Age Structure and Life Tables - Web.nmsu.edu | New Mexico ...web.nmsu.edu/~brook/courses/conservation-biology/lectures/life... · Age Structure and Life Tables Brook Milligan Department

Age Structure and Life Tables

Brook Milligan

Department of BiologyNew Mexico State University

Las Cruces, New Mexico [email protected]

Fall 2009

Brook Milligan Age Structure and Life Tables

mailto:[email protected]

Structured and Unstructured Populaions

Simplest model of a population

Population characterized by a single number: population sizeProjections: simply N(t)Rate of growth: depends only on current population size

Many populations are more complex

Rate of growth: depends on age of individuals in populationPopulation characterized by many numbers: e.g., number ofindividuals of each age Ni (t)Requires more complex models of population growth


Structured and Unstructured Populaions

Simplest model of a population

Population characterized by a single number: population sizeProjections: simply N(t)Rate of growth: depends only on current population size

Many populations are more complex

Rate of growth: depends on age of individuals in populationPopulation characterized by many numbers: e.g., number ofindividuals of each age Ni (t)Requires more complex models of population growth


Life Tables

A life table is a record of survival and reproductive rates in apopulation, broken down by age, size, or developmental stage(e.g. egg, hatchling, juvenile, adult).

Life tables are useful in predicting the growth and decline ofpopulations. For example:

The human population of a region depends in part on howmany children each person has and the the age at whichpeople die.Perhaps surprisingly, it also depends on the age at which theyhave their children.

Life tables help organize the effects of population structure onpopulation dynamics.


Life Tables


Life tables are useful in predicting the growth and decline ofpopulations.

For example:




Life Tables



The human population of a region depends in part on howmany children each person has and the the age at whichpeople die.

Perhaps surprisingly, it also depends on the age at which theyhave their children.



Life Tables






Life Tables






Types of Life Tables

Life tables come in two varieties

:

Cohort life tables follow the survival and reproduction of allmembers of a cohort from birth to death.Here, a cohort is the set of all individuals born, hatched, orrecruited into a population during a defined time interval.

A static life table records the number of living individuals ofeach age in a population and their reproductive output.



Life tables come in two varieties:

Cohort life tables follow the survival and reproduction of allmembers of a cohort from birth to death.

Here, a cohort is the set of all individuals born, hatched, orrecruited into a population during a defined time interval.













Age Structured Populations

Life tables that classify individuals by age are called age-basedlife tables.

Size-based and stage-based life tables classify individuals bysize or developmental stage.Such life tables are more useful when organisms are difficult toclassify by age or when the vital rates depend on size or stagerather than age.




Size-based and stage-based life tables classify individuals bysize or developmental stage.

Such life tables are more useful when organisms are difficult toclassify by age or when the vital rates depend on size or stagerather than age.




Size-based and stage-based life tables classify individuals bysize or developmental stage.Such life tables are more useful when organisms are difficult toclassify by age or when the vital rates depend on size or stagerather than age.


Cohort Life Tables

Suppose we wish to build a cohort life table for humans born in theUnited States in the 1900s.

We would record

The number of individuals born in the year 1900, and howmany survived to the beginning of 1901, 1902, etc. until therewere no more survivors.This record is called the survivorship schedule, denoted Sx .

The number of offspring born to individuals of each age.The total number of offspring is usually divided by thenumber of individuals in the age, giving an average number ofoffspring per individual of each age.This record is called the fecundity schedule, denoted bx .


Cohort Life Tables

Suppose we wish to build a cohort life table for humans born in theUnited States in the 1900s.We would record




Cohort Life Tables


The number of individuals born in the year 1900, and howmany survived to the beginning of 1901, 1902, etc. until therewere no more survivors.

This record is called the survivorship schedule, denoted Sx .



Cohort Life Tables





Cohort Life Tables



The number of offspring born to individuals of each age.

The total number of offspring is usually divided by thenumber of individuals in the age, giving an average number ofoffspring per individual of each age.This record is called the fecundity schedule, denoted bx .


Cohort Life Tables



The number of offspring born to individuals of each age.The total number of offspring is usually divided by thenumber of individuals in the age, giving an average number ofoffspring per individual of each age.

This record is called the fecundity schedule, denoted bx .


Cohort Life Tables





Quantities in a Life Table

Survivorship and fecundity schedules are the raw data of any lifetable.

