Lecture 10 Population Age and Size Structure

1
Lecture 10 Population Age and Size Structure

• I. Factors Affecting Birth and Death Rates in
  Populations
• A. Age
• B. Size
• C. Stage of development
• D. Gender

2
Lecture 10 Population Age and Size Structure

• I. Factors Affecting Birth and Death Rates in
  Populations
• A. Age a young rattlesnake or elephant or
  human or giant sequoia has a birth rate of zero.
  Very old individuals often have a high death
  rate.
• B. Size
• C. Stage of development
• D. Gender

3
Lecture 10 Population Age and Size Structure

• I. Factors Affecting Birth and Death Rates in
  Populations
• A. Age a young rattlesnake or elephant or
  human or giant sequoia has a birth rate of zero.
  Very old individuals often have a high death
  rate.
• B. Size small plants produce fewer offspring
  and have higher death rates than larger
  plants of the same age.
• C. Stage of development
• D. Gender

4
Lecture 10 Population Age and Size Structure

• I. Factors Affecting Birth and Death Rates in
  Populations
• A. Age a young rattlesnake or elephant or
  human or giant sequoia has a birth rate of zero.
  Very old individuals often have a high death
  rate.
• B. Size small plants produce fewer offspring
  and have higher death rates than larger
  plants of the same age. Small animals of many
  species may also not produce as many offspring
  or live as long as larger individuals of the
  same age.
• C. Stage of development
• D. Gender

5
Lecture 10 Population Age and Size Structure

• I. Factors Affecting Birth and Death Rates in
  Populations
• A. Age a young rattlesnake or elephant or
  human or giant sequoia has a birth rate of zero.
  Very old individuals often have a high death
  rate.
• B. Size small plants produce fewer offspring
  and have higher death rates than larger
  plants of the same age. Small animals of many
  species may also not produce as many offspring
  or live as long as larger individuals of the
  same age.
• C. Stage of development most insects go
  through dramatically different stages of
  development, but many other organisms also have
  distinct juvenile and adult stages.
• D. Gender

6
Lecture 10 Population Age and Size Structure

• I. Factors Affecting Birth and Death Rates in
  Populations
• A. Age a young rattlesnake or elephant or
  human or giant sequoia has a birth rate of zero.
  Very old individuals often have a high death
  rate.
• B. Size small plants produce fewer offspring
  and have higher death rates than larger
  plants of the same age. Small animals of many
  species may also not produce as many offspring
  or live as long as larger individuals of the
  same age.
• C. Stage of development most insects go
  through dramatically different stages of
  development, but many other organisms also have
  distinct juvenile and adult stages. Juveniles of
  many bird species may be as large as adults but
  have very different coloration (so stage isnt
  the same as size).
• D. Gender

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Juvenile mallard (Anas platyrhyncos)
Male and female mallards
prairiefrontier.com
8
Bald eagle juvenile (Haliaeetus leucocephalus)
Bald eagle adult (Haliaeetus leucocephalus)
Photo by Tim Knight (homepage.mac.com)
9
Lecture 10 Population Age and Size Structure

• I. Factors Affecting Birth and Death Rates in
  Populations
• C. Stage of development most insects go
  through dramatically different stages of
  development, but many other organisms also have
  distinct juvenile and adult stages. Juveniles of
  many bird species may be as large as adults but
  have very different coloration (so stage isnt
  the same as size). Seeds of many plants can
  live for hundreds of years without germinating
  (so stage isnt the same as age).
• D. Gender

10
Lecture 10 Population Age and Size Structure

• I. Factors Affecting Birth and Death Rates in
  Populations
• C. Stage of development most insects go
  through dramatically different stages of
  development, but many other organisms also have
  distinct juvenile and adult stages. Juveniles of
  many bird species may be as large as adults but
  have very different coloration (so stage isnt
  the same as size). Seeds of many plants can
  live for hundreds of years without germinating
  (so stage isnt the same as age).
• D. Gender only females give birth in animals
  so most analyses of animal populations are of
  females.

11
Lecture 10 Population Age and Size Structure

• I. Factors Affecting Birth and Death Rates in
  Populations
• C. Stage of development most insects go
  through dramatically different stages of
  development, but many other organisms also have
  distinct juvenile and adult stages. Juveniles of
  many bird species may be as large as adults but
  have very different coloration (so stage isnt
  the same as size). Seeds of many plants can
  live for hundreds of years without germinating
  (so stage isnt the same as age).
• D. Gender only females give birth in animals
  so most analyses of animal populations are of
  females. Females and males often have different
  death rates also. For example, human mortality
  rates are higher in men than in women in many
  countries

12
Lecture 10 Population Age and Size Structure

• I. Factors Affecting Birth and Death Rates in
  Populations
• C. Stage of development most insects go
  through dramatically different stages of
  development, but many other organisms also have
  distinct juvenile and adult stages. Juveniles of
  many bird species may be as large as adults but
  have very different coloration (so stage isnt
  the same as size). Seeds of many plants can
  live for hundreds of years without germinating
  (so stage isnt the same as age).
• D. Gender only females give birth in animals
  so most analyses of animal populations are of
  females. Females and males often have different
  death rates also. For example, human mortality
  rates are higher in men than in women in many
  countries but it hasnt always been that way.
  

