Age Structure and Life Tables Brook Milligan Department of Biology - - PowerPoint PPT Presentation

Age Structure and Life Tables

Brook Milligan

Department of Biology New Mexico State University Las Cruces, New Mexico 88003 brook@nmsu.edu

Fall 2009

Brook Milligan Age Structure and Life Tables

Structured and Unstructured Populaions

Simplest model of a population

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

Brook Milligan Age Structure and Life Tables

Structured and Unstructured Populaions

Simplest model of a population

Population characterized by a single number: population size Projections: 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 population Population characterized by many numbers: e.g., number of individuals of each age Ni(t) Requires more complex models of population growth

Brook Milligan Age Structure and Life Tables

Life Tables

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

Brook Milligan Age Structure and Life Tables

Life Tables

A life table is a record of survival and reproductive rates in a population, broken down by age, size, or developmental stage (e.g. egg, hatchling, juvenile, adult). Life tables are useful in predicting the growth and decline of populations.

Brook Milligan Age Structure and Life Tables

Life Tables

A life table is a record of survival and reproductive rates in a population, broken down by age, size, or developmental stage (e.g. egg, hatchling, juvenile, adult). Life tables are useful in predicting the growth and decline of

• populations. For example:

The human population of a region depends in part on how many children each person has and the the age at which people die.

Brook Milligan Age Structure and Life Tables

Life Tables

A life table is a record of survival and reproductive rates in a population, broken down by age, size, or developmental stage (e.g. egg, hatchling, juvenile, adult). Life tables are useful in predicting the growth and decline of

• populations. For example:

The human population of a region depends in part on how many children each person has and the the age at which people die. Perhaps surprisingly, it also depends on the age at which they have their children.

Brook Milligan Age Structure and Life Tables

Life Tables

A life table is a record of survival and reproductive rates in a population, broken down by age, size, or developmental stage (e.g. egg, hatchling, juvenile, adult). Life tables are useful in predicting the growth and decline of

• populations. For example:

The human population of a region depends in part on how many children each person has and the the age at which people die. Perhaps surprisingly, it also depends on the age at which they have their children.

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

Brook Milligan Age Structure and Life Tables

Types of Life Tables

Life tables come in two varieties

Brook Milligan Age Structure and Life Tables

Types of Life Tables

Life tables come in two varieties: Cohort life tables follow the survival and reproduction of all members of a cohort from birth to death.

Brook Milligan Age Structure and Life Tables

Types of Life Tables

Life tables come in two varieties: Cohort life tables follow the survival and reproduction of all members of a cohort from birth to death. Here, a cohort is the set of all individuals born, hatched, or recruited into a population during a defined time interval.

Brook Milligan Age Structure and Life Tables

Types of Life Tables

Life tables come in two varieties: Cohort life tables follow the survival and reproduction of all members of a cohort from birth to death. Here, a cohort is the set of all individuals born, hatched, or recruited into a population during a defined time interval. A static life table records the number of living individuals of each age in a population and their reproductive output.

Brook Milligan Age Structure and Life Tables

Age Structured Populations

Life tables that classify individuals by age are called age-based life tables.

Brook Milligan Age Structure and Life Tables

Age Structured Populations

Life tables that classify individuals by age are called age-based life tables. Size-based and stage-based life tables classify individuals by size or developmental stage.

Brook Milligan Age Structure and Life Tables

Age Structured Populations

Life tables that classify individuals by age are called age-based life tables. Size-based and stage-based life tables classify individuals by size or developmental stage. Such life tables are more useful when organisms are difficult to classify by age or when the vital rates depend on size or stage rather than age.

Brook Milligan Age Structure and Life Tables

Cohort Life Tables

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

Brook Milligan Age Structure and Life Tables

Cohort Life Tables

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

Brook Milligan Age Structure and Life Tables

Cohort Life Tables

Suppose we wish to build a cohort life table for humans born in the United States in the 1900s. We would record The number of individuals born in the year 1900, and how many survived to the beginning of 1901, 1902, etc. until there were no more survivors.

Brook Milligan Age Structure and Life Tables

Cohort Life Tables

Suppose we wish to build a cohort life table for humans born in the United States in the 1900s. We would record The number of individuals born in the year 1900, and how many survived to the beginning of 1901, 1902, etc. until there were no more survivors. This record is called the survivorship schedule, denoted Sx.

Brook Milligan Age Structure and Life Tables

Cohort Life Tables

Suppose we wish to build a cohort life table for humans born in the United States in the 1900s. We would record The number of individuals born in the year 1900, and how many survived to the beginning of 1901, 1902, etc. until there were no more survivors. This record is called the survivorship schedule, denoted Sx. The number of offspring born to individuals of each age.

