Age structured populations

Alfred James Lotka (1880-1949)

Vito Volterra (1860-1940)

Age N f d l0 1000 0 0.15 0.85

10 600 0 0.09 0.9120 700 0.1 0.15 0.8530 595 1.2 0.25 0.7540 446 0.6 0.35 0.6550 290 0.3 0.38 0.6260 180 0.05 0.42 0.5870 200 0 0.55 0.4580 90 0 0.75 0.2590 50 0 0.95 0.05

100 3 0 1 0

FecundityMortality

rateSurvival

rate

First steps in life tables

• 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.

Age t N D d l0 0 1000

10 5 850 150 0.150 0.85020 15 835 15 0.018 0.98230 25 812 23 0.028 0.97240 35 777 35 0.043 0.95750 45 737 40 0.051 0.94960 55 661 76 0.103 0.89770 65 551 110 0.166 0.83480 75 270 281 0.510 0.49090 85 167 103 0.381 0.619

100 95 7 160 0.958 0.042110 105 1 6 0.857 0.143120 115 0 1 1.000 0.000

Pivotal age

Number of deaths at each age classage

𝑑𝑥=𝐷𝑥

𝑁 𝑥

𝑙𝑥=1−𝑑𝑥=𝑁 𝑥

𝑁 𝑥− 1

The pivotal age is the averge age per age cohort class

The basic information needed is the total number of deaths per age cohort.

survival

mortality

Mortality rate survival rate

Age N0 f d l N1

0 1000 0 0.15 0.85 114810 600 0 0.09 0.91 85020 700 0.1 0.15 0.85 54630 595 1.2 0.25 0.75 59540 446 0.6 0.35 0.65 44650 290 0.3 0.38 0.62 29060 180 0.05 0.42 0.58 18070 200 0 0.55 0.45 10480 90 0 0.75 0.25 9090 50 0 0.95 0.05 23

100 3 0 1 0 =E27*H27

First steps in life tables

Initial age distribution

Age distribution of

the next generation

N1(0)00

70714268879000

=+E28*F28

Population size of each cohort after reproduction

𝑁1 (10 )=𝑁0 (0)×𝑙(0) 𝑁1 (0 )=∑ 𝑁 𝑖× 𝑓 𝑖

FecundityDeath

rateSurvival

rate

Age 0 10 20 30 40 50 60 70 80 90 1000 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 0 020 0 0.91 0 0 0 0 0 0 0 0 030 0 0 0.85 0 0 0 0 0 0 0 040 0 0 0 0.75 0 0 0 0 0 0 050 0 0 0 0 0.65 0 0 0 0 0 060 0 0 0 0 0 0.62 0 0 0 0 070 0 0 0 0 0 0 0.58 0 0 0 080 0 0 0 0 0 0 0 0.45 0 0 090 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

If the population is age structured and contains k age classes we get

Fecundities

Survival rates

)0()0(...)0()0()1(1

22110

k

ikkkk NbNbNbNbN

N0(0) = 1000N0(1) = 1148

Leslie matrix

595*0.75=446

The mutiplication of the abundance vector with each row of the Leslie matrix gives the abundance of the next generation.

N0 N11000 1148600 850700 546595 595446 446290 290180 180200 10490 9050 233 3

Leslie matrix

We have w-1 age classes, w is the maximum age of an individual. L is a square matrix.

1

2

1

0

...

n

n

n

n

tN

00000

0...............

0...000

0...000

0...000

...

2

2

1

0

13210

s

s

s

s

fffff

L

tt LNN 1

Numbers per age class at time t+1 are the dot product of the Leslie matrix with the abundance vector N at time t

01 NLN tt

The Leslie model is a linear approach.It assumes stable fecundity and mortality rates

Going Excel

• At the long run population size increases.• Diagonal waves in abundances occur.• The first age cohort increases fastest.