13
Lecture 10 Population Age and Size Structure

• II. Life Tables
• A. What is a life table?
• B. Life table parameters
• C. Classification methods for life tables
• D. Cohort vs static life tables

14
Lecture 10 Population Age and Size Structure

• II. Life Tables
• A. What is a life table? Table showing the
  number of individuals alive over time in a
  population and the mortality rates at different
  times.
• B. Life table parameters
• C. Classification methods for life tables
• D. Cohort vs static life tables

15
Lecture 10 Population Age and Size Structure

• II. Life Tables
• A. What is a life table? Table showing the
  number of individuals alive over time in a
  population and the mortality rates at different
  times.
• B. Life table parameters (FIGS. 1,2)
• 1.
• 2.
• 3.
• 4.
• 5.
• 6.
• C. Classification methods for life tables
• D. Cohort vs static life tables

16
Lecture 10 Population Age and Size Structure

• II. Life Tables
• A. What is a life table? Table showing the
  number of individuals alive over time in a
  population and the mortality rates at different
  times.
• B. Life table parameters (FIGS. 1,2)
• 1. Time interval corresponding to age, age
  class, size class, or stage of development
  x
• 2.
• 3.
• 4.
• 5.
• 6.
• C. Classification methods for life tables
• D. Cohort vs static life tables

17
Phlox drummondii
Larry Allain_at_ USGS NWRC Plants Database
18
Lecture 10 Population Age and Size Structure

• II. Life Tables
• A. What is a life table? Table showing the
  number of individuals alive over time in a
  population and the mortality rates at different
  times.
• B. Life table parameters (FIGS. 1,2)
• 1. Time interval corresponding to age, age
  class, size class, or stage of development
  x
• 2. Number of individuals surviving to time x
  ax or nx or Nx.
• 3.
• 4.
• 5.
• 6.
• C. Classification methods for life tables
• D. Cohort vs static life tables

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20
Western spruce budworm (Choristoneura
occidentalis)
Damage by spruce budworm
Spruce budworm adult
Entomology.umn.edu
Adult spruce budworm Climatology.umn.edu
21
Lecture 10 Population Age and Size Structure

• II. Life Tables
• A. What is a life table? Table showing the
  number of individuals alive over time in a
  population and the mortality rates at different
  times.
• B. Life table parameters (FIGS. 1,2)
• 1. Time interval corresponding to age, age
  class, size class, or stage of development
  x
• 2. Number of individuals surviving to time x
  ax or nx or Nx.
• 3. Survivorship, the proportion or
  standardized number of individuals surviving
  to time x lx.
• 4.
• 5.
• 6.

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24
Lecture 10 Population Age and Size Structure

• II. Life Tables
• A. What is a life table? Table showing the
  number of individuals alive over time in a
  population and the mortality rates at different
  times.
• B. Life table parameters (FIGS. 1,2)
• 1. Time interval corresponding to age, age
  class, size class, or stage of development
  x
• 2. Number of individuals surviving to time x
  ax or nx or Nx.
• 3. Survivorship, the proportion or
  standardized number of individuals
  surviving to time x lx.
• 4. Number of individuals dying between times
  x and x1 dx
• 5.
• 6.

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27
Lecture 10 Population Age and Size Structure

• II. Life Tables
• A. What is a life table? Table showing the
  number of individuals alive over time in a
  population and the mortality rates at different
  times.
• B. Life table parameters (FIGS. 1,2)
• 1. Time interval corresponding to age, age
  class, size class, or stage of development
  x
• 2. Number of individuals surviving to time x
  ax or nx or Nx.
• 3. Survivorship, the proportion or
  standardized number of individuals
  surviving to time x lx.
• 4. Number of individuals dying between times
  x and x1 dx
• 5. Mortality rate (proportion of individuals
  dying) between times x and x1 qx.
• 6.

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30
Lecture 10 Population Age and Size Structure

• II. Life Tables
• B. Life table parameters (FIGS. 1,2)
• 1. Time interval corresponding to age, age
  class, size class, or stage of
  development x
• 2. Number of individuals surviving to time x
  ax or nx or Nx.
• 3. Survivorship, the proportion or
  standardized number of individuals
  surviving to time x lx.
• 4. Number of individuals dying between times
  x and x1 dx
• 5. Mortality rate (proportion of individuals
  dying) between times x and x1 qx.
• 6. Mean life expectation (expectancy) for
  individuals reaching time x ex.

31
Lecture 10 Population Age and Size Structure

• II. Life Tables
• C. Classification methods for life tables (FIGS.
  1,2,3,4)
• 1.
• 2.
• 3.

32
Lecture 10 Population Age and Size Structure

• II. Life Tables
• C. Classification methods for life tables (FIGS.
  1,2,3,4)
• 1. Age classes or intervals (FIGS. 1,3,4)
• 2.
• 3.

