
age, seed banks
Type III, low survivorship of young
individuals, fish, many insects
Type II
Type III
Polish mortality
rates 2012
Newborns
New motocycle
and car drivers
Average life expectancy at birth in Poland
81 years
Women
Men
8 years
72 years
Average life expectancy at age 60 in Poland
84 years
5 years
78 years
Reproduction life tables
Age
0
10
20
30
40
50
60
70
80
90
100
110
120
Sum
Pivotal
age
t
0
5
15
25
35
45
55
65
75
85
95
105
115
Survival Number
rate offspring
N0
1000
850
835
812
777
737
661
551
270
167
7
1
0
𝑅𝑖 =
𝑙𝑖 𝑏𝑖
Net reproduction rate
R
D
l
B
b
lb
150
15
23
35
40
76
110
281
103
160
6
1
0.85
0.98
0.97
0.96
0.95
0.90
0.83
0.49
0.62
0.04
0.14
0.00
0
20
515
342
59
2
0
0
0
0
0
0
0.000
0.024
0.634
0.440
0.080
0.003
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.024
0.617
0.421
0.076
0.003
0.000
0.000
0.000
0.000
0.000
0.000
1.140
+I25/I24
𝑅0 =
Birth
rate
lbt
0.0
0.4
15.4
14.7
3.4
0.1
0.0
0.0
0.0
0.0
0.0
0.0
34.096
29.9
𝐺=
=L25/I25 +M25*K25 +N25*H25
𝑙𝑖 𝑏𝑖 𝑡𝑖
𝑙𝑖 𝑏𝑖
The mean generation length is the mean period elapsing
between the birth of parents and the birth of offspring.
It is the weighted mean of pivotal age weighted by the number
of offspring.
Species
Castor canadensis
Clethreonomys glareolus
Peromyscus leucopus
P. maniculatus
Sciurus carolinensis
Spermophilus armatus
S. beldingi
S. lateralis
S. parrylii
Tamias striatus
Tamiasciurus hudsonicus
Ochotona princeps
Sylvilagus floridanus
Lutra canadensis
Lynx rufus
Mephitis mephitis
Taxidea taxus
Equus burchelli
Aepycerus melampus
Cervus elaphus
Connochaetes taurinus
Hemitragus jemlahicus
Hippopotamus amphibicus
Kobus defassa
Ovis canadensis
Phacochoerus aethiopicus
Sus scrofa
Syncerus caffer
Loxodonta africana
Life
Age of
Life
Reproductive
Body mass
expectation
Litter size maturity expectanc
value at
(kg)
at maturity
(yr)
y (yr)
maturity
(yr)
18
0.025
0.02
0.02
0.6
0.35
0.25
0.155
0.7
0.1
0.189
0.13
1.25
7.2
7.5
2.25
7.15
270
44
175
165
100
2390
200
55
87
85
490
4000
6.6
5
5
3.6
2.9
5.3
7.4
5.2
7.3
4.2
4
2.8
5
2
2.8
6
2
1
1
1
1
1
1
1
1
4.8
5
1
1
2
0.11
0.15
0.15
1
1
1
1.3
1
1
1
1
1
3
1
1
1
4
2
4
3
3
10
2
4
2
2
4
15
1.52
0.16
0.21
0.23
1.37
1.38
1.3
1.47
1.28
1.24
1.5
1.37
1.48
2.88
1.72
1.33
1.45
3.84
3.44
4.9
3.84
3.97
7.62
3.35
3.81
1.6
1.79
4.47
17.9
2.22
0.48
0.28
0.43
2.17
1.72
1.78
2.12
1.71
1.63
2.45
2.33
1.25
3.79
2.48
1.9
2.33
7.95
4.8
3.85
4.79
4.71
16.4
5.87
5.48
2.82
1.91
4.82
19.1
Data from Millar and Zammuto 1983, Ecology 64: 631
5.63
7.9
4.52
5.04
5.95
4.52
5.89
5.08
6.17
6.84
4.9
6.51
2.62
3.79
3.48
5.71
2.48
4
2.42
1.73
2.56
2.12
3.98
2.94
2.74
6.76
4.82
2.41
2.24
Generation
length (yr)
4.87
0.33
0.27
0.35
2.07
1.78
1.56
2.45
1.59
1.59
1.95
2.07
1.29
5.07
2.87
1.78
1.24
8.74
4.36
5.7
6.29
5.43
19.82
5.08
6.52
4.28
3.15
6.98
25.8
Life history data and
body size
Reproductive value at age x
𝑚𝑎𝑥
𝑅𝑉𝑥 = 𝑓𝑥 +
𝑦=𝑥+1
𝑙𝑦
𝑓 𝑙 𝑏
𝑙𝑥 𝑦 𝑦 𝑦
Life history data are
allometrically related
to body size.