N0 N1 N2 N3 N4 N5 N61000 1148 1132 998 1183 1366 1413600 850 976 963 848 1005 1161700 546 774 888 876 772 915595 595 464 657 755 744 656446 446 446 348 493 566 558290 290 290 290 226 321 368180 180 180 180 180 140 199200 104 104 104 104 104 8190 90 47 47 47 47 4750 23 23 12 12 12 123 3 1 1 1 1 1

Age 0 10 20 30 40 50 60 70 80 90 1000 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 0 020 0 0.91 0 0 0 0 0 0 0 0 030 0 0 0.85 0 0 0 0 0 0 0 040 0 0 0 0.75 0 0 0 0 0 0 050 0 0 0 0 0.65 0 0 0 0 0 060 0 0 0 0 0 0.62 0 0 0 0 070 0 0 0 0 0 0 0.58 0 0 0 080 0 0 0 0 0 0 0 0.45 0 0 090 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

Age 0 10 20 30 40 50 60 70 80 90 1000 0 0 0.05 0.3 0.6 1.2 0.1 0 0 0 0

10 0.85 0 0 0 0 0 0 0 0 0 020 0 0.91 0 0 0 0 0 0 0 0 030 0 0 0.85 0 0 0 0 0 0 0 040 0 0 0 0.75 0 0 0 0 0 0 050 0 0 0 0 0.65 0 0 0 0 0 060 0 0 0 0 0 0.62 0 0 0 0 070 0 0 0 0 0 0 0.58 0 0 0 080 0 0 0 0 0 0 0 0.45 0 0 090 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

Age 0 10 20 30 40 50 60 70 80 90 1000 0 0 0.1 1.2 0.6 0.3 0.05 0 0 0 0

10 0.85 0 0 0 0 0 0 0 0 0 020 0 0.91 0 0 0 0 0 0 0 0 030 0 0 0.85 0 0 0 0 0 0 0 040 0 0 0 0.75 0 0 0 0 0 0 050 0 0 0 0 0.65 0 0 0 0 0 060 0 0 0 0 0 0.62 0 0 0 0 070 0 0 0 0 0 0 0.58 0 0 0 080 0 0 0 0 0 0 0 0.45 0 0 090 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

Reproduction in early age contributes more to population size than later reproduction.This is caused by the higher number of females in earlier cohorts.

The effect of age in reproduction

• 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 1000 0 0 0.1 1.2 0.6 0.3 0.05 0 0 0 0

10 0.25 0 0 0 0 0 0 0 0 0 020 0 0.91 0 0 0 0 0 0 0 0 030 0 0 0.85 0 0 0 0 0 0 0 040 0 0 0 0.75 0 0 0 0 0 0 050 0 0 0 0 0.65 0 0 0 0 0 060 0 0 0 0 0 0.62 0 0 0 0 070 0 0 0 0 0 0 0.58 0 0 0 080 0 0 0 0 0 0 0 0.45 0 0 090 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

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?

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.

0dtdN

Important properties:1. Eventually all age classes grow or shrink

at the same rate2. Initial growth depends on the age

structure3. Early reproduction contributes more to

population growth than late reproduction

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.

𝑵 𝑡+1=𝑅𝑵 𝑡

𝑵 𝑡+1=𝑳❑𝑡 𝑵0𝑵 𝑡+1=𝑅𝑡 𝑵0

𝑵 𝑡+1=𝑳𝑵𝑡=𝑵 𝑡

𝑳𝑼=𝜆𝑼𝑵 𝑡+1=𝑳𝑵𝑡=𝜆𝑵 𝑡=𝑅𝑵 𝑡

Age Eggs Larva 1 Larva 2 Larva 3 ImagoEggs 0 0 0 0 2000

Larva 1 0.25 0 0 0 0Larva 2 0 0.15 0 0 0Larva 3 0 0 0.15 0 0Imago 0 0 0 0.1 0

Largest eigenvalue r = l = 1.02

Age Eggs Larva 1 Larva 2 Larva 3 ImagoEggs 0 0 0 0 200

Larva 1 0.25 0 0 0 0Larva 2 0 0.15 0 0 0Larva 3 0 0 0.15 0 0Imago 0 0 0 0.1 0

Largest eigenvalue r = l = 0.65

The population steadily declines.