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34
Red deer hind (Cervus elaphus)
Paul Hobson photo
Red deer stag
Falconergame.co.uk
35
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36
Lecture 10 Population Age and Size Structure

• II. Life Tables
• C. Classification methods for life tables (FIGS.
  1,2,3,4)
• 1. Age classes or intervals (FIGS. 1,3,4)
• 2. Stage of development (FIG. 2)
• 3.

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Lecture 10 Population Age and Size Structure

• II. Life Tables
• C. Classification methods for life tables (FIGS.
  1,2,3,4)
• 1. Age classes or intervals (FIGS. 1,3,4)
• 2. Stage of development (FIG. 2)
• 3. Size classes or intervals. This is often
  used for plants.

39
Lecture 10 Population Age and Size Structure

• II. Life Tables
• C. Classification methods for life tables (FIGS.
  1,2,3,4)
• 1. Age classes or intervals (FIGS. 1,3,4)
• 2. Stage of development (FIG. 2)
• 3. Size classes or intervals. This is often
  used for plants.
• Forest tree example 10-20 cm, 20-40 cm

40
Lecture 10 Population Age and Size Structure

• II. Life Tables
• C. Classification methods for life tables (FIGS.
  1,2,3,4)
• 1. Age classes or intervals (FIGS. 1,3,4)
• 2. Stage of development (FIG. 2)
• 3. Size classes or intervals. This is often
  used for plants.
• Forest tree example 10-20 cm, 20-40 cm
• D. Cohort vs static life tables
• 1. Cohort (dynamic, age-specific) life tables
  (FIGS. 1,2,3)
• 2. Static (time-specific) life tables (FIG.
  4)

41
Lecture 10 Population Age and Size Structure

• II. Life Tables
• C. Classification methods for life tables (FIGS.
  1,2,3,4)
• 1. Age classes or intervals (FIGS. 1,3,4)
• 2. Stage of development (FIG. 2)
• 3. Size classes or intervals. This is often
  used for plants.
• Forest tree example 10-20 cm, 20-40 cm
• D. Cohort vs static life tables
• 1. Cohort (dynamic, age-specific) life tables
  (FIGS. 1,2,3)
• A cohort is a group of individuals of the
  same age (age-mates).
• 2. Static (time-specific) life tables (FIG.
  4)

42
Lecture 10 Population Age and Size Structure

• II. Life Tables
• C. Classification methods for life tables (FIGS.
  1,2,3,4)
• 1. Age classes or intervals (FIGS. 1,3,4)
• 2. Stage of development (FIG. 2)
• 3. Size classes or intervals. This is often
  used for plants.
• Forest tree example 10-20 cm, 20-40 cm
• D. Cohort vs static life tables
• 1. Cohort (dynamic, age-specific) life tables
  (FIGS. 1,2,3)
• A cohort is a group of individuals of the
  same age (age-mates). To develop a
  cohort life table, you follow all individuals of
  a cohort from birth until every individual
  has died.
• 2. Static (time-specific) life tables (FIG.
  4)

43
Lecture 10 Population Age and Size Structure

• II. Life Tables
• D. Cohort vs static life tables
• 1. Cohort (dynamic, age-specific) life tables
  (FIGS. 1,2,3)
• A cohort is a group of individuals of the
  same age (age-mates). To develop a
  cohort life table, you follow all individuals of
  a cohort from birth until every individual
  has died. The most useful and most accurate
  life table but . . .
• 2. Static (time-specific) life tables (FIG.
  4)

44
Lecture 10 Population Age and Size Structure

• II. Life Tables
• D. Cohort vs static life tables
• 1. Cohort (dynamic, age-specific) life tables
  (FIGS. 1,2,3)
• A cohort is a group of individuals of the
  same age (age-mates). To develop a
  cohort life table, you follow all individuals of
  a cohort from birth until every individual
  has died. The most useful and most accurate
  life table but it may be difficult to locate
  all individuals at birth and follow them until
  the last one has died.
• 2. Static (time-specific) life tables (FIG.
  4)

45
Lecture 10 Population Age and Size Structure

• II. Life Tables
• D. Cohort vs static life tables
• 1. Cohort (dynamic, age-specific) life tables
  (FIGS. 1,2,3)
• A cohort is a group of individuals of the
  same age (age-mates). To develop a
  cohort life table, you follow all individuals of
  a cohort from birth until every individual
  has died. The most useful and most accurate
  life table but it may be difficult to locate
  all individuals at birth and follow them until
  the last one has died. Its not good for
  long-lived species!
• 2. Static (time-specific) life tables (FIG.
  4)

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49
Lecture 10 Population Age and Size Structure

• II. Life Tables
• D. Cohort vs static life tables
• 1. Cohort (dynamic, age-specific) life tables
  (FIGS. 1,2,3)
• A cohort is a group of individuals of the
  same age (age-mates). To develop a
  cohort life table, you follow all individuals of
  a cohort from birth until every individual
  has died. The most useful and most accurate
  life table but it may be difficult to locate
  all individuals at birth and follow them until
  the last one has died. Its not good for
  long-lived species!
• 2. Static (time-specific) life tables (FIG.
  4)
• To develop a static life table, you first
  estimate the age of each individual in a
  population at a particular time.