We will work with the following example to illustrate the ideaspresented here.

x Sx bx

0 500 0

1 400 2

2 200 3

3 50 1

4 0 0



Survivorship and fecundity schedules are the raw data of any lifetable.We will work with the following example to illustrate the ideaspresented here.

x Sx bx

0 500 0

1 400 2

2 200 3

3 50 1

4 0 0



Survivorship and fecundity schedules are the raw data of any lifetable.We will work with the following example to illustrate the ideaspresented here.

x Sx bx

0 500 0

1 400 2

2 200 3

3 50 1

4 0 0


Survivorship Curves: Standardized Survival Schedule lx :

We standardize all cohorts to S0 (their initial size at time zero)because we want to compare cohorts of different initial sizes.

This proportion is calculated as lx =Sx

S0.

We can think of lx as the probability that an individualsurvives from birth to the beginning of age x .Note that lx always begins with a value of one, and can onlydecrease with time. At the last age, k, lk is zero (sinceSk = 0).

Please compute lx for the example.

x Sx bx lx0 500 0 1.01 400 2 0.82 200 3 0.43 50 1 0.14 0 0 0.0





S0.



x Sx bx lx0 500 0 1.01 400 2 0.82 200 3 0.43 50 1 0.14 0 0 0.0





S0.

We can think of lx as the probability that an individualsurvives from birth to the beginning of age x .

Note that lx always begins with a value of one, and can onlydecrease with time. At the last age, k, lk is zero (sinceSk = 0).


x Sx bx lx0 500 0 1.01 400 2 0.82 200 3 0.43 50 1 0.14 0 0 0.0





S0.



x Sx bx lx0 500 0 1.01 400 2 0.82 200 3 0.43 50 1 0.14 0 0 0.0





S0.



x Sx bx lx0 500 0 1.01 400 2 0.82 200 3 0.43 50 1 0.14 0 0 0.0





S0.



x Sx bx lx0 500 0 1.01 400 2 0.82 200 3 0.43 50 1 0.14 0 0 0.0


Survivorship Curves: Age-specific Survivorship gx :

This Age-specific survivorship gives us the probability that anindividual who has already survived to age x will survive to agex + 1.

This is calculated as gx =Sx+1

Sx.

Please compute gx for the example.

x Sx bx lx gx

0 500 0 1.0 0.80

1 400 2 0.8 0.50

2 200 3 0.4 0.25

3 50 1 0.1 0.00

4 0 0 0.0





Sx.


x Sx bx lx gx

0 500 0 1.0 0.80

1 400 2 0.8 0.50

2 200 3 0.4 0.25

3 50 1 0.1 0.00

4 0 0 0.0





Sx.


x Sx bx lx gx

0 500 0 1.0 0.80

1 400 2 0.8 0.50

2 200 3 0.4 0.25

3 50 1 0.1 0.00

4 0 0 0.0





Sx.


x Sx bx lx gx

0 500 0 1.0 0.80

1 400 2 0.8 0.50

2 200 3 0.4 0.25

3 50 1 0.1 0.00

4 0 0 0.0


Survivorship Curves: Life Expectancy ex :

The Life expectancy gives us the expected number of agecategories remaining until death for individuals surviving to thebeginning of age category x .

This is calculated as ex = lx+1+lx+2+···+lklx

=

∑ki=x+1 li

lx.

Please compute ex for the example.

x Sx bx lx gx ex

0 500 0 1.0 0.80 1.300

1 400 2 0.8 0.50 0.625

2 200 3 0.4 0.25 0.250

3 50 1 0.1 0.00 0.000

4 0 0 0.0





=

∑ki=x+1 li

lx.


x Sx bx lx gx ex

0 500 0 1.0 0.80 1.300

1 400 2 0.8 0.50 0.625

2 200 3 0.4 0.25 0.250

3 50 1 0.1 0.00 0.000

4 0 0 0.0





=

∑ki=x+1 li

lx.


x Sx bx lx gx ex

0 500 0 1.0 0.80 1.300

1 400 2 0.8 0.50 0.625

2 200 3 0.4 0.25 0.250

3 50 1 0.1 0.00 0.000

4 0 0 0.0





=

∑ki=x+1 li

lx.


x Sx bx lx gx ex

0 500 0 1.0 0.80 1.300

1 400 2 0.8 0.50 0.625

2 200 3 0.4 0.25 0.250

3 50 1 0.1 0.00 0.000

4 0 0 0.0


Net Reproductive Rate

The net reproductive rate R0 is defined as the average number ofoffspring produced per female over her lifetime.