Leslie matrices in insect populatons

N0 Eggs Larva 1 Larva 2 Larva 3 Imago Eggs100000 11250 11250 11250 11250 11250 1265.6

25000 25000 2813 2813 2813 2812.5 2812.53750 3750 3750 421.9 421.9 421.875 421.88

563 562.5 562.5 562.5 63.28 63.2813 63.28156 56.25 56.25 56.25 56.25 6.32813 6.3281

The diagonal matrix elements show how

many individuals survive.

𝑵1=𝑳𝑵0

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.

Age Eggs Larva 1 Larva 2 Larva 3 ImagoEggs 0 0 0 0 2000

Larva 1 0.25 0 0 0 0Larva 2 0 0.15 0 0 0Larva 3 0 0 0.15 0 0Imago 0 0 0 0.1 0

Stable age distribution

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.

𝑵 𝑡+1=𝑳𝑵𝑡=𝜆𝑵 𝑡

U =

AgeEggs

Larva 1Larva 2Larva 3Imago

Stable age class distribution

U Nstable

0.970859 0.7777820.237064 0.1899190.034732 0.0278250.005088 0.0040770.000497 0.000398

Sum 1.248241 1

U0.9708590.2370640.0347320.0050880.000497

l = 1.02

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.

Age Eggs Larva 1 Larva 2 Larva 3 ImagoEggs 0.10 0 0 0 2000

Larva 1 0.25 0.15 0 0 0Larva 2 0 0.15 0.05 0 0Larva 3 0 0 0.15 0.05 0Imago 0 0 0 0.1 0.5

Remaining in the same age class

𝑵 𝑡+1=𝑳𝑵𝑡=𝜆𝑵 𝑡

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=

AgeEggs

Larva 1Larva 2Larva 3Imago

Stable age class distribution

U Nstable

0.972869 0.7869070.229415 0.1855630.029662 0.0239920.003835 0.003102

0.00054 0.000437Sum 1.236321 1

U0.9728690.2294150.0296620.003835

0.00054

Nstable

0.7777820.1899190.0278250.0040770.000398

l = 1.02l = 1.21

Without staying

the same

Sensitivity analysis

Age Eggs Larva 1Larva 2 Larva 3 ImagoEggs 0 0 0 0 2000

Larva 1 0.25 0 0 0 0Larva 2 0 0.15 0 0 0Larva 3 0 0 0.15 0 0Imago 0 0 0 0.1 0

Age Eggs Larva 1Larva 2 Larva 3 ImagoEggs 0 0 0 0 2.5

Larva 1 0.95 0 0 0 0Larva 2 0 0.91 0 0 0Larva 3 0 0 0.93 0 0Imago 0 0 0 0.95 0

l = 1.02 l = 1.14

High mortality, high fecundityr strategist species

Low mortality, low fecundityK strategist species

l > 1 → effective population size increases

How robust is l with respect to changes in survival and fecundity rates?

Age Eggs Larva 1 Larva 2 Larva 3 ImagoEggs 0 0 0 0 1.4

Larva 1 0.95 0 0 0 0Larva 2 0 0.91 0 0 0Larva 3 0 0 0.93 0 0Imago 0 0 0 0.95 0

l = 1.01

The lowest possible fecundity is 1.4 female eggs per female.

Age Eggs Larva 1 Larva 2 Larva 3 ImagoEggs 0 0 0 0 2.5

Larva 1 0.86 0 0 0 0Larva 2 0 0.819 0 0 0Larva 3 0 0 0.837 0 0Imago 0 0 0 0.855 0

l = 1.05

Mortality rates might be 10% higher to remain effective population sizes still increasing.