50
Lecture 10 Population Age and Size Structure

• II. Life Tables
• D. Cohort vs static life tables
• 2. Static (time-specific) life tables
  (FIG. 4)
• To develop a static life table, you first
  estimate the age of each individual in a
  population at a particular time. Assuming that b
  and d have remained constant since the
  oldest individual was born, you work back
  from the oldest individuals to the youngest
  to estimate how many were born in each year.

51
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52
Lecture 10 Population Age and Size Structure

• II. Life Tables
• D. Cohort vs static life tables
• 2. Static (time-specific) life tables (FIG.
  4)
• To develop a static life table, you first
  estimate the age of each individual in a
  population at a particular time. Assuming that b
  and d have remained constant since the
  oldest individual was born, you work back
  from the oldest individuals to the youngest
  to estimate how many were born in each year.
  Sometimes these data are smoothed to
  eliminate oddities as we see for ages 6
  and 7 in FIG. 4.

53
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54
Lecture 10 Population Age and Size Structure

• II. Life Tables
• D. Cohort vs static life tables
• 2. Static (time-specific) life tables (FIG.
  4)
• To develop a static life table, you first
  estimate the age of each individual in a
  population at a particular time. Assuming that b
  and d have remained constant since the
  oldest individual was born, you work back
  from the oldest individuals to the youngest
  to estimate how many were born in each year.
  Sometimes these data are smoothed to
  eliminate oddities as we see for ages 6
  and 7 in FIG. 4. Static life tables are used for
  long-lived organisms like trees,
  humans, and other large mammals.

55
Lecture 10 Population Age and Size Structure

• III. Survivorship Curves
• A. How to develop survivorship curves
• B. Three standard survivorship curves (FIG. 5)
• C. Examples of survivorship curves in nature
  (FIGS. 6,7,8)

56
Lecture 10 Population Age and Size Structure

• III. Survivorship Curves
• A. How to develop survivorship curves
• Plot the logarithm of survivorship (log lx) on
  Y-axis and age, size, or stage on
  X-axis. Usually use natural logs but can use any
  base.
• B. Three standard survivorship curves (FIG. 5)
• C. Examples of survivorship curves in nature
  (FIGS. 6,7,8)

57
Lecture 10 Population Age and Size Structure

• III. Survivorship Curves
• A. How to develop survivorship curves
• Plot the logarithm of survivorship (log lx) on
  Y-axis and age, size, or stage on X-axis.
  Usually use natural logs but can use any base.
  These curves show the proportion of individuals
  dying (i.e. the mortality rate) at each age,
  size, or stage.
• B. Three standard survivorship curves (FIG. 5)
• C. Examples of survivorship curves in nature
  (FIGS. 6,7,8)

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59
Lecture 10 Population Age and Size Structure

• III. Survivorship Curves
• A. How to develop survivorship curves
• Plot the logarithm of survivorship (log lx) on
  Y-axis and age, size, or stage on X-axis.
  Usually use natural logs but can use any base.
  These curves show the proportion of individuals
  dying (i.e. the mortality rate) at each age,
  size, or stage.
• B. Three standard survivorship curves (FIG. 5)
• C. Examples of survivorship curves in nature
  (FIGS. 6,7,8)

60
Lecture 10 Population Age and Size Structure

• III. Survivorship Curves
• A. How to develop survivorship curves
• Plot the logarithm of survivorship (log lx) on
  Y-axis and age, size, or stage on X-axis.
  Usually use natural logs but can use any base.
  These curves show the proportion of individuals
  dying (i.e. the mortality rate) at each age,
  size, or stage.
• B. Three standard survivorship curves (FIG. 5)
• Developed by Pearl--often called Deevey
  curves. These show general patterns of
  mortality in natural populations.
• C. Examples of survivorship curves in nature
  (FIGS. 6,7,8)

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62
Lecture 10 Population Age and Size Structure

• III. Survivorship Curves
• B. Three standard survivorship curves (FIG. 5)
• Developed by Pearl--often called Deevey
  curves. These show general patterns of
  mortality in natural populations. In Type I
  curves, most mortality occurs late in life.
  Typical of humans and other large organisms that
  have few offspring and give much parental care.
• C. Examples of survivorship curves in nature
  (FIGS. 6,7,8)

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64
Lecture 10 Population Age and Size Structure

• III. Survivorship Curves
• B. Three standard survivorship curves (FIG. 5)
• Developed by Pearl--often called Deevey
  curves. These show general patterns of
  mortality in natural populations. In Type I
  curves, most mortality occurs late in life.
  Typical of humans and other large organisms that
  have few offspring and give much parental care.
  Type II curves have fairly constant mortality
  rate throughout life. Typical of many bird
  species and other organisms with intermediate
  number of offspring and parental care.
• C. Examples of survivorship curves in nature
  (FIGS. 6,7,8)