Brook Milligan Age Structure and Life Tables

Cohort Life Tables

Suppose we wish to build a cohort life table for humans born in the United States in the 1900s. We would record The number of individuals born in the year 1900, and how many survived to the beginning of 1901, 1902, etc. until there were 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 the number of individuals in the age, giving an average number of

• ffspring per individual of each age.

Brook Milligan Age Structure and Life Tables

Cohort Life Tables

Suppose we wish to build a cohort life table for humans born in the United States in the 1900s. We would record The number of individuals born in the year 1900, and how many survived to the beginning of 1901, 1902, etc. until there were 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 the number of individuals in the age, giving an average number of

• ffspring per individual of each age.

This record is called the fecundity schedule, denoted bx.

Brook Milligan Age Structure and Life Tables

Quantities in a Life Table

Survivorship and fecundity schedules are the raw data of any life table.

Brook Milligan Age Structure and Life Tables

Quantities in a Life Table

Survivorship and fecundity schedules are the raw data of any life table. We will work with the following example to illustrate the ideas presented here.

Brook Milligan Age Structure and Life Tables

Quantities in a Life Table

Survivorship and fecundity schedules are the raw data of any life table. We will work with the following example to illustrate the ideas presented here. x Sx bx 500 1 400 2 2 200 3 3 50 1 4

Brook Milligan Age Structure and Life Tables

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.

Brook Milligan Age Structure and Life Tables

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 .

Brook Milligan Age Structure and Life Tables

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 individual survives from birth to the beginning of age x.

Brook Milligan Age Structure and Life Tables

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 individual survives from birth to the beginning of age x. Note that lx always begins with a value of one, and can only decrease with time. At the last age, k, lk is zero (since Sk = 0).

Brook Milligan Age Structure and Life Tables

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 individual survives from birth to the beginning of age x. Note that lx always begins with a value of one, and can only decrease with time. At the last age, k, lk is zero (since Sk = 0). Please compute lx for the example.

Brook Milligan Age Structure and Life Tables

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 individual survives from birth to the beginning of age x. Note that lx always begins with a value of one, and can only decrease with time. At the last age, k, lk is zero (since Sk = 0). Please compute lx for the example.

x Sx bx lx 500 1.0 1 400 2 0.8 2 200 3 0.4 3 50 1 0.1 4 0.0

Brook Milligan Age Structure and Life Tables

Survivorship Curves: Age-specific Survivorship gx:

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

Brook Milligan Age Structure and Life Tables

Survivorship Curves: Age-specific Survivorship gx:

This Age-specific survivorship gives us the probability that an individual who has already survived to age x will survive to age x + 1. This is calculated as gx = Sx+1 Sx .

Brook Milligan Age Structure and Life Tables

Survivorship Curves: Age-specific Survivorship gx:

This Age-specific survivorship gives us the probability that an individual who has already survived to age x will survive to age x + 1. This is calculated as gx = Sx+1 Sx . Please compute gx for the example.

Brook Milligan Age Structure and Life Tables

Survivorship Curves: Age-specific Survivorship gx:

This Age-specific survivorship gives us the probability that an individual who has already survived to age x will survive to age x + 1. This is calculated as gx = Sx+1 Sx . Please compute gx for the example. x Sx bx lx gx 500 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

Brook Milligan Age Structure and Life Tables

Survivorship Curves: Life Expectancy ex:

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

Brook Milligan Age Structure and Life Tables

Survivorship Curves: Life Expectancy ex:

The Life expectancy gives us the expected number of age categories remaining until death for individuals surviving to the beginning of age category x. This is calculated as ex = lx+1+lx+2+···+lk

lx

= k

i=x+1 li

lx .

Brook Milligan Age Structure and Life Tables

Survivorship Curves: Life Expectancy ex:

The Life expectancy gives us the expected number of age categories remaining until death for individuals surviving to the beginning of age category x. This is calculated as ex = lx+1+lx+2+···+lk

lx

= k

i=x+1 li

lx . Please compute ex for the example.

Brook Milligan Age Structure and Life Tables

Survivorship Curves: Life Expectancy ex:

The Life expectancy gives us the expected number of age categories remaining until death for individuals surviving to the beginning of age category x. This is calculated as ex = lx+1+lx+2+···+lk

lx

= k

i=x+1 li

lx . Please compute ex for the example. x Sx bx lx gx ex 500 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

Brook Milligan Age Structure and Life Tables

Net Reproductive Rate

The net reproductive rate R0 is defined as the average number of

• ffspring produced per female over her lifetime.

Brook Milligan Age Structure and Life Tables

Net Reproductive Rate

The net reproductive rate R0 is defined as the average number of

• ffspring produced per female over her lifetime.

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

k

• x=0

lxbx.

Brook Milligan Age Structure and Life Tables

Net Reproductive Rate

The net reproductive rate R0 is defined as the average number of

• ffspring produced per female over her lifetime.