Age Eggs Larva 1Larva 2 Larva 3 ImagoEggs 0 0 0 0 2.5

Larva 1 0.95 0 0 0 0Larva 2 0 0.91 0 0 0Larva 3 0 0 0.93 0 0Imago 0 0 0 0.95 0

l = 1.14

Sensitivity analysis

Increasing mortality rates until the population stops increasing

Age N0 1000

10 85020 83530 81240 77750 73760 66170 55180 27090 167

100 7110 1120 0

l

0.850.980.970.960.950.900.830.490.620.040.140.00

+H25/H24

D

1501523354076

110281103160

61

d

0.150.020.030.040.050.100.170.510.380.960.861.00+J25/H24

L9258438247957576996064112198741 +

(H25+H24)/2

SL61685243440135772783202613277213109251

+SUMA(L$24:L24)

e61.761.752.744.135.827.520.113.111.55.56.45.0

+M24/

H24*$G$14

Number of death

Survival rate

Death rate

Average number alive in a cohort

Cumulative number alive in a cohort

Average life expectation

Survivorship tables

𝑒𝑥=Σ 𝐿𝑥

𝑁 𝑥

𝑘

k = length of cohort (10 years)

𝐿 (𝑥 )=𝑁 (𝑥 )+𝑁 (𝑥+1)

2

Σ 𝐿=∑𝑥

𝑚𝑎𝑥

𝐿(𝑥 )

Age N D l d L SL e0 100000 99787 8095890 80.95891 99574 426 0.99574 0.00426 99563 7996103 80.303122 99552 21 0.99978 0.00021 99544.5 7896540 79.320763 99537 15 0.99985 0.00015 99531 7796996 78.332644 99525 11 0.99988 0.00011 99520 7697465 77.342025 99515 10 0.99990 0.00010 99510.5 7597945 76.349746 99506 10 0.99991 0.00010 99501 7498434 75.35667 99496 9 0.99990 0.00009 99491.5 7398933 74.364138 99487 9 0.99991 0.00009 99483 7299442 73.370819 99479 9 0.99992 0.00009 99474.5 7199959 72.37667

10 99470 8 0.99991 0.00008 99465.5 7100484 71.3831711 99461 9 0.99991 0.00009 99456.5 7001019 70.3895812 99452 10 0.99991 0.00010 99446 6901562 69.3959113 99440 11 0.99988 0.00011 99433.5 6802116 68.4042214 99427 13 0.99987 0.00013 99419.5 6702683 67.413115 99412 16 0.99985 0.00016 99402.5 6603263 66.423216 99393 18 0.99981 0.00018 99383 6503861 65.435817 99373 21 0.99980 0.00021 99361.5 6404478 64.4488718 99350 23 0.99977 0.00023 99338 6305116 63.4636719 99326 24 0.99976 0.00024 99314 6205778 62.4788920 99302 24 0.99976 0.00024 99290 6106464 61.49387

98 4317 1524 0.739086 0.260914 3710 8889.5 2.05918599 3103 1214 0.718786 0.281214 2634 5179.5 1.669191

100 2165 938 0.697712 0.302288 1814 2545.5 1.175751>100 1463 702 0.675751 0.324249 731.5 731.5 0.5>120 0 1463 0 1 0 0 0

The female life table of Polish women 2012 (GUS 2013)Average life

expectancy at birth

𝑒𝑥=Σ 𝐿𝑥

𝑁 𝑥

𝑘

𝐿 (𝑥 )=𝑁 (𝑥 )+𝑁 (𝑥+1)

2

Σ 𝐿=∑𝑥

𝑚𝑎𝑥

𝐿(𝑥 )

Polish survivorship curve 2012

Type I

Type II

Type III

Type I, high survivorship of young individuals: large mammals, birdsType II, survivorship independent of age, seed banksType III, low survivorship of young individuals, fish, many insects