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66
Lecture 10 Population Age and Size Structure

• III. Survivorship Curves
• B. Three standard survivorship curves (FIG. 5)
• Developed by Pearl--often called Deevey
  curves. These show general patterns of
  mortality in natural populations. In Type I
  curves, most mortality occurs late in life.
  Typical of humans and other large organisms that
  have few offspring and give much parental care.
  Type II curves have fairly constant mortality
  rate throughout life. Typical of many bird
  species and other organisms with intermediate
  number of offspring and parental care. In Type
  III curves, most mortality occurs early in life.
  Typical of insects, marine invertebrates,
  plants, and other organisms that produce many
  offspring but few survive because there is
  little parental care for individual offspring.
• C. Examples of survivorship curves in nature
  (FIGS. 6,7,8)

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68
Lecture 10 Population Age and Size Structure

• III. Survivorship Curves
• C. Examples of survivorship curves in nature
  (FIGS. 6,7,8)

69
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70
Lecture 10 Population Age and Size Structure

• III. Survivorship Curves
• C. Examples of survivorship curves in nature
  (FIGS. 6,7,8)
• Dall sheep in FIG. 6 have Type I survivorship.

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72
Lecture 10 Population Age and Size Structure

• III. Survivorship Curves
• C. Examples of survivorship curves in nature
  (FIGS. 6,7,8)
• Dall sheep in FIG. 6 have Type I survivorship.
• Various birds in FIG. 7 have Type II
  survivorship.

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74
Lecture 10 Population Age and Size Structure

• III. Survivorship Curves
• C. Examples of survivorship curves in nature
  (FIGS. 6,7,8)
• Dall sheep in FIG. 6 have Type I survivorship.
• Various birds in FIG. 7 have Type II
  survivorship.
• The tropical palms in FIG. 8 have Type III
  survivorship.

75
Lecture 10 Population Age and Size Structure

• IV. Fecundity Schedules
• A. What is a fecundity schedule?
• B. Fecundity schedule parameters
• C. What does net reproductive rate represent?
• D. Examples

76
Lecture 10 Population Age and Size Structure

• IV. Fecundity Schedules
• A. What is a fecundity schedule? A table
  showing the number of offspring produced at each
  age (or size or stage) and also showing the
  survival of the parent.
• B. Fecundity schedule parameters
• C. What does net reproductive rate represent?
• D. Examples

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78
Lecture 10 Population Age and Size Structure

• IV. Fecundity Schedules
• A. What is a fecundity schedule? A table
  showing the number of offspring produced at each
  age (or size or stage) and also showing the
  survival of the parent.
• B. Fecundity schedule parameters
• 1. Number of offspring produced per
  individual from age (or time) x to x1
  mx or bx.

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80
Lecture 10 Population Age and Size Structure

• IV. Fecundity Schedules
• A. What is a fecundity schedule? A table
  showing the number of offspring produced at each
  age (or size or stage) and also showing the
  survival of the parent.
• B. Fecundity schedule parameters
• 1. Number of offspring produced per
  individual from age (or time) x to x1
  mx or bx.
• 2. Number of offspring produced by all
  individuals from age (or time) x to x1
  Fx.

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82
Lecture 10 Population Age and Size Structure

• IV. Fecundity Schedules
• A. What is a fecundity schedule? A table
  showing the number of offspring produced at each
  age (or size or stage) and also showing the
  survival of the parent.
• B. Fecundity schedule parameters
• 1. Number of offspring produced per
  individual from age (or time) x to x1
  mx or bx.
• 2. Number of offspring produced by all
  individuals from age (or time) x to x1
  Fx.
• 3. Net reproductive rate R0 sum of
  products of lx and mx values.

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84
Lecture 10 Population Age and Size Structure

• IV. Fecundity Schedules
• B. Fecundity schedule parameters
• 1. Number of offspring produced per
  individual from age (or time) x to x1
  mx or bx.
• 2. Number of offspring produced by all
  individuals from age (or time) x to x1
  Fx.
• 3. Net reproductive rate R0 sum of
  products of lx and mx values.
• C. What does the net reproductive rate represent?

85
Lecture 10 Population Age and Size Structure

• IV. Fecundity Schedules
• B. Fecundity schedule parameters
• 1. Number of offspring produced per
  individual from age (or time) x to x1
  mx or bx.
• 2. Number of offspring produced by all
  individuals from age (or time) x to x1
  Fx.
• 3. Net reproductive rate R0 sum of
  products of lx and mx values.
• C. What does the net reproductive rate
  represent? R0 measures the growth (or
  decline) in a population from one generation to
  the next. Its similar to ? but ? measures
  growth in the population from one year to the
  next.