To compute R0, multiply each value of lx by the correspondingvalue of bx and sum these products across all ages:

R0 =k∑

x=0

lxbx .

R0 > 1

R0 < 1

R0 = 1

Please compute lxbx and R0 for the example.



The net reproductive rate R0 is defined as the average number ofoffspring produced per female over her lifetime.To compute R0, multiply each value of lx by the correspondingvalue of bx and sum these products across all ages:

R0 =k∑

x=0

lxbx .

R0 > 1

R0 < 1

R0 = 1





R0 =k∑

x=0

lxbx .

R0 > 1

R0 < 1

R0 = 1





R0 =k∑

x=0

lxbx .

R0 > 1 =⇒ a net surplus of offspring produced eachgeneration, so the population increases.

R0 < 1

R0 = 1





R0 =k∑

x=0

lxbx .


R0 < 1 =⇒ the mortality is so great that the populationcannot replace itself, so the population declines.

R0 = 1





R0 =k∑

x=0

lxbx .



R0 = 1 =⇒ the offspring production exactly balances themortality each generation, and the population size is constant.





R0 =k∑

x=0

lxbx .



R0 = 1 =⇒ the offspring production exactly balances themortality each generation, and the population size is constant.



x Sx bx lx gx ex lxbx

0 500 0 1.0 0.80 1.300 0.0

1 400 2 0.8 0.50 0.625 1.6

2 200 3 0.4 0.25 0.250 1.2

3 50 1 0.1 0.00 0.000 0.1

4 0 0 0.0 0.0

R0 =k∑

x=0

lxbx = 2.9 offspring


Rate of Population Growth

Consider the following three populations that differ in bx but havethe same survivorship schedule lx .

x lx bx bx bx

0 1.0 0 2.9 0

1 0.8 2 0 0

2 0.4 3 0 0

3 0.1 1 0 29

4 0.0 0 0 0

R0 =k∑

x=0

lxbxoffspring

What is the net reproductive rate, R0, for each population?

Which population increases at the fastest rate?


Generation Time

Suppose we followed a cohort from birth and kept track of all theoffspring it produced as well as the age of the parents of theoffspring. Then generation time G is the average age of theparents of all the offspring produced by a single cohort.

G is computed as follows:

G =

∑kx=0 lxbx(x + 1)∑k

x=0 lxbx

.

The units of lx and bx cancel in the numerator and denominator,leaving us with units of time.Note also that the numerator will always be greater than or equalto the denominator. Consequently, the generation time will alwaysbe greater than or equal 1.0 for populations with age structure.Please compute lxbx(x + 1) and G for the example.


Generation Time

Suppose we followed a cohort from birth and kept track of all theoffspring it produced as well as the age of the parents of theoffspring. Then generation time G is the average age of theparents of all the offspring produced by a single cohort.G is computed as follows:

G =


x=0 lxbx

.



Generation Time


G =


x=0 lxbx

.

The units of lx and bx cancel in the numerator and denominator,leaving us with units of time.

Note also that the numerator will always be greater than or equalto the denominator. Consequently, the generation time will alwaysbe greater than or equal 1.0 for populations with age structure.Please compute lxbx(x + 1) and G for the example.


Generation Time


G =


x=0 lxbx

.

The units of lx and bx cancel in the numerator and denominator,leaving us with units of time.Note also that the numerator will always be greater than or equalto the denominator. Consequently, the generation time will alwaysbe greater than or equal 1.0 for populations with age structure.

Please compute lxbx(x + 1) and G for the example.


Generation Time


G =


x=0 lxbx

.



x Sx bx lx gx ex lxbx lxbx(x + 1)

0 500 0 1.0 0.80 1.300 0.0 0.0

1 400 2 0.8 0.50 0.625 1.6 3.2

2 200 3 0.4 0.25 0.250 1.2 3.6

3 50 1 0.1 0.00 0.000 0.1 0.4

4 0 0 0.0 0.0 0.0

R0 =k∑

x=0

lxbx = 2.9 offspring

G =7.2

2.9= 2.483 years


x Sx bx

0 500 0

1 400 2

2 200 3

3 50 1

4 0 0

R0 =k∑

x=0

lxbx = offspring

G =


x=0 lxbx

= years


Documents

Age Structure and Life Tables - Web.nmsu.edu | New Mexico ...web.nmsu.edu/~brook/courses/conservation-biology/lectures/life... · Age Structure and Life Tables Brook Milligan Department