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

k

• x=0

lxbx. R0 > 1 R0 < 1 R0 = 1

Brook Milligan Age Structure and Life Tables

Net Reproductive Rate

The net reproductive rate R0 is defined as the average number of

• ffspring produced per female over her lifetime.

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

k

• x=0

lxbx. R0 > 1 = ⇒ a net surplus of offspring produced each generation, so the population increases. R0 < 1 R0 = 1

Brook Milligan Age Structure and Life Tables

Net Reproductive Rate

The net reproductive rate R0 is defined as the average number of

• ffspring produced per female over her lifetime.

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

k

• x=0

lxbx. R0 > 1 = ⇒ a net surplus of offspring produced each generation, so the population increases. R0 < 1 = ⇒ the mortality is so great that the population cannot replace itself, so the population declines. R0 = 1

Brook Milligan Age Structure and Life Tables

Net Reproductive Rate

The net reproductive rate R0 is defined as the average number of

• ffspring produced per female over her lifetime.

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

k

• x=0

lxbx. R0 > 1 = ⇒ a net surplus of offspring produced each generation, so the population increases. R0 < 1 = ⇒ the mortality is so great that the population cannot replace itself, so the population declines. R0 = 1 = ⇒ the offspring production exactly balances the mortality each generation, and the population size is constant.

Brook Milligan Age Structure and Life Tables

Net Reproductive Rate

The net reproductive rate R0 is defined as the average number of

• ffspring produced per female over her lifetime.

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

k

• x=0

lxbx. R0 > 1 = ⇒ a net surplus of offspring produced each generation, so the population increases. R0 < 1 = ⇒ the mortality is so great that the population cannot replace itself, so the population declines. R0 = 1 = ⇒ the offspring production exactly balances the mortality each generation, and the population size is constant. Please compute lxbx and R0 for the example.

Brook Milligan Age Structure and Life Tables

x Sx bx lx gx ex lxbx 500 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 R0 =

k

• x=0

lxbx = 2.9 offspring

Brook Milligan Age Structure and Life Tables

Rate of Population Growth

Consider the following three populations that differ in bx but have the same survivorship schedule lx. x lx bx bx bx 1.0 2.9 1 0.8 2 2 0.4 3 3 0.1 1 29 4 0.0 R0 =

k

• x=0

lxbxoffspring What is the net reproductive rate, R0, for each population? Which population increases at the fastest rate?

Brook Milligan Age Structure and Life Tables

Generation Time

Suppose we followed a cohort from birth and kept track of all the

• ffspring it produced as well as the age of the parents of the
• ffspring. Then generation time G is the average age of the

parents of all the offspring produced by a single cohort.

Brook Milligan Age Structure and Life Tables

Generation Time

Suppose we followed a cohort from birth and kept track of all the

• ffspring it produced as well as the age of the parents of the
• ffspring. Then generation time G is the average age of the

parents of all the offspring produced by a single cohort. G is computed as follows: G = k

x=0 lxbx(x + 1)

k

x=0 lxbx

.

Brook Milligan Age Structure and Life Tables

Generation Time

Suppose we followed a cohort from birth and kept track of all the

• ffspring it produced as well as the age of the parents of the
• ffspring. Then generation time G is the average age of the

parents of all the offspring produced by a single cohort. G is computed as follows: G = k

x=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.

Brook Milligan Age Structure and Life Tables

Generation Time

Suppose we followed a cohort from birth and kept track of all the

• ffspring it produced as well as the age of the parents of the
• ffspring. Then generation time G is the average age of the

parents of all the offspring produced by a single cohort. G is computed as follows: G = k

x=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 equal to the denominator. Consequently, the generation time will always be greater than or equal 1.0 for populations with age structure.

Brook Milligan Age Structure and Life Tables

Generation Time

Suppose we followed a cohort from birth and kept track of all the

• ffspring it produced as well as the age of the parents of the
• ffspring. Then generation time G is the average age of the

parents of all the offspring produced by a single cohort. G is computed as follows: G = k

x=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 equal to the denominator. Consequently, the generation time will always be greater than or equal 1.0 for populations with age structure. Please compute lxbx(x + 1) and G for the example.

Brook Milligan Age Structure and Life Tables

x Sx bx lx gx ex lxbx lxbx(x + 1) 500 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 R0 =

k

• x=0

lxbx = 2.9 offspring G = 7.2 2.9 = 2.483 years

Brook Milligan Age Structure and Life Tables

x Sx bx 500 1 400 2 2 200 3 3 50 1 4 R0 =

k

• x=0

lxbx =

• ffspring

G = k

x=0 lxbx(x + 1)

k

x=0 lxbx

= years

Brook Milligan Age Structure and Life Tables