86
Lecture 10 Population Age and Size Structure

• IV. Fecundity Schedules
• C. What does the net reproductive rate
  represent? R0 measures the growth (or
  decline) in a population from one generation to
  the next. Its similar to ? but ? measures
  growth in the population from one year to
  the next.
• D. Examples
• 1. Phlox (FIG. 9)
• 2. Red deer (hinds)(FIG. 10)
• 3. Field grasshopper (FIG. 11)
• 4. Human females (FIG. 12)

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Lecture 10 Population Age and Size Structure

• IV. Fecundity Schedules
• C. What does the net reproductive rate
  represent? R0 measures the growth (or
  decline) in a population from one generation to
  the next. Its similar to ? but ? measures
  growth in the population from one year to
  the next.
• D. Examples
• 1. Phlox (FIG. 9). R0 2.4177 so each
  individual in the previous generation
  replaces itself with more than 2 individuals in
  the next generation. The population will
  increase.
• 2. Red deer (hinds)(FIG. 10)
• 3. Field grasshopper (FIG. 11)
• 4. Human females (FIG. 12)

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Lecture 10 Population Age and Size Structure

• IV. Fecundity Schedules
• D. Examples
• 1. Phlox (FIG. 9). R0 2.4177 so each
  individual in the previous generation
  replaces itself with more than 2 individuals in
  the next generation. The population will
  increase.
• 2. Red deer (hinds)(FIG. 10). R0 1.316 so
  the population should increase.
• 3. Field grasshopper (FIG. 11)
• 4. Human females (FIG. 12)

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Lecture 10 Population Age and Size Structure

• IV. Fecundity Schedules
• D. Examples
• 1. Phlox (FIG. 9). R0 2.4177 so each
  individual in the previous generation
  replaces itself with more than 2 individuals in
  the next generation. The population will
  increase.
• 2. Red deer (hinds)(FIG. 10). R0 1.316 so
  the population should increase.
• 3. Field grasshopper (FIG. 11). R0 0.51 so
  the population will decline.
• 4. Human females (FIG. 12)

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Lecture 10 Population Age and Size Structure

• IV. Fecundity Schedules
• D. Examples
• 1. Phlox (FIG. 9). R0 2.4177 so each
  individual in the previous generation
  replaces itself with about 2.4 individuals in the
  next generation. The population will
  increase.
• 2. Red deer (hinds)(FIG. 10). R0 1.316 so
  the population should increase.
• 3. Field grasshopper (FIG. 11). R0 0.51 so
  the population will decline.
• 4. Human females (FIG. 12). R0 1.0061 so
  the population of women in the U.S. is
  increasing at a rate of 0.61.

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Lecture 10 Population Age and Size Structure

• IV. Fecundity Schedules
• D. Examples
• 1. Phlox (FIG. 9). R0 2.4177 so each
  individual in the previous generation
  replaces itself with about 2.4 individuals in the
  next generation. The population will
  increase.
• 2. Red deer (hinds)(FIG. 10). R0 1.316 so
  the population should increase.
• 3. Field grasshopper (FIG. 11). R0 0.51 so
  the population will decline.
• 4. Human females (FIG. 12). R0 1.0061 so
  the population of women in the U.S. is
  increasing at a rate of 0.61. That is
  currently the approximate growth rate in the
  U.S.

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Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• A. Age-classified populations (Leslie matrix
  models)(FIG. 13A)
• B. Size-classified populations (FIG. 13B)
• C. Stage-classified populations (FIG. 14)
• D. Assumptions of matrix models
• E. Projecting population growth using matrix
  models (FIGS. 15,16,17)
• F. Including density-dependence to make matrix
  models more realistic.

97
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• A. Age-classified populations (Leslie matrix
  models)(FIG. 13A)
• 1. Life-cycle graphs
• 2. Transition matrix models

98
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• A. Age-classified populations (Leslie matrix
  models)(FIG. 13A)
• 1. Life-cycle graphs - show all age classes,
  the probability of surviving from one age
  to the next (P), and the fecundity at each
  age (F).
• 2. Transition matrix models

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Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• A. Age-classified populations (Leslie matrix
  models)(FIG. 13A)
• 1. Life-cycle graphs - show all age classes,
  the probability of surviving from one
  age to the next (P), and the fecundity at
  each age (F).
• 2. Transition matrix models

101
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• A. Age-classified populations (Leslie matrix
  models)(FIG. 13A)
• 1. Life-cycle graphs - show all age classes,
  the probability of surviving from one age
  to the next (P), and the fecundity at each
  age (F).
• 2. Transition matrix models. Matrix Aa
  contains the same information as the
  life-cycle graph. Columns indicate age
  classes at the present (time t) and rows indicate
  the same age classes at time t 1.

