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Age structured populations. Alfred James Lotka (1880-1949). Vito Volterra (1860-1940). First steps in life tables. Mortality rate. Survival rate. Fecundity. N 0 is the number of newborns. N is the number of females per age cohort.

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Age structured populations

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Age structured populations

Alfred James Lotka (1880-1949)

Vito Volterra (1860-1940)


First steps in life tables

Mortality rate

Survival rate


  • 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.

Number of deaths at each age classage

The pivotal age is the averge age per age cohort class

Pivotal age

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



survival rate

Mortality rate


First steps in life tables

Death rate

Survival rate


Population size of each cohort after reproduction

Initial age distribution

Age distribution of the next generation


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


Survival rates

Leslie matrix

N0(0) = 1000

N0(1) = 1148


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.

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 linearapproach.

It assumesstablefecundity and mortalityrates


Going Excel

Demographic low

  • At the long run populationsizeincreases.
  • Diagonal waves in abundancesoccur.
  • The firstagecohortincreasesfastest.

The effect of age in reproduction

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 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

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 eigenvectorU of the matrix.

Eigenvectors are only defined for square matrices.

Important properties:

Eventually all age classes grow or shrink at the same rate

Initial growth depends on the age structure

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.


Leslie matrices in insect populatons

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.

The diagonal matrix elements show how many individuals survive.


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.

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.

U =

l = 1.02

Stable age class distribution


Remaining in the same age class

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

l = 1.21

l = 1.02


Without staying the same

Stable age class distribution


Sensitivity analysis

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.


Sensitivity analysis

l = 1.14

Increasing mortality rates until the population stops increasing

l = 1.05

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


Survivorship tables

Average number alive in a cohort

Cumulative number alive in a cohort

Death rate

Number of death

Average life expectation

Survival rate

k = length of cohort (10 years)


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


New motocycle and car drivers


Average life expectancy at birth in Poland

81 years


8 years

72 years


Average life expectancy at age 60 in Poland

84 years

5 years

78 years


Reproduction life tables

Survival rate

Number offspring


Birth rate

Pivotal age

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 thenumber of offspring.

Net reproduction rate


Life history data and body size

Reproductive value at age x

Life history data are allometrically related to body size.

Data from Millar and Zammuto 1983, Ecology 64: 631


The characteristic life expectancy

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





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

The two parameter Weibull probability density function

Characteristic life expectancy T

; t = T


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?







b = 0.95

C = -2.54

Type III survivorshipcurve


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

Mortalities at younger age do not follow a Weibull distribution

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


Mortality of newborns