𝑥−𝑦
The characteristic life expectancy
The Weibull distribution is particularly used in the
analysis of life expectancies and mortality rates
𝑓 𝛼, 𝛽, 𝑡 = 𝛼𝛽𝑡 𝛽−1 𝑒 −𝛼𝑡
𝛽
𝑓 𝛼, 𝛽 𝑑𝑡 = 𝐹 𝛼, 𝛽, 𝑡 = 1 − 𝑒 −𝛼𝑡
a=1
b=0.1
b=0.5
b=1.0
b=2.0
b=3.0
𝛽
𝑓 𝛼, 𝛽, 𝑡 = 𝛼𝛽𝑡 𝛽−1 𝑒 −𝛼𝑡
𝛼=
𝛽
1
𝑇𝛽
𝛽 𝑡
𝑓 𝛽, 𝑡 =
𝑇 𝑇
𝛽−1
We interpret the time t as the time to death.
b > 1: Probability of death increases with time
b = 1: Probability of death is constant over time
b < 1: Probability of death decreases with time
𝑡 𝛽
−
𝑒 𝑇
𝑓 𝛽 𝑑𝑡 = 𝐹 𝛽, 𝑡 = 1 − 𝑒
−
𝑡 𝛽
𝑇
The two parameter Weibull probability density function
Characteristic life expectancy T
2.2
𝑡 𝛽
𝐹 𝛽, 𝑡 = 1 −
−
𝑒 𝑇
;t=T
𝐹 𝛽, 𝑡 = 1 − 𝑒 −1 ≈ 0.632
The characteristic life expectancy T is the
age at which 63.2% of the population
already died.
F is the cumulative number of deaths.
How to estimate the characteristic life expectancy?
𝐹 =1−𝑒
−
𝑡 𝛽
𝑇
𝑡 𝛽
ln(1 − 𝐹) = −
𝑇
ln − ln 1 − 𝐹 = 𝛽 ln 𝑡 − 𝛽ln(𝑇)
Y
Age
t
0
0
10
5
20 15
30 25
40 35
50 45
60 55
70 65
80 75
90 85
100 95
110 105
N0
D
SD
1000
630
420
250
110
60
34
15
5
3
1
0
0
370
210
170
140
50
26
19
10
2
2
1
0
370
580
750
890
940
966
985
995
997
999
1000
F
=
bX
+
Linear function
C
𝑇=
𝐶
𝑒 −𝛽
ln(ln(t)
ln(1-F))
0
0.37 -0.772
0.58 -0.142
0.75 0.327
0.89 0.792
0.94 1.034
0.966 1.218
0.985 1.435
0.995 1.667
0.997 1.759
0.999 1.933
1
+U25/S$13
1.609
2.708
3.219
3.555
3.807
4.007
4.174
4.317
4.443
4.554
b = 0.95
𝑇=
C = -2.54
−2.54
𝑒 −0.95
= 14.4
Type III survivorship curve
The female life table of Polish women 2012 (GUS 2013)
ln(-ln(1ln(age)
F))
Age
N
D
F
0
1
2
3
4
5
6
7
8
9
10
100000
99574
99552
99537
99525
99515
99506
99496
99487
99479
99470
426
21
15
11
10
10
9
9
9
8
0.00426
0.00447
0.00462
0.00473
0.00483
0.00493
0.00502
0.00511
0.0052
0.00528
-5.456
-5.408
-5.375
-5.351
-5.330
-5.310
-5.292
-5.274
-5.256
-5.241
0.000
0.693
1.099
1.386
1.609
1.792
1.946
2.079
2.197
2.303
86
87
88
89
90
91
92
93
94
95
96
41622
37586
33550
29573
25710
22020
18551
15352
12461
9906
7699
3983
4037
4035
3978
3863
3690
3469
3199
2891
2556
2207
0.58378
0.62415
0.6645
0.70428
0.74291
0.77981
0.8145
0.84649
0.8754
0.90096
0.92303
-0.132
-0.022
0.088
0.197
0.306
0.414
0.522
0.628
0.734
0.838
0.942
4.454
4.466
4.477
4.489
4.500
4.511
4.522
4.533
4.543
4.554
4.564
Mortalities at younger age do not follow
a Weibull distribution
𝑇=
𝐶
𝑒 −𝛽
T = 86.8 years
The characteristic life
expectancy of Polish woman in
2012 was 87 years
The female life table of Polish women 2012 (GUS 2013)
Maximum mortality
87
Mortality of
newborns