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103
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• A. Age-classified populations (Leslie matrix
  models)(FIG. 13A)
• 1. Life-cycle graphs - show all age classes,
  the probability of surviving from one age
  to the next (P), and the fecundity at each
  age (F).
• 2. Transition matrix models. Matrix Aa
  contains the same information as the
  life-cycle graph. Columns indicate age
  classes at the present (time t) and rows indicate
  the same age classes at time t 1.
• B. Size-classified populations (FIG. 13B)
• 1. Life-cycle graphs
• 2. Transition matrix models

104
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• A. Age-classified populations (Leslie matrix
  models)(FIG. 13A)
• 1. Life-cycle graphs - show all age classes,
  the probability of surviving from one age
  to the next (P), and the fecundity at each
  age (F).
• 2. Transition matrix models. Matrix Aa
  contains the same information as
  the life-cycle graph. Columns indicate age
  classes at the present (time t) and rows
  indicate the same age classes at time t
  1.
• B. Size-classified populations (FIG. 13B)
• 1. Life-cycle graphs - show all size classes,
  the probability of surviving but remaining
  in the same size class (P), the
  probability of growing into the next size class
  (G), and fecundity of each size class
  (F).
• 2. Transition matrix models

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106
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• B. Size-classified populations (FIG. 13B)
• 1. Life-cycle graphs - show all size classes,
  the probability of surviving but remaining
  in the same size class (P), the
  probability of growing into the next size class
  (G), and fecundity of each size class
  (F).
• 2. Transition matrix models. Contain the
  same P, G, and F values as the life-cycle
  graphs.

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108
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• B. Size-classified populations (FIG. 13B)
• 1. Life-cycle graphs - show all size classes,
  the probability of surviving but remaining
  in the same size class (P), the
  probability of growing into the next size class
  (G), and fecundity of each size class
  (F).
• 2. Transition matrix models. Contain the
  same P, G, and F values as the life-cycle
  graphs. Notice that the age-classified matrix
  has all zeroes on the diagonal, whereas the
  size-classified matrix may have values
  greater than zero.

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Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• B. Size-classified populations (FIG. 13B)
• 2. Transition matrix models. Contain the
  same P, G, and F values as the life-cycle
  graphs, with arrows indicating probability of
  staying in the same size class (P) or of
  growing (G) and the fecundity (F). Notice
  that the age-classified matrix has all zeroes
  on the diagonal, whereas the size-classified
  matrix may have values greater than zero.
• C. Stage-classified populations (FIG. 14)
• 1. Insect stages of development (FIG. 14a)
• 2. Stages of development in a tree population
  (FIG. 14b)
• 3. Coral life stages with sexual and asexual
  reproduction (FIG. 14c)

111
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• C. Stage-classified populations (FIG. 14)
• 1. Insect stages of development (FIG. 14a)
• 2. Stages of development in a tree population
  (FIG. 14b)
• 3. Coral life stages with sexual and asexual
  reproduction (FIG. 14c)

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113
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• C. Stage-classified populations (FIG. 14)
• 1. Insect stages of development (FIG. 14a)
• Each circle now represents a different
  stage of development. Notice the different
  notation for P and F.
• 2. Stages of development in a tree population
  (FIG. 14b)
• 3. Coral life stages with sexual and asexual
  reproduction (FIG. 14c)

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115
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• C. Stage-classified populations (FIG. 14)
• 1. Insect stages of development (FIG. 14a)
• Each circle now represents a different
  stage of development. Notice the different
  notation for P and F.
• 2. Stages of development in a tree population
  (FIG. 14b)
• 3. Coral life stages with sexual and asexual
  reproduction (FIG. 14c)

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117
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• C. Stage-classified populations (FIG. 14)
• 1. Insect stages of development (FIG. 14a)
• Each circle now represents a different
  stage of development. Notice the different
  notation for P and F.
• 2. Stages of development in a tree population
  (FIG. 14b)
• Numbered stages could be seed, seedling,
  sapling, mature tree, and senescent tree
  (past its prime).
• 3. Coral life stages with sexual and asexual
  reproduction (FIG. 14c)

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119
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• C. Stage-classified populations (FIG. 14)
• 1. Insect stages of development (FIG. 14a)
• Each circle now represents a different
  stage of development. Notice the different
  notation for P and F.
• 2. Stages of development in a tree population
  (FIG. 14b)
• Numbered stages could be seed, seedling,
  sapling, mature tree, and senescent tree
  (past its prime).
• 3. Coral life stages with sexual and asexual
  reproduction (FIG. 14c)

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121
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• C. Stage-classified populations (FIG. 14)
• 1. Insect stages of development (FIG. 14a)
• Each circle now represents a different
  stage of development. Notice the
  different notation for P and F.
• 2. Stages of development in a tree population
  (FIG. 14b)
• Numbered stages could be seed, seedling,
  sapling, mature tree, and senescent tree
  (past its prime).
• 3. Coral life stages with sexual and asexual
  reproduction (FIG. 14c)
• The coral life cycle is more complicated.
  It can grow slowly or quickly and it can
  reproduce sexually (F) or asexually (P) by
  fragmenting into smaller pieces.

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Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• C. Stage-classified populations (FIG. 14)
• 3. Coral life stages with sexual and asexual
  reproduction (FIG. 14c)
• The coral life cycle is more complicated.
  It can grow slowly or quickly and it can
  reproduce sexually (F) or asexually (P) by
  fragmenting into smaller pieces.
• Remember that stage-classified and
  size-classified matrices can both have positive
  values along the diagonal. Age-classified
  matrices can only have zeroes on the diagonal
  (because you cant stay the same age from one
  year to the next). Age-classified matrices are
  often called Leslie matrices in honor of Patrick
  Leslie. Also remember that fecundity (number of
  offspring) is shown on the first row of the
  matrix.