Polish mortality rates 2012

Newborns

New motocycle and car drivers

Average life expectancy at birth in Poland

8 years

Average life expectancy at age 60 in Poland

5 years

81 years

72 years

84 years

78 years

Men

Women

Reproduction life tables

Age t N0 D l0 0 1000

10 5 850 150 0.8520 15 835 15 0.9830 25 812 23 0.9740 35 777 35 0.9650 45 737 40 0.9560 55 661 76 0.9070 65 551 110 0.8380 75 270 281 0.4990 85 167 103 0.62

100 95 7 160 0.04110 105 1 6 0.14120 115 0 1 0.00Sum

+I25/I24

B b

0 0.00020 0.024

515 0.634342 0.44059 0.0802 0.0030 0.0000 0.0000 0.0000 0.0000 0.0000 0.000

=L25/I25

lb

0.0000.0240.6170.4210.0760.0030.0000.0000.0000.0000.0000.0001.140

+M25*K25

lbt

0.00.4

15.414.73.40.10.00.00.00.00.00.0

34.09629.9

+N25*H25

Pivotal age

Survival rate

Birth rate

Number offspring

R

𝑅0=∑ 𝑅𝑖=¿∑ 𝑙𝑖𝑏𝑖¿

𝐺=∑ 𝑙𝑖𝑏𝑖𝑡 𝑖∑ 𝑙𝑖𝑏𝑖

Net reproduction rate

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 Body mass (kg) Litter size

Age of maturity

(yr)

Life expectanc

y (yr)

Life expectation at maturity

(yr)

Reproductive value at maturity

Generation length (yr)

Castor canadensis 18 6.6 2 1.52 2.22 5.63 4.87Clethreonomys glareolus 0.025 5 0.11 0.16 0.48 7.9 0.33Peromyscus leucopus 0.02 5 0.15 0.21 0.28 4.52 0.27P. maniculatus 0.02 3.6 0.15 0.23 0.43 5.04 0.35Sciurus carolinensis 0.6 2.9 1 1.37 2.17 5.95 2.07Spermophilus armatus 0.35 5.3 1 1.38 1.72 4.52 1.78S. beldingi 0.25 7.4 1 1.3 1.78 5.89 1.56S. lateralis 0.155 5.2 1.3 1.47 2.12 5.08 2.45S. parrylii 0.7 7.3 1 1.28 1.71 6.17 1.59Tamias striatus 0.1 4.2 1 1.24 1.63 6.84 1.59Tamiasciurus hudsonicus 0.189 4 1 1.5 2.45 4.9 1.95Ochotona princeps 0.13 2.8 1 1.37 2.33 6.51 2.07Sylvilagus floridanus 1.25 5 1 1.48 1.25 2.62 1.29Lutra canadensis 7.2 2 3 2.88 3.79 3.79 5.07Lynx rufus 7.5 2.8 1 1.72 2.48 3.48 2.87Mephitis mephitis 2.25 6 1 1.33 1.9 5.71 1.78Taxidea taxus 7.15 2 1 1.45 2.33 2.48 1.24Equus burchelli 270 1 4 3.84 7.95 4 8.74Aepycerus melampus 44 1 2 3.44 4.8 2.42 4.36Cervus elaphus 175 1 4 4.9 3.85 1.73 5.7Connochaetes taurinus 165 1 3 3.84 4.79 2.56 6.29Hemitragus jemlahicus 100 1 3 3.97 4.71 2.12 5.43Hippopotamus amphibicus 2390 1 10 7.62 16.4 3.98 19.82Kobus defassa 200 1 2 3.35 5.87 2.94 5.08Ovis canadensis 55 1 4 3.81 5.48 2.74 6.52Phacochoerus aethiopicus 87 4.8 2 1.6 2.82 6.76 4.28Sus scrofa 85 5 2 1.79 1.91 4.82 3.15Syncerus caffer 490 1 4 4.47 4.82 2.41 6.98Loxodonta africana 4000 1 15 17.9 19.1 2.24 25.8

Life history data and body size

Data from Millar and Zammuto 1983, Ecology 64: 631

Life history data are allometrically related

to body size.