124
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• Remember that stage-classified and
  size-classified matrices can both have positive
  values along the diagonal. Age-classified
  matrices can only have zeroes on the diagonal
  (because you cant stay the same age from one
  year to the next). Age-classified matrices are
  often called Leslie matrices in honor of Patrick
  Leslie. Also remember that fecundity (number of
  offspring) is shown on the first row of the
  matrix.
• D. Assumptions of transition matrix models
• 1. Stationarity
• 2. Markov property

125
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• D. Assumptions of transition matrix models
• 1. Stationarity - P, G, and F values dont
  change over time. No stochastic effects of
  weather or disturbance and no resource
  limitation.
• 2. Markov property

126
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• D. Assumptions of transition matrix models
• 1. Stationarity - P, G, and F values dont
  change over time. No stochastic effects of
  weather or disturbance and no resource
  limitation.
• 2. Markov property - P, G, and F values only
  depend on the current age, size, or stage of
  development and not on the past history of
  the individual. In the coral example, the
  probability of a medium-sized individual
  becoming a large individual doesnt depend
  on how long it has been in the medium-sized
  stage or how it got to that stage.

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128
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• D. Assumptions of transition matrix models
• 2. Markov property - P, G, and F values only
  depend on the current age, size, or stage of
  development and not on the past history of
  the individual. In the coral example, the
  probability of a medium-sized individual
  becoming a large individual dont depend on
  how long it has been in the medium-sized stage or
  how it got to that stage.
• E. Projecting population growth using matrix
  models (FIGS. 15,16,17)

129
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• D. Assumptions of transition matrix models
• 2. Markov property - P, G, and F values only
  depend on the current age, size, or stage of
  development and not on the past history of the
  individual. In the coral example, the
  probability of a medium-sized individual
  becoming a large individual dont depend on
  how long it has been in the medium-sized stage or
  how it got to that stage.
• E. Projecting population growth using matrix
  models (FIGS. 15,16,17)
• We use matrix multiplication to project
  population growth into the future. We
  multiply the matrix by an initial population
  vector that shows how many individuals are in
  each age or size class or stage of
  development.

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Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• E. Projecting population growth using matrix
  models (FIGS. 15,16,17)
• We use matrix multiplication to project
  population growth into the future. We multiply
  the matrix by an initial population vector that
  shows how many individuals are in each age or
  size class or stage of development. By repeated
  multiplication, we can predict the growth rate
  (?), future N, and the expected proportion of
  individuals in each age or size class or stage
  of development (see handout).

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133
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• E. Projecting population growth using matrix
  models (FIGS. 15,16,17)
• We use matrix multiplication to project
  population growth into the future. We multiply
  the matrix by an initial population vector that
  shows how many individuals are in each age or
  size class or stage of development. By repeated
  multiplication, we can predict the growth rate
  (?), future N, and the expected proportion of
  individuals in each age or size class or stage
  of development (see handout). We can also
  determine what part of the life cycle is most
  important for maintaining the population at a
  reasonable size and what type of conservation
  efforts might be most effective.

134
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• E. Projecting population growth using matrix
  models (FIGS. 15,16,17)
• Properties of matrix models
• 1. The proportion of individuals in each age or
  size class or stage of development eventually
  stabilizes (FIG. 15).

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136
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• E. Projecting population growth using matrix
  models (FIGS. 15,16,17)
• Properties of matrix models
• 1. The proportion of individuals in each age or
  size class or stage of development eventually
  stabilizes (FIG. 15).
• 2. The stable population growth rate, ?,
  depends only on the matrix, not on the
  starting population (FIG. 16).

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138
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• E. Projecting population growth using matrix
  models (FIGS. 15, 16, 17)
• Properties of matrix models
• 1. The proportion of individuals in each age or
  size class or stage of development eventually
  stabilizes (FIG. 15).
• 2. The stable population growth rate, ?,
  depends only on the matrix, not on the
  starting population (FIG. 16).
• 3. If a model has only one positive fecundity
  value, the population will cycle (FIG. 17).

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140
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• F. Including density-dependence to make matrix
  models more realistic (FIGS. 18, 19, 20).

141
Lecture 10 Population Age and Size Structure

• V. Life Cycle Graphs and Transition Matrix
  Models
• F. Including density-dependence to make matrix
  models more realistic (FIGS. 18, 19, 20).
  Instead of having constant probabilities and
  fecundities in the matrix, you can use functions
  that depend on population density to account for
  resource limitations. This makes the
  models similar to logistic models and much more
  realistic.

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Lecture 10 Population Age and Size Structure

• VI. Application of Matrix Population Models in
  Conservation and Management
• A. Three important applications
• B. Procedure
• C. Exam