Reproductive value at age x

𝑅𝑉 𝑥= 𝑓 𝑥+ ∑𝑦=𝑥+1

𝑚𝑎𝑥 𝑙𝑦𝑙𝑥𝑓 𝑦 ( 𝑙𝑦𝑏𝑦 )𝑥−𝑦

The characteristic life expectancy

𝑓 (𝛼 , 𝛽 , 𝑡 )=𝛼𝛽𝑡𝛽−1𝑒−𝛼𝑡𝛽

The Weibull distribution is particularly used in the analysis of life expectancies and mortality rates

∫ 𝑓 (𝛼 , 𝛽 )𝑑𝑡=𝐹 (𝛼 , 𝛽 , 𝑡 )=1−𝑒−𝛼 𝑡𝛽

a=1b=0.1b=0.5b=1.0b=2.0b=3.0

𝑓 (𝛼 , 𝛽 , 𝑡 )=𝛼𝛽𝑡𝛽−1𝑒−𝛼𝑡𝛽

𝑓 (𝛽 , 𝑡 )= 𝛽𝑇 ( 𝑡𝑇 )

𝛽− 1

𝑒−( 𝑡𝑇 )

𝛽

𝛼=1

𝑇 𝛽

The two parameter Weibull probability density function

We interpret the time t as the time to death.b > 1: Probability of death increases with timeb = 1: Probability of death is constant over timeb < 1: Probability of death decreases with time

∫ 𝑓 (𝛽 )𝑑𝑡=𝐹 (𝛽 , 𝑡 )=1−𝑒−( 𝑡𝑇 )

𝛽

2.2

Characteristic life expectancy 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−( 𝑡𝑇 )

𝛽

Y = bX

+ C

Linear function

Age t N0 D SD F ln(-ln(1-F)) ln(t)

0 0 1000 0 0 010 5 630 370 370 0.37 -0.772 1.60920 15 420 210 580 0.58 -0.142 2.70830 25 250 170 750 0.75 0.327 3.21940 35 110 140 890 0.89 0.792 3.55550 45 60 50 940 0.94 1.034 3.80760 55 34 26 966 0.966 1.218 4.00770 65 15 19 985 0.985 1.435 4.17480 75 5 10 995 0.995 1.667 4.31790 85 3 2 997 0.997 1.759 4.443

100 95 1 2 999 0.999 1.933 4.554110 105 0 1 1000 1

+U25/S$13

b = 0.95

C = -2.54

𝑇=𝑒𝐶−𝛽

𝑇=𝑒−2.54− 0.95=14.4

Type III survivorship curve

The female life table of Polish women 2012 (GUS 2013)

𝑇=𝑒𝐶−𝛽

T = 86.8 years

The characteristic life expectancy of Polish woman in

2012 was 87 years

Mortalities at younger age do not follow a Weibull distribution

Age N D F ln(-ln(1-F)) ln(age)

0 1000001 99574 426 0.00426 -5.456 0.0002 99552 21 0.00447 -5.408 0.6933 99537 15 0.00462 -5.375 1.0994 99525 11 0.00473 -5.351 1.3865 99515 10 0.00483 -5.330 1.6096 99506 10 0.00493 -5.310 1.7927 99496 9 0.00502 -5.292 1.9468 99487 9 0.00511 -5.274 2.0799 99479 9 0.0052 -5.256 2.197

10 99470 8 0.00528 -5.241 2.303

86 41622 3983 0.58378 -0.132 4.45487 37586 4037 0.62415 -0.022 4.46688 33550 4035 0.6645 0.088 4.47789 29573 3978 0.70428 0.197 4.48990 25710 3863 0.74291 0.306 4.50091 22020 3690 0.77981 0.414 4.51192 18551 3469 0.8145 0.522 4.52293 15352 3199 0.84649 0.628 4.53394 12461 2891 0.8754 0.734 4.54395 9906 2556 0.90096 0.838 4.55496 7699 2207 0.92303 0.942 4.564

The female life table of Polish women 2012 (GUS 2013)

Maximum mortality

Mortality of newborns

87